Daily Papers

Knowledge distillation (KD) is a powerful paradigm for compressing large language models (LLMs), whose effectiveness depends on intertwined choices of divergence direction, optimization strategy, and data regime. We break down the design of existing KD methods and present a unified view that establishes connections between them, reformulating KD as a reweighted log-likelihood objective at the token level. We further propose Hybrid Policy Distillation (HPD), which integrates the complementary advantages of forward and reverse KL to balance mode coverage and mode-seeking, and combines off-policy data with lightweight, approximate on-policy sampling. We validate HPD on long-generation math reasoning as well as short-generation dialogue and code tasks, demonstrating improved optimization stability, computational efficiency, and final performance across diverse model families and scales. The code related to this work is available at https://github.com/zwhong714/Hybrid-Policy-Distillation.

Reinforcement learning for large language models (LLMs) typically relies on trust-region masks to stabilize off-policy updates. The dominant PPO-style approach uses the sampled-token importance ratio for two criteria: a proximity criterion, which asks whether the policy has moved too far from the behavior policy, and a direction criterion, which asks whether the update pushes it farther away. Recent work DPPO improves the proximity criterion by replacing PPO's ratio-based test with a probability divergence between the behavior and training policies. However, its direction criterion is still inherited from PPO. A token can be masked only when the sampled-token importance ratio moves away from one. We observe that this ratio-based direction criterion is a single-sample proxy that can disagree in sign with the change of the divergence that defines the proximity criterion. We therefore propose the predictive divergence mask, which asks whether the next policy-gradient step will increase or decrease the same divergence used by the trust region. For the discrete softmax policies used in LLM RL, we derive this prediction in closed form. Because production rollout engines expose only a truncated (top-K) view of the vocabulary, we develop two lightweight top-K estimators for this prediction. Detailed analysis shows the divergence-based direction is better aligned with the realized change of the divergence than the sampled ratio, and the resulting masks improve RL training across model scales and precision settings.

Large language models (LLMs) generalize smoothly across continuous semantic spaces, yet strict logical reasoning demands the formation of discrete decision boundaries. Prevailing theories relying on linear isometric projections fail to resolve this fundamental tension. In this work, we argue that task context operates as a non-isometric dynamical operator that enforces a necessary "topological distortion." By applying Gram-Schmidt decomposition to residual-stream activations , we reveal a dual-modulation mechanism driving this process: a class-agnostic topological preservation that anchors global structure to prevent semantic collapse, and a specific algebraic divergence that directionally tears apart cross-class concepts to forge logical boundaries. We validate this geometric evolution across a gradient of tasks, from simple mapping to complex primality testing. Crucially, targeted specific vector ablation establishes a strict causal binding between this topology and model function: algebraically erasing the divergence component collapses parity classification accuracy from 100% to chance levels (38.57%). Furthermore, we uncover a three-phase layer-wise geometric dynamic and demonstrate that under social pressure prompts, models fail to generate sufficient divergence. This results in a "manifold entanglement" that geometrically explains sycophancy and hallucination. Ultimately, our findings revise the linear-isometric presumption, demonstrating that the emergence of discrete logic in LLMs is purchased at an irreducible cost of topological deformation.

Since the advent of knowledge distillation, KL divergence has been the standard loss in distillation. Recently, on-policy distillation (OPD) has emerged as an efficient post-training paradigm for LLMs. As a distillation method, OPD naturally inherits KL divergence as its standard loss. However, in this work, we find that KL divergence may not be necessary for OPD. We show that simply preserving the update direction is sufficient for effective OPD. As long as the update direction is toward the teacher, OPD works. More precisely, it is not the direction of every token, but the direction of a small subset of tokens where the teacher and student disagree strongly. We first show that simply assigning a reward of (+1) to tokens where the teacher probability is higher than the student probability and (-1) where it is lower, which merely encourages updates toward the teacher, reproduces almost the same training mode as OPD with reverse KL. We further show that only the direction of a small subset of tokens with large teacher-student disagreement is critical, and training works as long as their update direction is toward the teacher, even if other tokens are pulled away from the teacher. And as an application of these findings, we introduce Consensus Multi-Teacher On-Policy Distillation (C-MOPD) to improve Multi-Teacher On-Policy Distillation (MOPD). Unlike MOPD, which routes each sample to a single teacher and may cause capability conflicts across domains, C-MOPD lets every sample be supervised by all teachers. Experiments show that C-MOPD consistently outperforms MOPD on both math and code benchmarks. Our code is available at https://github.com/LeapLabTHU/KL-Free-OPD.

Reinforcement learning with verifiable rewards (RLVR) has substantially improved the reasoning capabilities of large language models. While existing analyses identify that RLVR-induced changes are sparse, they primarily focus on the magnitude of these updates, largely overlooking their direction. In this work, we argue that the direction of updates is a more critical lens for understanding RLVR's effects, which can be captured by the signed, token-level log probability difference Δlog p between the base and final RLVR models. Through statistical analysis and token-replacement interventions, we demonstrate that Δlog p more effectively identifies sparse, yet reasoning-critical updates than magnitude-based metrics (\eg divergence or entropy). Building on this insight, we propose two practical applications: (1) a test-time extrapolation method that amplifies the policy along the learned Δlog p direction to improve reasoning accuracy without further training; (2) a training-time reweighting method that focuses learning on low-probability (corresponding to higher Δlog p) tokens, which improves reasoning performance across models and benchmarks. Our work establishes the direction of change as a key principle for analyzing and improving RLVR.

Reinforcement learning (RL) has become a key component of post-training large language models (LLMs). In practice, LLM RL is often off-policy because of training-inference mismatch and policy staleness, making trust-region control essential for stable optimization. Mainstream methods such as PPO and GRPO approximate this control with a ratio-clipping mechanism, but the importance ratio can be a poor proxy for distributional shift in long-tailed vocabularies. Recent work such as DPPO addresses this mismatch by replacing ratio-based clipping with a divergence-based mask, yielding a trust region defined by the sampled token's absolute probability shift. However, DPPO still relies on a hard mask: once a token crosses the trust-region boundary in a harmful direction, its gradient is discarded rather than corrected. To address this, we propose Divergence Regularized Policy Optimization (DRPO), which replaces the hard mask with a smooth advantage-weighted quadratic regularizer on policy shift. DRPO preserves the same trust-region geometry as DPPO while inducing bounded, continuous gradient weights that attenuate diverging updates and provide corrective signals beyond the boundary. Experiments across model scales, architectures, and precision settings show that DRPO improves the stability and efficiency of LLM RL training.

Knowledge distillation (KD) exploits a large well-trained model (i.e., teacher) to train a small student model on the same dataset for the same task. Treating teacher features as knowledge, prevailing methods of knowledge distillation train student by aligning its features with the teacher's, e.g., by minimizing the KL-divergence between their logits or L2 distance between their intermediate features. While it is natural to believe that better alignment of student features to the teacher better distills teacher knowledge, simply forcing this alignment does not directly contribute to the student's performance, e.g., classification accuracy. In this work, we propose to align student features with class-mean of teacher features, where class-mean naturally serves as a strong classifier. To this end, we explore baseline techniques such as adopting the cosine distance based loss to encourage the similarity between student features and their corresponding class-means of the teacher. Moreover, we train the student to produce large-norm features, inspired by other lines of work (e.g., model pruning and domain adaptation), which find the large-norm features to be more significant. Finally, we propose a rather simple loss term (dubbed ND loss) to simultaneously (1) encourage student to produce large-norm features, and (2) align the direction of student features and teacher class-means. Experiments on standard benchmarks demonstrate that our explored techniques help existing KD methods achieve better performance, i.e., higher classification accuracy on ImageNet and CIFAR100 datasets, and higher detection precision on COCO dataset. Importantly, our proposed ND loss helps the most, leading to the state-of-the-art performance on these benchmarks. The source code is available at https://github.com/WangYZ1608/Knowledge-Distillation-via-ND.

Instruction tuning relies on large instruction-response corpora whose quality and composition strongly affect downstream performance. We propose Answer Divergence-Guided Selection (ADG), which selects instruction data based on the geometric structure of multi-sample outputs. ADG draws several high-temperature generations per instruction, maps responses into an embedding space, and computes an output divergence score that jointly encodes dispersion magnitude and shape anisotropy. High scores correspond to instructions whose answers are both far apart and multi-modal, rather than clustered paraphrases along a single direction. Across two backbones and three public instruction pools, fine-tuning on only 10K ADG-selected examples consistently outperforms strong selectors on six benchmarks spanning reasoning, knowledge, and coding. Analyses further show that both dispersion magnitude and shape anisotropy are necessary, supporting answer divergence as a practical signal for instruction data selection. Code and appendix are included in the supplementary materials.

Open-weight language models can be rendered unsafe through several parameter-level interventions, yet models with matched harmful compliance can exhibit fundamentally different failure modes. We compare harmful supervised fine-tuning (SFT), harmful reinforcement learning with verifiable rewards (RLVR), and refusal-feature abliteration in Qwen2.5-7B and Llama-3.1-8B using harmfulness, capability, self-audit, safety reflection, representation similarity, and refusal-direction repair. All three routes reach near-ceiling harmfulness, but SFT causes the broadest capability loss and representational drift; abliteration yields localized, family-dependent refusal-feature suppression; and RLVR largely preserves base-model capability, explicit safety judgments, and representation geometry while retargeting behavior toward compliance. RLVR models consequently remain unusually responsive to safety-reflection prompts despite complying under direct prompting. We further find that harmful RLVR induces capability-blind compliance: models claim to complete unavailable actions and fabricate information about nonexistent entities. Targeted RLVR calibration substantially reduces this false acceptance without restoring safety or degrading general capability. These results show that harmful compliance, harm recognition, and capability awareness are separable behavioral axes, and that self-audit and hallucination patterns are not necessarily robust safety signals under adaptive post-training.

We develop a discipline-agnostic emergence calculus that treats theories as fixed points of idempotent operators acting on descriptions. We show that, once processes are composable but access to the underlying system is mediated by a bounded observational interface, a canonical toolkit of six closure-changing primitives (P1--P6) is unavoidable. The framework unifies order-theoretic closure operators with dynamics-induced endomaps E_{τ,f} built from a Markov kernel, a coarse-graining lens, and a time scale τ. We introduce a computable total-variation idempotence defect for E_{τ,f}; small retention error implies approximate idempotence and yields stable "objects" packaged at the chosen τ within a fixed lens. For directionality, we define an arrow-of-time functional as the path-space KL divergence between forward and time-reversed trajectories and prove it is monotone under coarse-graining (data processing); we also formalize a protocol-trap audit showing that protocol holonomy alone cannot sustain asymmetry without a genuine affinity in the lifted dynamics. Finally, we prove a finite forcing-style counting lemma: relative to a partition-based theory, definable predicate extensions are exponentially rare, giving a clean anti-saturation mechanism for strict ladder climbing.

Reinforcement learning with verifiable rewards has become a scalable post-training paradigm for large language models. However, training agents over long-horizon interactions presents two challenges: using only terminal rewards makes turn-level credit assignment sparse and high-variance, while sequential mini-batch updates on a fixed rollout batch introduce policy staleness as the current policy drifts from the behavior policy. Existing approaches obtain denser credit via learned critics, process reward models, or on-policy distillation, but they rely on a value network, step-level annotations, or a stronger external teacher that increase training complexity. Meanwhile, importance sampling and clipping mitigate policy drift, but they do not specify a finite target that stops sequential updates from overshooting. In this work, we introduce TreeOPSD, a critic-free, tree-aware posterior self-distillation framework for long-horizon agentic reinforcement learning. Treating each interaction turn as a tree node, TreeOPSD branches alternative continuations from shared prefixes and recursively backs up terminal rewards into implicit turn-level credit. It then maps each credit to a closed-form, bounded teacher distribution parameterized by the likelihood ratio relative to the frozen behavior policy. Minimizing the divergence between this teacher and the student policy recovers the policy-gradient direction as the tangent at the behavior policy while bounding updates toward a finite target across sequential mini-batches. Empirically, TreeOPSD improves agentic capabilities across several competitive benchmarks and model scales, especially for long-horizon tasks, e.g., raising the average accuracy of Qwen3-30B-A3B-Thinking-2507 by 4.97 points over GRPO on the Deep Research setting.

Neural end-to-end TTS can generate very high-quality synthesized speech, and even close to human recording within similar domain text. However, it performs unsatisfactory when scaling it to challenging test sets. One concern is that the encoder-decoder with attention-based network adopts autoregressive generative sequence model with the limitation of "exposure bias" To address this issue, we propose two novel methods, which learn to predict future by improving agreement between forward and backward decoding sequence. The first one is achieved by introducing divergence regularization terms into model training objective to reduce the mismatch between two directional models, namely L2R and R2L (which generates targets from left-to-right and right-to-left, respectively). While the second one operates on decoder-level and exploits the future information during decoding. In addition, we employ a joint training strategy to allow forward and backward decoding to improve each other in an interactive process. Experimental results show our proposed methods especially the second one (bidirectional decoder regularization), leads a significantly improvement on both robustness and overall naturalness, as outperforming baseline (the revised version of Tacotron2) with a MOS gap of 0.14 in a challenging test, and achieving close to human quality (4.42 vs. 4.49 in MOS) on general test.

Background: Agent skills are increasingly deployed as modular, reusable capability units in AI agent systems. Medical research agent skills require safeguards beyond general-purpose evaluation, including scientific integrity, methodological validity, reproducibility, and boundary safety. This study developed and preliminarily evaluated a domain-specific audit framework for medical research agent skills, with a focus on reliability against expert review. Methods: We developed MedSkillAudit (skill-auditor@1.0), a layered framework assessing skill release readiness before deployment. We evaluated 75 skills across five medical research categories (15 per category). Two experts independently assigned a quality score (0-100), an ordinal release disposition (Production Ready / Limited Release / Beta Only / Reject), and a high-risk failure flag. System-expert agreement was quantified using ICC(2,1) and linearly weighted Cohen's kappa, benchmarked against the human inter-rater baseline. Results: The mean consensus quality score was 72.4 (SD = 13.0); 57.3% of skills fell below the Limited Release threshold. MedSkillAudit achieved ICC(2,1) = 0.449 (95% CI: 0.250-0.610), exceeding the human inter-rater ICC of 0.300. System-consensus score divergence (SD = 9.5) was smaller than inter-expert divergence (SD = 12.4), with no directional bias (Wilcoxon p = 0.613). Protocol Design showed the strongest category-level agreement (ICC = 0.551); Academic Writing showed a negative ICC (-0.567), reflecting a structural rubric-expert mismatch. Conclusions: Domain-specific pre-deployment audit may provide a practical foundation for governing medical research agent skills, complementing general-purpose quality checks with structured audit workflows tailored to scientific use cases.

Numerous capability and safety techniques of Large Language Models (LLMs), including RLHF, automated red-teaming, prompt engineering, and infilling, can be cast as sampling from an unnormalized target distribution defined by a given reward or potential function over the full sequence. In this work, we leverage the rich toolkit of Sequential Monte Carlo (SMC) for these probabilistic inference problems. In particular, we use learned twist functions to estimate the expected future value of the potential at each timestep, which enables us to focus inference-time computation on promising partial sequences. We propose a novel contrastive method for learning the twist functions, and establish connections with the rich literature of soft reinforcement learning. As a complementary application of our twisted SMC framework, we present methods for evaluating the accuracy of language model inference techniques using novel bidirectional SMC bounds on the log partition function. These bounds can be used to estimate the KL divergence between the inference and target distributions in both directions. We apply our inference evaluation techniques to show that twisted SMC is effective for sampling undesirable outputs from a pretrained model (a useful component of harmlessness training and automated red-teaming), generating reviews with varied sentiment, and performing infilling tasks.

Vision-language (VL) models have demonstrated strong performance across various tasks. However, these models often rely on a specific modality for predictions, leading to "dominant modality bias.'' This bias significantly hurts performance, especially when one modality is impaired. In this study, we analyze model behavior under dominant modality bias and theoretically show that unaligned gradients or differences in gradient magnitudes prevent balanced convergence of the loss. Based on these findings, we propose a novel framework, BalGrad to mitigate dominant modality bias. Our approach includes inter-modality gradient reweighting, adjusting the gradient of KL divergence based on each modality's contribution, and inter-task gradient projection to align task directions in a non-conflicting manner. Experiments on UPMC Food-101, Hateful Memes, and MM-IMDb datasets confirm that BalGrad effectively alleviates over-reliance on specific modalities when making predictions.

This study proposes a behaviorally-informed multi-factor stock selection framework that integrates short-cycle technical alpha signals with deep learning. We design a dual-task multilayer perceptron (MLP) that jointly predicts five-day future returns and directional price movements, thereby capturing nonlinear market behaviors such as volume-price divergence, momentum-driven herding, and bottom reversals. The model is trained on 40 carefully constructed factors derived from price-volume patterns and behavioral finance insights. Empirical evaluation demonstrates that the dual-task MLP achieves superior and stable performance across both predictive accuracy and economic relevance, as measured by information coefficient (IC), information ratio (IR), and portfolio backtesting results. Comparative experiments further show that deep learning methods outperform linear baselines by effectively capturing structural interactions between factors. This work highlights the potential of structure-aware deep learning in enhancing multi-factor modeling and provides a practical framework for short-horizon quantitative investment strategies.

Direct On-Policy Distillation (Direct-OPD) transfers reinforcement-learning-induced policy improvements from a small model to a larger student by using the token-level log-ratio between post-RL and pre-RL checkpoints as dense supervision on the student's own rollouts. This transfer rewards the policy shift at every state, yet the log-ratio measures only relative change: it can stay fixed even as the probability mass that both checkpoints assign to the student's candidate tokens vanishes. Through an exact construction, we show that the Direct-OPD reward and its update can remain unchanged while the Jensen-Shannon divergence (JSD) and both KL directions between the checkpoints vanish with this mass, and we note that a small JSD bounds how much the teacher's behavior changed. Motivated by this analysis, we propose Selective Supervision for Direct-OPD (S^2D-OPD), which ranks student-sampled states by their teacher-reference JSD and masks Direct-OPD supervision at low-divergence states, retaining only the top 10% of states per response. Across two teacher pairs and four student models ranging from 1.7B to 8B parameters, S^2D-OPD improves held-out accuracy over dense Direct-OPD on AIME and HMMT benchmarks in seven of eight settings and matches it in the eighth, without extra forward passes.

Cross-entropy pretraining and preference alignment update the same transformer weights, but leave geometrically distinct traces. We characterise this asymmetry with a relative-subspace-fraction probe that tracks how weight deltas align with residual-stream activation subspaces and with the prediction subspace defined by the unembedding. Alignment deltas concentrate in the read pathway (W_Q, W_K), along principal directions of attention-input activations, while remaining near-isotropic in the write pathway (W_O, W_2) relative to the prediction subspace. We explain this pattern through anisotropic gradient accumulation: updates to a matrix W are sums of outer products δ_t a_t^top, and inherit directional structure from whichever side has concentrated covariance. For read-pathway matrices, this side is the input activation a_t, whose covariance is spiked in trained transformers and therefore produces objective-agnostic concentration. For write-pathway matrices, the relevant side is the upstream gradient δ_t, whose anisotropy depends on the loss. Cross-entropy supplies the canonical sharp per-sample signal, inducing write-pathway prediction geometry during pretraining; alignment objectives typically add little further write-side concentration. We support this explanation with a within-checkpoint trajectory, a graded contrastive-objective control, and a closed-form rank-1 intervention with matched direction controls, providing causal evidence for the proposed weight-space geometry.

Recently, a series of papers proposed deep learning-based approaches to sample from unnormalized target densities using controlled diffusion processes. In this work, we identify these approaches as special cases of the Schr\"odinger bridge problem, seeking the most likely stochastic evolution between a given prior distribution and the specified target. We further generalize this framework by introducing a variational formulation based on divergences between path space measures of time-reversed diffusion processes. This abstract perspective leads to practical losses that can be optimized by gradient-based algorithms and includes previous objectives as special cases. At the same time, it allows us to consider divergences other than the reverse Kullback-Leibler divergence that is known to suffer from mode collapse. In particular, we propose the so-called log-variance loss, which exhibits favorable numerical properties and leads to significantly improved performance across all considered approaches.

Forward Gradients - the idea of using directional derivatives in forward differentiation mode - have recently been shown to be utilizable for neural network training while avoiding problems generally associated with backpropagation gradient computation, such as locking and memorization requirements. The cost is the requirement to guess the step direction, which is hard in high dimensions. While current solutions rely on weighted averages over isotropic guess vector distributions, we propose to strongly bias our gradient guesses in directions that are much more promising, such as feedback obtained from small, local auxiliary networks. For a standard computer vision neural network, we conduct a rigorous study systematically covering a variety of combinations of gradient targets and gradient guesses, including those previously presented in the literature. We find that using gradients obtained from a local loss as a candidate direction drastically improves on random noise in Forward Gradient methods.

Sampling from diffusion models involves a slow iterative process that hinders their practical deployment, especially for interactive applications. To accelerate generation speed, recent approaches distill a multi-step diffusion model into a single-step student generator via variational score distillation, which matches the distribution of samples generated by the student to the teacher's distribution. However, these approaches use the reverse Kullback-Leibler (KL) divergence for distribution matching which is known to be mode seeking. In this paper, we generalize the distribution matching approach using a novel f-divergence minimization framework, termed f-distill, that covers different divergences with different trade-offs in terms of mode coverage and training variance. We derive the gradient of the f-divergence between the teacher and student distributions and show that it is expressed as the product of their score differences and a weighting function determined by their density ratio. This weighting function naturally emphasizes samples with higher density in the teacher distribution, when using a less mode-seeking divergence. We observe that the popular variational score distillation approach using the reverse-KL divergence is a special case within our framework. Empirically, we demonstrate that alternative f-divergences, such as forward-KL and Jensen-Shannon divergences, outperform the current best variational score distillation methods across image generation tasks. In particular, when using Jensen-Shannon divergence, f-distill achieves current state-of-the-art one-step generation performance on ImageNet64 and zero-shot text-to-image generation on MS-COCO. Project page: https://research.nvidia.com/labs/genair/f-distill

We propose a gradient flow procedure for generative modeling by transporting particles from an initial source distribution to a target distribution, where the gradient field on the particles is given by a noise-adaptive Wasserstein Gradient of the Maximum Mean Discrepancy (MMD). The noise-adaptive MMD is trained on data distributions corrupted by increasing levels of noise, obtained via a forward diffusion process, as commonly used in denoising diffusion probabilistic models. The result is a generalization of MMD Gradient Flow, which we call Diffusion-MMD-Gradient Flow or DMMD. The divergence training procedure is related to discriminator training in Generative Adversarial Networks (GAN), but does not require adversarial training. We obtain competitive empirical performance in unconditional image generation on CIFAR10, MNIST, CELEB-A (64 x64) and LSUN Church (64 x 64). Furthermore, we demonstrate the validity of the approach when MMD is replaced by a lower bound on the KL divergence.

How strongly do the remaining diffusion-sampling steps amplify a perturbation at a late latent state? Standard measurements answer this question with newly sampled isotropic noise, even though perturbations encountered during sampling have already been transformed by earlier steps. We compare these two cases directly. For each trajectory, we transport a centered perturbation from an earlier step to a late state, then replay its direction at the same magnitude as a newly sampled isotropic perturbation; both then undergo the same remaining updates. Across the samplers we study, median paired shaped-to-fresh angular-gain ratios range from 1.14 to 2.35. Shaped angular gain exceeds its matched fresh counterpart in every trajectory in the original main cohorts. The same pattern appears when endpoint change is measured by latent RMS. The effect also appears in perceptual feature representations: earlier-sampling directions cause larger feature changes at the endpoint, even when the corresponding pixel-space change is comparable. Permuting shaped directions across trajectories weakens the effect, including within class, indicating that the advantage depends on alignment with the receiving trajectory as well as on shared directional structure.

Modern neural network training relies on optimizers such as Adam and Muon which act on each weight matrix as a single object. Yet every weight matrix carries two distinct quantities -- a magnitude and a direction -- and all optimizers stepping in the matrix as a whole couple their dynamics: the directional change from an update depends on the current magnitude, while the magnitude drifts as a byproduct of learning the direction, so neither is governed directly by the learning rate. Typical training therefore leans on surrounding recipes such as weight decay and warmup to keep learning stable at scale, though these regulate the coupling only indirectly; other recent methods instead constrain the weight to a fixed-norm sphere, but add no learnable magnitude, leaving scale control to normalization layers alone. We propose Magnitude--Direction (MD) Decoupling, an optimizer modification that factorizes each weight into a fixed-norm direction on a hypersphere and learnable per-row and per-column magnitude gains, updated at separate learning rates, all while the model still sees a single fused weight tensor. The method is agnostic to the base optimizer and removes the need for weight decay and warmup. Across both Adam and Muon, MD Decoupling improves on well-tuned baselines, transfers the optimal LR across model width without retuning, and continues to help at scale on large Mixture-of-Experts (MoE) models. Treating magnitude and direction as separately controlled quantities thus yields more predictable training dynamics and a simple, broadly applicable improvement to modern optimizers.

We propose a direct estimation method for R\'{e}nyi and f-divergence measures based on a new graph theoretical interpretation. Suppose that we are given two sample sets X and Y, respectively with N and M samples, where eta:=M/N is a constant value. Considering the k-nearest neighbor (k-NN) graph of Y in the joint data set (X,Y), we show that the average powered ratio of the number of X points to the number of Y points among all k-NN points is proportional to R\'{e}nyi divergence of X and Y densities. A similar method can also be used to estimate f-divergence measures. We derive bias and variance rates, and show that for the class of gamma-H\"{o}lder smooth functions, the estimator achieves the MSE rate of O(N^{-2gamma/(gamma+d)}). Furthermore, by using a weighted ensemble estimation technique, for density functions with continuous and bounded derivatives of up to the order d, and some extra conditions at the support set boundary, we derive an ensemble estimator that achieves the parametric MSE rate of O(1/N). Our estimators are more computationally tractable than other competing estimators, which makes them appealing in many practical applications.

We introduce a rank-statistic approximation of f-divergences that avoids explicit density-ratio estimation by working directly with the distribution of ranks. For a resolution parameter K, we map the mismatch between two univariate distributions μ and ν to a rank histogram on { 0, ldots, K} and measure its deviation from uniformity via a discrete f-divergence, yielding a rank-statistic divergence estimator. We prove that the resulting estimator of the divergence is monotone in K, is always a lower bound of the true f-divergence, and we establish quantitative convergence rates for Ktoinfty under mild regularity of the quantile-domain density ratio. To handle high-dimensional data, we define the sliced rank-statistic f-divergence by averaging the univariate construction over random projections, and we provide convergence results for the sliced limit as well. We also derive finite-sample deviation bounds along with asymptotic normality results for the estimator. Finally, we empirically validate the approach by benchmarking against neural baselines and illustrating its use as a learning objective in generative modeling experiments.

Deciding whether to call a tool is a core competence of an LLM agent, and a costly one to get wrong: needless calls add latency, accrue cost, and may trigger irreversible side effects, while missing calls leave the model confidently wrong on questions it could only answer through tool-calls. Models manage this balance poorly, both over-using and under-using tools. Existing methods such as post-training and prompt engineering are expensive and difficult to modify at inference time. We show that whether an instruction-tuned model calls a tool can be controlled by a single linear direction in its residual stream, extracted without any training from the model's own tool-use preference signal and turned into an inference-time intervention with no prompt change. Adding the direction with strength α moves the call rate monotonically from near 0% to over 90% while keeping calls well-formed. The steering works in both directions: dialing it down suppresses calls, and dialing it up induces new calls that land precisely on the questions the model cannot answer from its own knowledge. We also show that the direction generalizes to unseen tools with strength comparable to each tool's own direction and without favoring any specific tool choice. With live tool execution, a single sweep of the steering traces a cost/accuracy Pareto frontier and nearly doubles open-domain QA accuracy (0.29 ! rightarrow ! 0.56); the same recipe transfers across a diverse range of models spanning dense, MoE, and multimodal architectures, without any training. Our code is publicly available at https://github.com/YuqiChen4188/Steering-Tool-Use-Propensity.

This paper studies the problem of training a two-layer ReLU network for binary classification using gradient flow with small initialization. We consider a training dataset with well-separated input vectors: Any pair of input data with the same label are positively correlated, and any pair with different labels are negatively correlated. Our analysis shows that, during the early phase of training, neurons in the first layer try to align with either the positive data or the negative data, depending on its corresponding weight on the second layer. A careful analysis of the neurons' directional dynamics allows us to provide an O(log n{mu}) upper bound on the time it takes for all neurons to achieve good alignment with the input data, where n is the number of data points and mu measures how well the data are separated. After the early alignment phase, the loss converges to zero at a O(1{t}) rate, and the weight matrix on the first layer is approximately low-rank. Numerical experiments on the MNIST dataset illustrate our theoretical findings.

Optimal transport (OT) theory has been been used in machine learning to study and characterize maps that can push-forward efficiently a probability measure onto another. Recent works have drawn inspiration from Brenier's theorem, which states that when the ground cost is the squared-Euclidean distance, the ``best'' map to morph a continuous measure in P(Rd) into another must be the gradient of a convex function. To exploit that result, [Makkuva+ 2020, Korotin+2020] consider maps T=nabla f_theta, where f_theta is an input convex neural network (ICNN), as defined by Amos+2017, and fit theta with SGD using samples. Despite their mathematical elegance, fitting OT maps with ICNNs raises many challenges, due notably to the many constraints imposed on theta; the need to approximate the conjugate of f_theta; or the limitation that they only work for the squared-Euclidean cost. More generally, we question the relevance of using Brenier's result, which only applies to densities, to constrain the architecture of candidate maps fitted on samples. Motivated by these limitations, we propose a radically different approach to estimating OT maps: Given a cost c and a reference measure rho, we introduce a regularizer, the Monge gap M^c_{rho}(T) of a map T. That gap quantifies how far a map T deviates from the ideal properties we expect from a c-OT map. In practice, we drop all architecture requirements for T and simply minimize a distance (e.g., the Sinkhorn divergence) between Tsharpmu and nu, regularized by M^c_rho(T). We study M^c_{rho}, and show how our simple pipeline outperforms significantly other baselines in practice.

We give an improved theoretical analysis of score-based generative modeling. Under a score estimate with small L^2 error (averaged across timesteps), we provide efficient convergence guarantees for any data distribution with second-order moment, by either employing early stopping or assuming smoothness condition on the score function of the data distribution. Our result does not rely on any log-concavity or functional inequality assumption and has a logarithmic dependence on the smoothness. In particular, we show that under only a finite second moment condition, approximating the following in reverse KL divergence in epsilon-accuracy can be done in tilde Oleft(d log (1/delta){epsilon}right) steps: 1) the variance-delta Gaussian perturbation of any data distribution; 2) data distributions with 1/delta-smooth score functions. Our analysis also provides a quantitative comparison between different discrete approximations and may guide the choice of discretization points in practice.

We point out a few pitfalls in implementing gradient estimation for KL divergence in RL training for LLM, as seen in a number of open source projects and papers. The first major pitfall is to differentiate through the KL estimate as loss functions to minimize KL divergence. We show that such implementations are generally incorrect and do not produce the desired KL gradient. Secondly, we show that some implementations do not account for the sequential nature of the estimation problem and produce a partial gradient at best. We demonstrate the impact of such issues with illustrative tabular and LLM experiments, and show the correct way to implement the KL gradient.

Using backpropagation to compute gradients of objective functions for optimization has remained a mainstay of machine learning. Backpropagation, or reverse-mode differentiation, is a special case within the general family of automatic differentiation algorithms that also includes the forward mode. We present a method to compute gradients based solely on the directional derivative that one can compute exactly and efficiently via the forward mode. We call this formulation the forward gradient, an unbiased estimate of the gradient that can be evaluated in a single forward run of the function, entirely eliminating the need for backpropagation in gradient descent. We demonstrate forward gradient descent in a range of problems, showing substantial savings in computation and enabling training up to twice as fast in some cases.

Optimization analyses for cross-entropy training rely on local Taylor models of the loss to predict whether a proposed step will decrease the objective. These surrogates are reliable only inside the Taylor convergence radius of the true loss along the update direction. That radius is set not by real-line curvature alone but by the nearest complex singularity. For cross-entropy, the softmax partition function F=sum_j exp(z_j) has complex zeros -- ``ghosts of softmax'' -- that induce logarithmic singularities in the loss and cap this radius. To make this geometry usable, we derive closed-form expressions under logit linearization along the proposed update direction. In the binary case, the exact radius is ρ^*=δ^2+ π^2/Δ_a. In the multiclass case, we obtain the lower bound ρ_a=π/Δ_a, where Δ_a=max_k a_k-min_k a_k is the spread of directional logit derivatives a_k=nabla z_kcdot v. This bound costs one Jacobian-vector product and reveals what makes a step fragile: samples that are both near a decision flip and highly sensitive to the proposed direction tighten the radius. The normalized step size r=τ/ρ_a separates safe from dangerous updates. Across six tested architectures and multiple step directions, no model fails for r<1, yet collapse appears once rge 1. Temperature scaling confirms the mechanism: normalizing by ρ_a shrinks the onset-threshold spread from standard deviation 0.992 to 0.164. A controller that enforces τleρ_a survives learning-rate spikes up to 10{,} 000times in our tests, where gradient clipping still collapses. Together, these results identify a geometric constraint on cross-entropy optimization that operates through Taylor convergence rather than Hessian curvature.

When a language model is fed a wrong answer, what happens inside the network? Current understanding treats truthfulness as a static property of individual-layer representations-a direction to be probed, a feature to be extracted. Less is known about the dynamics: how internal representations diverge across the full depth of the network when the model processes correct versus incorrect continuations. We introduce forced-completion probing, a method that presents identical queries with known correct and incorrect single-token continuations and tracks five geometric measurements across every layer of four decoder-only models(1.5B-13B parameters). We report three findings. First, correct and incorrect paths diverge through rotation, not rescaling: displacement vectors maintain near-identical magnitudes while their angular separation increases, meaning factual selection is encoded in direction on an approximate hypersphere. Second, the model does not passively fail on incorrect input-it actively suppresses the correct answer, driving internal probability away from the right token. Third, both phenomena are entirely absent below a parameter threshold and emerge at 1.6B, suggesting a phase transition in factual processing capability. These results show that factual constraint processing has a specific geometric character-rotational, not scalar; active, not passive-that is invisible to methods based on single-layer probes or magnitude comparisons.

Low-precision number formats are widely used in modern machine learning systems due to their efficiency. Accurate direction representation is key to the accuracy of vector operations. This work precisely explores the extent to which the direction of a vector can be represented by selecting its scalar elements from a common finite alphabet of a given size. This is standard practice in machine learning, where low-precision significands may be narrow-width floating-point or integer values. A geometric framework is introduced for analyzing the directional coverage of such product-structured codes. This work analytically quantifies the suboptimality gap between such product-structured codes and spherical codes for the vector as a whole, in both low and asymptotically high dimensions. Furthermore, within the product code class, it is proven that the standard formats of two's complement, fixed-point, and floating-point are suboptimal, again with quantified gap, pointing to the potential to develop new scalar number formats. Such scalar alphabets are numerically optimized across multiple block dimensions for directional coverage, including the dimension used in NVIDIA's NVFP4 format. Experimental results are presented comparing the performance of standard formats and the optimized alphabet. We find that for four bits, NVIDIA's choice of E2M1 closely approximates the optimized alphabet, providing a geometric explanation for its strong performance in low-precision machine learning workloads and an analytical understanding of the link between that superiority and block size. We provide open-source formal proofs in Lean for the theorems in this work, along with the experimental code and the optimized alphabets obtained.

The annealing phase is a pivotal convergence stage in LLM pre-training that ultimately determines final model quality. However, effectively selecting training data during this phase remains a key challenge. Current strategies rely on empirical heuristics, such as domain filtering or context extension, which lack a principled grounding in optimization theory. In this work, we characterize the annealing phase through the lens of the loss landscape's spectral geometry. We argue that optimal convergence requires gradient updates to satisfy heterogeneous constraints across different eigen-directions. Building on this insight, we formulate data selection as a problem of satisfying these directional constraints. To this end, we propose DiReCT (Directionally-Restrained Constrained Training), a novel framework that reformulates sample selection in the annealing stage as a constrained optimization problem. By imposing explicit directional constraints on per-sample gradients based on the spectral properties of the Hessian, DiReCT identifies samples that align with the optimal curvature-aware descent path. Extensive experiments across various model scales demonstrate that DiReCT consistently achieves state-of-the-art performance. For future research, code is available at https://github.com/xuyj233/Direct.

Autoregressive models accumulate error over long rollouts, yet at deployment there is no ground truth to measure it against. We train a single conditional latent diffusion model that steps a dynamical system forward or backward in time via a direction flag, and show that this bidirectionality supplies a measurement-free test-time error signal: rolling forward i steps and then backward i steps must return the model to its start, so the round-trip discrepancy C_i is a self-supervised proxy for the unobservable rollout error: no ensembles, no held-out data, no governing equations, for one extra rollout. We validate on compressible magnetohydrodynamics (MHD), an astrophysical turbulent radiative mixing layer, and natural face videos (CelebV-HQ). On held-out MHD trajectories, C_i ranks rollout error (Spearman 0.91-0.98 at fixed depth; 0.69 pm 0.16 within trajectories), and a simple calibrator fit on training rollouts predicts its magnitude to within 1.14times (68%) and 1.29times (95%) with near-nominal coverage - one nat beyond a depth-only predictor, transferring to all six decoded physical fields. The same signal flags the out-of-distribution Orszag-Tang vortex (AUROC 0.98; 1.0 by depth 10) exactly where sampling-dispersion baselines invert, and it cuts incurred error by 15% at 80% coverage - three times the depth-only baseline. Bidirectional training comes at negative cost, beating direction specialists in both directions, and the backward direction doubles as a fast inverse solver. On LE-PDE-UQ's turbulent Navier-Stokes benchmark, a single bidirectional model reaches accuracy within 1.3times of their ten-model ensemble at a tenth of the training cost, with the best training-free pixel-level calibration. Round-trip consistency turns reversibility into a practical trust signal for generative models.

Decision APIs that return a probability for each option are routinely used as if those probabilities were properties of the options themselves. That reading is valid only if the model satisfies independence of irrelevant alternatives (IIA): adding an option that is neither the right answer nor its strongest rival should leave the odds between those two unchanged. We test this in a pre-registered study of jev-1.13.0, a hosted decision model, using 299,463 calls on intent classification (BANKING77, CLINC150) and topic classification (AG News). Every added option moves the log-odds between the gold label and the strongest wrong label: the root-mean-square systematic shift is 0.20-0.52 nats, 1.2 to 3 times the run-to-run variation, even when the added option takes at most 0.1% of the probability mass. The shifts are directional. An option similar to the wrong answer draws probability from it (+0.10 nats toward the gold label). A paraphrase of the wrong answer produces an intermediate pattern: the gold label's probability is almost conserved, but the paraphrase receives only about a third of the wrong answer's share rather than the half that strict substitution implies. The distortion grows steadily with the logarithm of the number of options, reaching about 1 nat at 255 options, and off-domain padding pushes the odds against the gold label. In contrast, questions packed into one call do not interact (bias < 0.001). Scoring each option with an independent yes/no question removes the option-set dependence but picks the right answer 8.7 points less often. Jev sits between two local baselines, an option-independent scorer and a prompt readout. Its probabilities are best read as relative to the option list they were computed over.

A popular approach to post-training control of large language models (LLMs) is the steering of intermediate latent representations. Namely, identify a well-chosen direction depending on the task at hand and perturbs representations along this direction at inference time. While many propositions exist to pick this direction, considerably less is understood about how to choose the magnitude of the move, whereas its importance is clear: too little and the intended behavior does not emerge, too much and the model's performance degrades beyond repair. In this work, we propose the first theoretical analysis of steering strength. We characterize its effect on next token probability, presence of a concept, and cross-entropy, deriving precise qualitative laws governing these quantities. Our analysis reveals surprising behaviors, including non-monotonic effects of steering strength. We validate our theoretical predictions empirically on eleven language models, ranging from a small GPT architecture to modern models.

Most modern optimizers form their momentum as an exponential moving average (EMA) of past gradients, forgetting every direction at one fixed rate. However, the inputs a deep network sees during training can be highly anisotropic, with a few directions queried frequently while most are seen rarely. Preconditioning methods address this anisotropy by wrapping extra processing around this buffer and leave the momentum update itself unchanged. We propose Activation-Keyed Momentum (AK-Momentum), which builds direction-awareness into the momentum update rule. The gradient of a linear layer splits into an input activation that acts as a key and an output-side error that acts as a value. Keying on that activation, AK-Momentum updates the momentum buffer by the canonical delta rule, so each direction is forgotten at a rate set by how often it appears. We prove that it is a valid momentum, that it applies the input-side curvature correction without matrix inversion, and that it clears stale directions faster than EMA under both a fixed and a drifting optimum. It is a drop-in replacement for the momentum buffer of any optimizer, its coefficient transfers across widths under μP, and its extra compute stays between 22.2% and 25.0% of a gated-MLP block's linear cost with no persistent memory. In FineWeb-Edu pretraining, AdamW with AK-Momentum (AK-AdamW) reaches AdamW's validation loss in up to 46.39 pm 4.32% fewer steps at 67M and 22.12 pm 0.80% at 370M over three seeds, and the gain persists at 1B on a Chinchilla-optimal budget. A Muon baseline tuned under the same protocol sits above AK-AdamW at both language-model scales, and the gain holds for SGD, ResNet-18, and ViT-Tiny on CIFAR-10. Training-time diagnostics confirm the predicted mechanism, better gradient tracking and healthier input directions.

Construction of ambiguity set in robust optimization relies on the choice of divergences between probability distributions. In distribution learning, choosing appropriate probability distributions based on observed data is critical for approximating the true distribution. To improve the performance of machine learning models, there has recently been interest in designing objective functions based on Lp-Wasserstein distance rather than the classical Kullback-Leibler (KL) divergence. In this paper, we derive concentration and asymptotic results using Bregman divergence. We propose a novel asymmetric statistical divergence called Wasserstein-Bregman divergence as a generalization of L2-Wasserstein distance. We discuss how these results can be applied to the construction of ambiguity set in robust optimization.

Data imbalance is a common problem in machine learning that can have a critical effect on the performance of a model. Various solutions exist but their impact on the convergence of the learning dynamics is not understood. Here, we elucidate the significant negative impact of data imbalance on learning, showing that the learning curves for minority and majority classes follow sub-optimal trajectories when training with a gradient-based optimizer. This slowdown is related to the imbalance ratio and can be traced back to a competition between the optimization of different classes. Our main contribution is the analysis of the convergence of full-batch (GD) and stochastic gradient descent (SGD), and of variants that renormalize the contribution of each per-class gradient. We find that GD is not guaranteed to decrease the loss for each class but that this problem can be addressed by performing a per-class normalization of the gradient. With SGD, class imbalance has an additional effect on the direction of the gradients: the minority class suffers from a higher directional noise, which reduces the effectiveness of the per-class gradient normalization. Our findings not only allow us to understand the potential and limitations of strategies involving the per-class gradients, but also the reason for the effectiveness of previously used solutions for class imbalance such as oversampling.

Aligning language models with preferences can be posed as approximating a target distribution representing some desired behavior. Existing approaches differ both in the functional form of the target distribution and the algorithm used to approximate it. For instance, Reinforcement Learning from Human Feedback (RLHF) corresponds to minimizing a reverse KL from an implicit target distribution arising from a KL penalty in the objective. On the other hand, Generative Distributional Control (GDC) has an explicit target distribution and minimizes a forward KL from it using the Distributional Policy Gradient (DPG) algorithm. In this paper, we propose a new approach, f-DPG, which allows the use of any f-divergence to approximate any target distribution that can be evaluated. f-DPG unifies both frameworks (RLHF, GDC) and the approximation methods (DPG, RL with KL penalties). We show the practical benefits of various choices of divergence objectives and demonstrate that there is no universally optimal objective but that different divergences present different alignment and diversity trade-offs. We show that Jensen-Shannon divergence strikes a good balance between these objectives, and frequently outperforms forward KL divergence by a wide margin, leading to significant improvements over prior work. These distinguishing characteristics between divergences persist as the model size increases, highlighting the importance of selecting appropriate divergence objectives.

We introduce a novel method for solving density-based topology optimization problems: Sigmoidal Mirror descent with a Projected Latent variable (SiMPL). The SiMPL method (pronounced as ``the simple method'') optimizes a design using only first-order derivative information of the objective function. The bound constraints on the density field are enforced with the help of the (negative) Fermi--Dirac entropy, which is also used to define a non-symmetric distance function called a Bregman divergence on the set of admissible designs. This Bregman divergence leads to a simple update rule that is further simplified with the help of a so-called latent variable. Because the SiMPL method involves discretizing the latent variable, it produces a sequence of pointwise-feasible iterates, even when high-order finite elements are used in the discretization. Numerical experiments demonstrate that the method outperforms other popular first-order optimization algorithms. To outline the general applicability of the technique, we include examples with (self-load) compliance minimization and compliant mechanism optimization problems.

On-policy distillation (OPD) adapts diffusion models by querying a teacher along trajectories generated by the current student, but how it should behave under classifier-free guidance (CFG), a default component of modern diffusion systems, remains poorly understood. Existing OPD methods naturally extend velocity matching to the CFG-composed prediction, directly matching teacher and student guided velocities. We show that this objective is under-identified at the branch level: positive- and negative-branch errors can compensate in the guided prediction. Through two contrasting cases, we find that naive matching remains effective under shared negative conditioning, where both branch errors decrease jointly. When the model's native CFG schema retains privileged information in the teacher's negative branch that is unavailable to the student, however, this joint reduction breaks down and the composed objective induces antagonistic branch-error dynamics, reducing the positive-branch error while increasing the negative-branch error. We term this failure mode Negative Branch Asymmetry (NBA). To address NBA, we introduce Positive--Direction Matching (PDM), a branch-aware OPD objective that separately constrains the positive prediction and the CFG conditional direction. We apply PDM to dense-to-sparse video control, where naive guided matching is highly sensitive to inference guidance scales, while branch-aware supervision enables more robust and effective knowledge transfer.

A central aspiration of mechanistic interpretability is controllability: if we know where a behavior is represented in a model's activations, we should be able to modify it. This rests on a hidden premise -- that the direction which detects a behavior and the direction which controls it are the same, or close. We test this geometrically: what is the angle between the direction that best detects a behavior and the one that best causes it? If detection implies control the cosine is near 1; otherwise it quantifies a detection-intervention gap. On Gemma 2-2B-it, output format (clean JSON vs markdown fencing) collapses both roles onto one axis. Hallucination does not: the model detects fake entities with perfect linear separability (AUC = 1.000 from layer 5), yet that direction sits at cos = 0.12 (about 83 degrees) from the direction producing a refusal -- a small, reproducible alignment, far from the cos = 1 that "detection is control" would require. A detector built from activations, with no chosen tokens, likewise fails to align (cos = -0.06). The gap generalizes: across four models from three families and two scales (1B-9B), cos stays in [0.12, 0.20], identical before and after instruction tuning (0.1197 vs 0.1200), placing its origin in pretraining. A 15-degree rotation toward the refusal direction partially bridges it -- 73% and 60% refusal on two held-out fake-entity categories at 1.8% false positives. We then ask whether this cosine predicts steerability, and it does not: detection is a high-dimensional class, not a single direction, and what separates the steerable case is functional, not readable from a static angle. The cosine is a weight-computable signature of the dissociation between knowing and steering, not a predictor of it.

We tackle the problem of estimating a Manhattan frame, i.e. three orthogonal vanishing points, and the unknown focal length of the camera, leveraging a prior vertical direction. The direction can come from an Inertial Measurement Unit that is a standard component of recent consumer devices, e.g., smartphones. We provide an exhaustive analysis of minimal line configurations and derive two new 2-line solvers, one of which does not suffer from singularities affecting existing solvers. Additionally, we design a new non-minimal method, running on an arbitrary number of lines, to boost the performance in local optimization. Combining all solvers in a hybrid robust estimator, our method achieves increased accuracy even with a rough prior. Experiments on synthetic and real-world datasets demonstrate the superior accuracy of our method compared to the state of the art, while having comparable runtimes. We further demonstrate the applicability of our solvers for relative rotation estimation. The code is available at https://github.com/cvg/VP-Estimation-with-Prior-Gravity.

Recent advancements in Text-to-Image (T2I) diffusion models have demonstrated impressive success in generating high-quality images with zero-shot generalization capabilities. Yet, current models struggle to closely adhere to prompt semantics, often misrepresenting or overlooking specific attributes. To address this, we propose a simple, training-free approach that modulates the guidance direction of diffusion models during inference. We first decompose the prompt semantics into a set of concepts, and monitor the guidance trajectory in relation to each concept. Our key observation is that deviations in model's adherence to prompt semantics are highly correlated with divergence of the guidance from one or more of these concepts. Based on this observation, we devise a technique to steer the guidance direction towards any concept from which the model diverges. Extensive experimentation validates that our method improves the semantic alignment of images generated by diffusion models in response to prompts. Project page is available at: https://korguy.github.io/

Despite the impressive performance of diffusion models such as Stable Diffusion (SD) in image generation, their slow inference limits practical deployment. Recent works accelerate inference by distilling multi-step diffusion into one-step generators. To better understand the distillation mechanism, we analyze U-Net/DiT weight changes between one-step students and their multi-step teacher counterparts. Our analysis reveals that changes in weight direction significantly exceed those in weight norm, highlighting it as the key factor during distillation. Motivated by this insight, we propose the Low-rank Rotation of weight Direction (LoRaD), a parameter-efficient adapter tailored to one-step diffusion distillation. LoRaD is designed to model these structured directional changes using learnable low-rank rotation matrices. We further integrate LoRaD into Variational Score Distillation (VSD), resulting in Weight Direction-aware Distillation (WaDi)-a novel one-step distillation framework. WaDi achieves state-of-the-art FID scores on COCO 2014 and COCO 2017 while using only approximately 10% of the trainable parameters of the U-Net/DiT. Furthermore, the distilled one-step model demonstrates strong versatility and scalability, generalizing well to various downstream tasks such as controllable generation, relation inversion, and high-resolution synthesis.

The problem of estimating an unknown discrete distribution from its samples is a fundamental tenet of statistical learning. Over the past decade, it attracted significant research effort and has been solved for a variety of divergence measures. Surprisingly, an equally important problem, estimating an unknown Markov chain from its samples, is still far from understood. We consider two problems related to the min-max risk (expected loss) of estimating an unknown k-state Markov chain from its n sequential samples: predicting the conditional distribution of the next sample with respect to the KL-divergence, and estimating the transition matrix with respect to a natural loss induced by KL or a more general f-divergence measure. For the first measure, we determine the min-max prediction risk to within a linear factor in the alphabet size, showing it is Omega(kloglog n / n) and O(k^2loglog n / n). For the second, if the transition probabilities can be arbitrarily small, then only trivial uniform risk upper bounds can be derived. We therefore consider transition probabilities that are bounded away from zero, and resolve the problem for essentially all sufficiently smooth f-divergences, including KL-, L_2-, Chi-squared, Hellinger, and Alpha-divergences.

Moment restrictions and their conditional counterparts emerge in many areas of machine learning and statistics ranging from causal inference to reinforcement learning. Estimators for these tasks, generally called methods of moments, include the prominent generalized method of moments (GMM) which has recently gained attention in causal inference. GMM is a special case of the broader family of empirical likelihood estimators which are based on approximating a population distribution by means of minimizing a varphi-divergence to an empirical distribution. However, the use of varphi-divergences effectively limits the candidate distributions to reweightings of the data samples. We lift this long-standing limitation and provide a method of moments that goes beyond data reweighting. This is achieved by defining an empirical likelihood estimator based on maximum mean discrepancy which we term the kernel method of moments (KMM). We provide a variant of our estimator for conditional moment restrictions and show that it is asymptotically first-order optimal for such problems. Finally, we show that our method achieves competitive performance on several conditional moment restriction tasks.

In this work we present a new method for the estimation of Mutual Information (MI) between random variables. Our approach is based on an original interpretation of the Girsanov theorem, which allows us to use score-based diffusion models to estimate the Kullback Leibler divergence between two densities as a difference between their score functions. As a by-product, our method also enables the estimation of the entropy of random variables. Armed with such building blocks, we present a general recipe to measure MI, which unfolds in two directions: one uses conditional diffusion process, whereas the other uses joint diffusion processes that allow simultaneous modelling of two random variables. Our results, which derive from a thorough experimental protocol over all the variants of our approach, indicate that our method is more accurate than the main alternatives from the literature, especially for challenging distributions. Furthermore, our methods pass MI self-consistency tests, including data processing and additivity under independence, which instead are a pain-point of existing methods.

For many evaluation metrics commonly used as benchmarks for unconditional image generation, trivially memorizing the training set attains a better score than models which are considered state-of-the-art; we consider this problematic. We clarify a necessary condition for an evaluation metric not to behave this way: estimating the function must require a large sample from the model. In search of such a metric, we turn to neural network divergences (NNDs), which are defined in terms of a neural network trained to distinguish between distributions. The resulting benchmarks cannot be "won" by training set memorization, while still being perceptually correlated and computable only from samples. We survey past work on using NNDs for evaluation and implement an example black-box metric based on these ideas. Through experimental validation we show that it can effectively measure diversity, sample quality, and generalization.

The LogSumExp function, dual to the Kullback-Leibler (KL) divergence, plays a central role in many important optimization problems, including entropy-regularized optimal transport (OT) and distributionally robust optimization (DRO). In practice, when the number of exponential terms inside the logarithm is large or infinite, optimization becomes challenging since computing the gradient requires differentiating every term. We propose a novel convexity- and smoothness-preserving approximation to LogSumExp that can be efficiently optimized using stochastic gradient methods. This approximation is rooted in a sound modification of the KL divergence in the dual, resulting in a new f-divergence called the Safe KL divergence. Our experiments and theoretical analysis of the LogSumExp-based stochastic optimization, arising in DRO and continuous OT, demonstrate the advantages of our approach over existing baselines.

Since its inception in "Attention Is All You Need", transformer architecture has led to revolutionary advancements in NLP. The attention layer within the transformer admits a sequence of input tokens X and makes them interact through pairwise similarities computed as softmax(XQK^top X^top), where (K,Q) are the trainable key-query parameters. In this work, we establish a formal equivalence between the optimization geometry of self-attention and a hard-margin SVM problem that separates optimal input tokens from non-optimal tokens using linear constraints on the outer-products of token pairs. This formalism allows us to characterize the implicit bias of 1-layer transformers optimized with gradient descent: (1) Optimizing the attention layer with vanishing regularization, parameterized by (K,Q), converges in direction to an SVM solution minimizing the nuclear norm of the combined parameter W=KQ^top. Instead, directly parameterizing by W minimizes a Frobenius norm objective. We characterize this convergence, highlighting that it can occur toward locally-optimal directions rather than global ones. (2) Complementing this, we prove the local/global directional convergence of gradient descent under suitable geometric conditions. Importantly, we show that over-parameterization catalyzes global convergence by ensuring the feasibility of the SVM problem and by guaranteeing a benign optimization landscape devoid of stationary points. (3) While our theory applies primarily to linear prediction heads, we propose a more general SVM equivalence that predicts the implicit bias with nonlinear heads. Our findings are applicable to arbitrary datasets and their validity is verified via experiments. We also introduce several open problems and research directions. We believe these findings inspire the interpretation of transformers as a hierarchy of SVMs that separates and selects optimal tokens.

Large language models are increasingly used as decision aids whose probability judgments shape downstream choices. Whether those judgments carry a systematic directional tilt has been hard to detect: calibration metrics aggregate unsigned errors, and naturalistic uncertainty offers no ground-truth probability. When an LLM rates a startup's success at 70% but its failure at 15%, the missing 15 points expose a distortion no aggregate score flags. We introduce OptimismBench, which detects directional bias with inverted pairs: each scenario elicits both P(success) and P(failure), and asymmetry between the two framings yields a signed bias score without ground truth. Across 16 models from 8 providers, fourteen are optimistic; pessimism appears only in Anthropic's frontier tier. Eleven matched base-versus-chat pairs across four families show post-training sets the sign of the bias, with opposite shifts in different families. The pattern survives prompt, temperature, perspective, and self-debiasing ablations. A seventeen-model six-language comparison further shows model identity dominates language, with inter-model variance at 4.7x inter-language variance. We release 3,870 items across 10 languages for per-model directional-bias auditing. When alignment makes a model more helpful, it also tilts its probabilities; downstream pipelines inherit the tilt by default.

Adaptive gradient methods, e.g. Adam, have achieved tremendous success in machine learning. Scaling the learning rate element-wisely by a certain form of second moment estimate of gradients, such methods are able to attain rapid training of modern deep neural networks. Nevertheless, they are observed to suffer from compromised generalization ability compared with stochastic gradient descent (SGD) and tend to be trapped in local minima at an early stage during training. Intriguingly, we discover that substituting the gradient in the second raw moment estimate term with its momentumized version in Adam can resolve the issue. The intuition is that gradient with momentum contains more accurate directional information and therefore its second moment estimation is a more favorable option for learning rate scaling than that of the raw gradient. Thereby we propose AdaMomentum as a new optimizer reaching the goal of training fast while generalizing much better. We further develop a theory to back up the improvement in generalization and provide convergence guarantees under both convex and nonconvex settings. Extensive experiments on a wide range of tasks and models demonstrate that AdaMomentum exhibits state-of-the-art performance and superior training stability consistently.

The core learning signal used in language model distillation is the standard Kullback-Leibler (KL) divergence between the student and teacher distributions. Traditional KL divergence tends to be dominated by the next tokens with the highest probabilities, i.e., the teacher's modes, thereby diminishing the influence of less probable yet potentially informative components of the output distribution. We propose a new tail-aware divergence that decouples the contribution of the teacher model's top-K predicted probabilities from that of lower-probability predictions, while maintaining the same computational profile as the KL Divergence. Our decoupled approach reduces the impact of the teacher modes and, consequently, increases the contribution of the tail of the distribution. Experimental results demonstrate that our modified distillation method yields competitive performance in both pre-training and supervised distillation of decoder models across various datasets. Furthermore, the distillation process is efficient and can be performed with a modest academic budget for large datasets, eliminating the need for industry-scale computing.

A new sharp inequality featuring the differential Rényi entropy, the Rényi divergence and the Rényi cross-entropy of a pair of probability density functions is established. The equality is reached when one of the probability density function is an escort density of the other. This inequality is applied, together with a general framework of a pair of transformations reciprocal to each other, to derive a number of further inequalities involving both classical and new informational functionals. A remarkable fact is that, in all these inequalities, the Rényi divergence of two probability density functions is sharply bounded by quotients of informational functionals of cross-type and single type. More precisely, we derive sharp inequalities composed by relative and cross versions of the absolute moments, or of the Fisher information measures (among others), and involving two and three probability density functions.

We present a faster algorithm to generate a warm start for sampling an arbitrary logconcave density specified by an evaluation oracle, leading to the first sub-cubic sampling algorithms for inputs in (near-)isotropic position. A long line of prior work incurred a warm-start penalty of at least linear in the dimension, hitting a cubic barrier, even for the special case of uniform sampling from convex bodies. Our improvement relies on two key ingredients of independent interest. (1) We show how to sample given a warm start in weaker notions of distance, in particular q-R\'enyi divergence for q=mathcal{O}(1), whereas previous analyses required stringent infty-R\'enyi divergence (with the exception of Hit-and-Run, whose known mixing time is higher). This marks the first improvement in the required warmness since Lov\'asz and Simonovits (1991). (2) We refine and generalize the log-Sobolev inequality of Lee and Vempala (2018), originally established for isotropic logconcave distributions in terms of the diameter of the support, to logconcave distributions in terms of a geometric average of the support diameter and the largest eigenvalue of the covariance matrix.

While previous distribution shift detection approaches can identify if a shift has occurred, these approaches cannot localize which specific features have caused a distribution shift -- a critical step in diagnosing or fixing any underlying issue. For example, in military sensor networks, users will want to detect when one or more of the sensors has been compromised, and critically, they will want to know which specific sensors might be compromised. Thus, we first define a formalization of this problem as multiple conditional distribution hypothesis tests and propose both non-parametric and parametric statistical tests. For both efficiency and flexibility, we then propose to use a test statistic based on the density model score function (i.e. gradient with respect to the input) -- which can easily compute test statistics for all dimensions in a single forward and backward pass. Any density model could be used for computing the necessary statistics including deep density models such as normalizing flows or autoregressive models. We additionally develop methods for identifying when and where a shift occurs in multivariate time-series data and show results for multiple scenarios using realistic attack models on both simulated and real world data.

Ensembles of independently trained neural networks are a state-of-the-art approach to estimate predictive uncertainty in Deep Learning, and can be interpreted as an approximation of the posterior distribution via a mixture of delta functions. The training of ensembles relies on non-convexity of the loss landscape and random initialization of their individual members, making the resulting posterior approximation uncontrolled. This paper proposes a novel and principled method to tackle this limitation, minimizing an f-divergence between the true posterior and a kernel density estimator (KDE) in a function space. We analyze this objective from a combinatorial point of view, and show that it is submodular with respect to mixture components for any f. Subsequently, we consider the problem of greedy ensemble construction. From the marginal gain on the negative f-divergence, which quantifies an improvement in posterior approximation yielded by adding a new component into the KDE, we derive a novel diversity term for ensemble methods. The performance of our approach is demonstrated on computer vision out-of-distribution detection benchmarks in a range of architectures trained on multiple datasets. The source code of our method is made publicly available at https://github.com/Oulu-IMEDS/greedy_ensembles_training.

Many policy optimization approaches in reinforcement learning incorporate a Kullback-Leilbler (KL) divergence to the previous policy, to prevent the policy from changing too quickly. This idea was initially proposed in a seminal paper on Conservative Policy Iteration, with approximations given by algorithms like TRPO and Munchausen Value Iteration (MVI). We continue this line of work by investigating a generalized KL divergence -- called the Tsallis KL divergence -- which use the q-logarithm in the definition. The approach is a strict generalization, as q = 1 corresponds to the standard KL divergence; q > 1 provides a range of new options. We characterize the types of policies learned under the Tsallis KL, and motivate when q >1 could be beneficial. To obtain a practical algorithm that incorporates Tsallis KL regularization, we extend MVI, which is one of the simplest approaches to incorporate KL regularization. We show that this generalized MVI(q) obtains significant improvements over the standard MVI(q = 1) across 35 Atari games.

Textual Inversion (TI) is an efficient approach to text-to-image personalization but often fails on complex prompts. We trace these failures to embedding norm inflation: learned tokens drift to out-of-distribution magnitudes, degrading prompt conditioning in pre-norm Transformers. Empirically, we show semantics are primarily encoded by direction in CLIP token space, while inflated norms harm contextualization; theoretically, we analyze how large magnitudes attenuate positional information and hinder residual updates in pre-norm blocks. We propose Directional Textual Inversion (DTI), which fixes the embedding magnitude to an in-distribution scale and optimizes only direction on the unit hypersphere via Riemannian SGD. We cast direction learning as MAP with a von Mises-Fisher prior, yielding a constant-direction prior gradient that is simple and efficient to incorporate. Across personalization tasks, DTI improves text fidelity over TI and TI-variants while maintaining subject similarity. Crucially, DTI's hyperspherical parameterization enables smooth, semantically coherent interpolation between learned concepts (slerp), a capability that is absent in standard TI. Our findings suggest that direction-only optimization is a robust and scalable path for prompt-faithful personalization.

We construct nets of curves in a rectangle R with special properties. The net is formed by two families of curves, "horizontal" and "vertical", each family foliates R, and every horizontal curve meets every vertical curve orthogonally. Moreover, every horizontal curve started at the left edge of R arrives to the right edge of R at the same height, and every vertical curve started at the bottom edge of R arrives to the top edge of R at a prescribed in advance longitude. The latter longitude function as a map from bottom to top can be chosen in advance as any small perturbation of the identity. Our motivation for this was to design some elliptic divergence form operators, for instance on one of the domains bounded by a Reifenberg flat curve in the plane, with the scalar coefficient and the corresponding elliptic measure being absolutely continuous with respect to the natural measure on the boundary (an object of a separate paper). The construction of our net of curves in a rectangle can also be interpreted as control problem, has an interest on its own, and, likely, has other applications.

In recent years, Muon has emerged as the dominant method for training large language models, and transformers more broadly. The essential difference, when compared to standard gradient descent methods, is to replace the usual update matrix M=UΣV^top with its polar factor UV^top. In this work, we consider a class of Muon-like updates, where we replace the update M with UΣ^p V^top for some parameter p. We call this a "spectral-shaping" operation, and develop a theory of how to pick p which depends on (a) local curvature of the loss function, (b) noise stemming from stochastic gradients and label noise, and (c) training stage. Our theory and experimentation reveal a previously overlooked behavior: positive p helps early by emphasizing high-curvature directions and accelerating signal contraction, while mildly negative p helps later by reallocating update strength toward low-curvature directions that still contain useful training signals. Building on the insight, we propose DynMuon, an efficient dynamic spectral shaping method that schedules p from positive to mildly negative over training. Extensive experiments across model sizes, architectures, and training settings show that DynMuon consistently achieves lower validation loss than Muon, while requiring 10.6-26.5% fewer steps to reach the same target loss.

We study the implicit bias of momentum-based optimizers on smooth homogeneous models. We show that momentum steepest descent algorithms like Muon (spectral norm), MomentumGD (ell_2 norm), and Signum (ell_infty norm) are approximate steepest descent trajectories under a decaying learning rate schedule, proving that these algorithms have a bias towards KKT points of the corresponding margin maximization problem. We extend the analysis to Adam (without the stability constant), which maximizes the ell_infty margin, and to Muon-Signum and Muon-Adam, which maximize a hybrid norm. Our experiments corroborate the theory and show that the identity of the margin maximized depends on the choice of optimizer. Overall, our results extend earlier lines of work on steepest descent in homogeneous models and momentum-based optimizers in linear models.

On-policy distillation (OPD) trains a student on its own trajectories under token-level teacher supervision, but existing methods are capped by a single-teacher capability ceiling: when the teacher errs, the student inherits the error. OPD also remains largely unexplored in agentic tasks, where per-step errors compound across long trajectories and destabilize training. We propose MAD-OPD (Multi-Agent Debate-driven On-Policy Distillation), which breaks this ceiling by recasting the distillation teacher as a deliberative collective of teachers that debate over the student's on-policy state; the debate produces an emergent collective intelligence that supplies token-level supervision, with each teacher's contribution weighted by its post-debate confidence. To extend OPD to agentic tasks, we also introduce On-Policy Agentic Distillation (OPAD), which adds step-level sampling to stabilize training under multi-step error compounding. We additionally derive a task-adaptive divergence principle, selecting JSD (Jensen-Shannon divergence) for agentic stability and reverse KL (Kullback-Leibler) divergence for code generation, and verify it both theoretically and empirically. Across six teacher-student configurations (Qwen3 and Qwen3.5; 1.7B-14B students, 8B-32B teachers) and five agentic and code benchmarks, MAD-OPD ranks first across all six configurations; on the 14B+8Bto4B setting it lifts the agentic average by +2.4% and the code average by +3.7% over the stronger single-teacher OPD.

The increasing capabilities of large language models (LLMs) raise opportunities for artificial general intelligence but concurrently amplify safety concerns, such as potential misuse of AI systems, necessitating effective AI alignment. Reinforcement Learning from Human Feedback (RLHF) has emerged as a promising pathway towards AI alignment but brings forth challenges due to its complexity and dependence on a separate reward model. Direct Preference Optimization (DPO) has been proposed as an alternative, and it remains equivalent to RLHF under the reverse KL regularization constraint. This paper presents f-DPO, a generalized approach to DPO by incorporating diverse divergence constraints. We show that under certain f-divergences, including Jensen-Shannon divergence, forward KL divergences and alpha-divergences, the complex relationship between the reward and optimal policy can also be simplified by addressing the Karush-Kuhn-Tucker conditions. This eliminates the need for estimating the normalizing constant in the Bradley-Terry model and enables a tractable mapping between the reward function and the optimal policy. Our approach optimizes LLMs to align with human preferences in a more efficient and supervised manner under a broad set of divergence constraints. Empirically, adopting these divergences ensures a balance between alignment performance and generation diversity. Importantly, f-DPO outperforms PPO-based methods in divergence efficiency, and divergence constraints directly influence expected calibration error (ECE).

We investigate diffusion approximations for a class of multivariate inhomogeneous finite-velocity random motions motivated by models of active particles. A particle alternates cyclically among prescribed velocity fields V_1,dots,V_p with random run times and evolves according to either a flight dynamics, consisting of piecewise-linear motion, or a gliding dynamics, in which the particle follows the corresponding velocity flow. Under a Kac-type scaling that couples vanishing run times with diverging particle speed, we prove weak convergence of both processes to multidimensional diffusions. It turns out that the order of successive cyclic motions affects the diffusion limit. In addition to the second-order term generated by fluctuations of the random run times, the limiting drift contains directional derivatives DV_i[V_j] of the underlying vector fields. In Stratonovich form, part of this drift is expressed through their Lie brackets [V_i,V_j]. Thus, the generic non-commutativity of the microscopic motions survives the diffusive scaling and generates a macroscopic drift which depends on the cyclic order. The limiting drift also distinguishes the flight and gliding dynamics, despite their being driven by the same vector fields and run times. Several examples, including run-and-reverse motion and cyclic dynamics generated by multiple vector fields, illustrate how the cyclic switching protocol and the geometry of the underlying velocity fields shape the limiting diffusions.

While recent flow-based image editing models demonstrate general-purpose capabilities across diverse tasks, they often struggle to specialize in challenging scenarios -- particularly those involving large-scale shape transformations. When performing such structural edits, these methods either fail to achieve the intended shape change or inadvertently alter non-target regions, resulting in degraded background quality. We propose Follow-Your-Shape, a training-free and mask-free framework that supports precise and controllable editing of object shapes while strictly preserving non-target content. Motivated by the divergence between inversion and editing trajectories, we compute a Trajectory Divergence Map (TDM) by comparing token-wise velocity differences between the inversion and denoising paths. The TDM enables precise localization of editable regions and guides a Scheduled KV Injection mechanism that ensures stable and faithful editing. To facilitate a rigorous evaluation, we introduce ReShapeBench, a new benchmark comprising 120 new images and enriched prompt pairs specifically curated for shape-aware editing. Experiments demonstrate that our method achieves superior editability and visual fidelity, particularly in tasks requiring large-scale shape replacement.

We analyze cumulative parameter trajectories of transformer training under AdamW and identify a dominant low-dimensional drift direction ("backbone") that captures 60--80% of long-horizon displacement from initialization. This direction is highly stable over rolling training windows yet reorients gradually across phases, particularly following objective reweighting. Per-batch gradients exhibit near-noise-floor alignment with the backbone, whereas optimizer-integrated updates align strongly with it, indicating that the structure emerges from accumulated optimizer dynamics rather than instantaneous gradient geometry. Replacing AdamW with SGD-family optimizers eliminates this structure, and reducing β_2 smoothly degrades backbone dominance and reheating recoverability. Reheating experiments show that transverse probe modes can be transiently re-excited without substantially altering accumulated backbone drift. These results provide a trajectory-level characterization of optimizer-induced geometric structure in transformer training and shift attention from instantaneous gradient properties to cumulative update dynamics.

We propose an algorithm for change point monitoring in linear causal models that accounts for interventions. Through a special centralization technique, we can concentrate the changes arising from causal propagation across nodes into a single dimension. Additionally, by selecting appropriate intervention nodes based on Kullback-Leibler divergence, we can amplify the change magnitude. We also present an algorithm for selecting the intervention values, which aids in the identification of the most effective intervention nodes. Two monitoring methods are proposed, each with an adaptive intervention policy to make a balance between exploration and exploitation. We theoretically demonstrate the first-order optimality of the proposed methods and validate their properties using simulation datasets and two real-world case studies.

Within-dataset class-split evaluation is widely used as a proxy for fully unconditional out-of-distribution anomaly detection. We show that this protocol can become ill-posed when the held-out anomaly class overlaps the normal mixture in representation space. In this regime, anomaly scores may collapse toward chance or even invert, and the preferred score direction can depend on the unknown anomaly class. We introduce a simple training-free diagnostic, neighborhood class leakage, and show that it predicts score-direction instability across Fashion-MNIST, CIFAR-10, and Imagenette, in both pixel and VAE latent spaces. Our results suggest that class-split AD benchmarks should be treated as geometry-dependent stress tests rather than unconditional evidence of anomaly-detection ability.

Westerlund 1 (Wd1) is one of the Milky Way's most massive young clusters, in which heavy extinction and field contamination have historically hampered its kinematic characterization. We aim to assess Wd1's internal kinematics, dynamical state, and formation scenario. Using a robust 5D clustering approach, we identified 559 bona fide members. We modeled their proper motions via a 2D first-order Taylor expansion to derive the velocity gradient tensor, mapping expansion, rotation, and shear. Restricting the kinematic analysis to 524 stars with proper motion uncertainties <0.5~mas,yr^{-1}, we find at least 103 gravitationally unbound members, including 14 walkaway stars. The 2D analysis reveals a statistically significant divergence (>4σ) highlighting preferred expansion along the Galactic latitude (position angle 164^circpm10.1^circ). A detected cross-gradient (partial v_b / partial l = -0.41 pm 0.08 km s^{-1} pc^{-1}) provides a clear signature of shear on the plane of the sky. Wd1 does not behave as a single cohesive population, but comprises two decoupled subsystems: a stable, bound core in dynamical equilibrium and an escaping, unbound envelope. This prominent anisotropic expansion perpendicular to the Galactic plane has a kinematic timescale (Δt_{kin} = 2.08 pm 0.44 Myr) matching the independent orbital mid-plane crossing time (Δt_{orbit} = 2.10 pm 0.05 Myr). This temporal alignment indicates the expansion onset coincided with Wd1's Galactic disk passage, triggered either by internal dynamics (e.g., early supernova feedback) or external tidal perturbations. Finally, the robust unbound population demonstrates that active cluster dissolution is underway, driven by discrete dynamical ejection events rather than long-term stellar evaporation.

Transformers were originally proposed as a sequence-to-sequence model for text but have become vital for a wide range of modalities, including images, audio, video, and undirected graphs. However, transformers for directed graphs are a surprisingly underexplored topic, despite their applicability to ubiquitous domains including source code and logic circuits. In this work, we propose two direction- and structure-aware positional encodings for directed graphs: (1) the eigenvectors of the Magnetic Laplacian - a direction-aware generalization of the combinatorial Laplacian; (2) directional random walk encodings. Empirically, we show that the extra directionality information is useful in various downstream tasks, including correctness testing of sorting networks and source code understanding. Together with a data-flow-centric graph construction, our model outperforms the prior state of the art on the Open Graph Benchmark Code2 relatively by 14.7%.

Where inside a language model does refusal live, and does that place change when the architecture does? In a transformer, refusal is governed by a single direction in the residual stream, a finding that safety and interpretability tooling now depend on. State-space models (SSMs) route information through a recurrent update instead of attention, sharing no token-mixing mechanism with a transformer. Does the same safety representation survive this shift, or must it be rediscovered per architecture? It survives. A single rigid rotation, which can only reorient a space and not reshape it, aligns one model's representation space with another's, so the two genuinely share the representation. A harm probe trained on a transformer then flags an SSM's harmful inputs, and removing the aligned direction makes a model answer attacks it would otherwise refuse, while a random direction of the same size does far less. What is architecture-specific is not where the direction is steered but where it must be read. Each layer computes a fresh output that is then added into the residual stream, and harm is cleanly readable at this output, the write site, before the addition. A control that holds the intervention's strength fixed shows that what matters is where the direction is estimated, not where it is applied. Applied through a detector-triggered gate, this direction lowers jailbreak success in all four architecture families we test (SSM, transformer, recurrent, hybrid), and on the SSM it holds against an attacker that tunes its prompt against the defense. The gate only matches a trivial rule that returns a fixed refusal whenever the same detector fires, so what transfers across architectures is the direction itself, not defense strength. Safety tooling built on refusal therefore ports to a new architecture by re-estimating the direction at that architecture's write site, not by rebuilding it.

For a parametric model of distributions, the closest distribution in the model to the true distribution located outside the model is considered. Measuring the closeness between two distributions with the Kullback-Leibler (K-L) divergence, the closest distribution is called the "information projection." The estimation risk of the maximum likelihood estimator (MLE) is defined as the expectation of K-L divergence between the information projection and the predictive distribution with plugged-in MLE. Here, the asymptotic expansion of the risk is derived up to n^{-2}-order, and the sufficient condition on the risk for the Bayes error rate between the true distribution and the information projection to be lower than a specified value is investigated. Combining these results, the "p-n criterion" is proposed, which determines whether the MLE is sufficiently close to the information projection for the given model and sample. In particular, the criterion for an exponential family model is relatively simple and can be used for a complex model with no explicit form of normalizing constant. This criterion can constitute a solution to the sample size or model acceptance problem. Use of the p-n criteria is demonstrated for two practical datasets. The relationship between the results and information criteria is also studied.

Credit risk modeling relies extensively on Weight of Evidence (WoE) and Information Value (IV) for feature engineering, and Population Stability Index (PSI) for drift monitoring, yet their theoretical foundations remain disconnected. We establish a unified information-theoretic framework revealing these industry-standard metrics as instances of classical information divergences. Specifically, we prove that IV exactly equals PSI (Jeffreys divergence) computed between good and bad credit outcomes over identical bins. Through the delta method applied to WoE transformations, we derive standard errors for IV and PSI, enabling formal hypothesis testing and probabilistic fairness constraints for the first time. We formalize credit modeling's inherent performance-fairness trade-off as maximizing IV for predictive power while minimizing IV for protected attributes. Using automated binning with depth-1 XGBoost stumps, we compare three encoding strategies: logistic regression with one-hot encoding, WoE transformation, and constrained XGBoost. All methods achieve comparable predictive performance (AUC 0.82-0.84), demonstrating that principled, information-theoretic binning outweighs encoding choice. Mixed-integer programming traces Pareto-efficient solutions along the performance-fairness frontier with uncertainty quantification. This framework bridges theory and practice, providing the first rigorous statistical foundation for widely-used credit risk metrics while offering principled tools for balancing accuracy and fairness in regulated environments.

A central paradox in fine-tuning Large Language Models (LLMs) with Reinforcement Learning with Verifiable Reward (RLVR) is the frequent degradation of multi-attempt performance (Pass@k) despite improvements in single-attempt accuracy (Pass@1). This is often accompanied by catastrophic forgetting, where models lose previously acquired skills. While various methods have been proposed, the choice and function of the divergence term have been surprisingly unexamined as a proactive solution. We argue that standard RLVR objectives -- both those using the mode-seeking reverse KL-divergence and those forgoing a divergence term entirely -- lack a crucial mechanism for knowledge retention. The reverse-KL actively accelerates this decay by narrowing the policy, while its absence provides no safeguard against the model drifting from its diverse knowledge base. We propose a fundamental shift in perspective: using the divergence term itself as the solution. Our framework, Diversity-Preserving Hybrid RL (DPH-RL), leverages mass-covering f-divergences (like forward-KL and JS-divergence) to function as a rehearsal mechanism. By continuously referencing the initial policy, this approach forces the model to maintain broad solution coverage. Extensive experiments on math and SQL generation demonstrate that DPH-RL not only resolves the Pass@k degradation but improves both Pass@1 and Pass@k in- and out-of-domain. Additionally, DPH-RL is more training-efficient because it computes f-divergence using generator functions, requiring only sampling from the initial policy and no online reference model. Our work highlights a crucial, overlooked axis for improving RLVR, demonstrating that the proper selection of a divergence measure is a powerful tool for building more general and diverse reasoning models.

Activation steering presupposes that task-relevant behaviors correspond to linear directions in activation space -- directions that should both steer the model and be readable along the unembedding. Function vectors (FVs), extracted as mean differences across ICL demonstrations, are the canonical test case; the prediction: steering and decoding succeed or fail together. Across 12 tasks, 6 models from 3 families, and 4,032 directed cross-template pairs, we find the opposite. FV steering routinely succeeds where the logit lens cannot decode the correct answer at any intermediate layer, while the converse -- decodable without steerable -- is nearly empty (3 of 72). The gap is not representational dialect. A diagonal tuned lens closes 1 of 14 steerable-not-decodable cases; a 2-layer MLP probe with a Hewitt \& Liang control closes 5 of 10 via nonlinearly encoded structure but leaves 5 invisible to every decoder tested. Even at > 0.90 steering accuracy, projecting the FV through the unembedding yields incoherent token distributions: FVs encode computational instructions, not answer directions. A model-family asymmetry sharpens the picture. Mistral FVs rewrite intermediate representations, while Llama and Gemma FVs steer the final output without leaving a logit-lens-visible trace, corroborated by three signals (post-steering deltas, activation-patching recovery, FV norm-transfer correlations). A previously reported negative cosine-transfer correlation dissolves at scale, adding at most ΔR^2 = 0.011 beyond task identity. These results decompose the linear representation hypothesis into linear decodability and linear steerability and show they come apart opposite to intuition, with implications for safety monitoring: vocabulary-projection tools are blind to FV-style interventions on widely deployed model families.

Reinforcement learning with verifiable rewards (RLVR) improves the reasoning capabilities of large language models using automatically verifiable outcome signals, but these signals are typically sparse and at the sequence-level. On-policy self-distillation (OPSD) mitigates this sparsity by querying a privileged teacher at student-visited prefixes and providing dense token-level distributional supervision. Although this dense supervision alleviates signal sparsity, we find that standard OPSD still underexploits the temporal structure of the rollout. It assigns every local divergence the same coefficient, regardless of its position or the divergence sequence in which it occurs. In on-policy autoregressive generation, the same divergence magnitude can follow different discrepancy histories, reflecting different evolutions of the mismatch between the teacher and student. Since the local scalar alone cannot distinguish these temporal contexts, standard OPSD cannot adapt its token-level weights to the realized discrepancy sequence. To address this limitation, we propose Divergence-Adaptive Supervision Horizons (DASH). DASH maps the gap between each local distillation signal and the sequence-level mean to an adaptive propagation gate and then uses these gates to control backward multi-step aggregation. By doing so, DASH adjusts token-level supervision weights according to how local divergences evolve during generation. Experiments on three mathematical reasoning benchmarks across three model scales show that DASH improves over our matched vanilla OPSD reruns on every benchmark at all three scales. DASH reuses the teacher and student distributions that OPSD already computes, so the gains require no additional teacher or student forward pass. Code: https://github.com/DBtxy/DASH-OPSD

Visual on-policy distillation relies heavily on an informative teacher-student asymmetry, through either a larger, stronger teacher or privileged supervision, such as reference answers or ground-truth regions of interest. This raises a fundamental question: where can informative asymmetry come from when nothing privileged is available? We answer this by inverting where the asymmetry comes from. Rather than adding privileged information to the teacher, we subtract information from the student. This asymmetry creates the same effective learning signal for free as a teacher with access to information unavailable to the student, without ground-truth annotations, rewards, or a separate stronger teacher model. Building on this principle, we introduce Self-Supervised Visual On-Policy Distillation (S^2VOPD), a simple yet effective method that constructs on-policy learning signals from asymmetric augmented views. S^2VOPD distills the teacher's distribution conditioned on the original image on-policy into the student distribution conditioned on a strongly augmented view of the same image. We systematically explore a broad design space of visual augmentations and uncover that (1) asymmetry matters: all four augmentation families improve performance, while symmetric self-distillation degrades it; (2) strength matters: performance peaks at a moderate strength; and (3) the gap must remain task-consistent: augmentations that completely remove the question-relevant evidence can induce large but uninformative discrepancies. Across six fine-grained perception benchmarks, S^2VOPD improves Qwen3.5-4B from 70.7% to 77.4%, above all open-source models compared, up to Qwen3-VL at 235B, and surpasses GPT-5.4. While holding training data the same, it recovers 96% of the improvement achieved by methods with privileged information. Website is at https://williamium3000.github.io/s2vopd

Fine-tuning on harmless data can partially undo behaviors acquired earlier in training. Safety can erode under benign post-alignment updates, unlearned capabilities can re-emerge, latent traits can transfer through apparently unrelated supervision, and related post-alignment fragility appears in other generative settings. We argue these phenomena are usefully viewed through a common training-history lens. Our hypothesis is geometric: large early training phases create dominant behavioral manifolds, while later alignment or specialization phases are shallower displacements from them. Subsequent fine-tuning can therefore inherit a persistent reversion component pointing back toward a witness of the dominant manifold. We call this the gravitational interpretation of fine-tuning reversion. Across our main settings, representational drift rapidly acquires a component along a history-defined reversion direction (v_rev). In our main track, alignment with v_rev rises from cos = 0.429 +/- 0.052 after the first update to 0.647 +/- 0.021 by step 20. Across 24 run-step pairs, every observed alignment exceeds the p99 of an isotropic activation-space null. We demonstrate that selectively blocking motion along v_rev changes the final alignment at T=100 from 0.648 +/- 0.009 to -0.211 +/- 0.021 and reduces harmfulness from 19.0% +/- 4.0% to 8.5% +/- 1.5% with little task cost. These results support v_rev as a causally relevant mediator of early post-alignment reversion in our setup. Importantly, we do not claim that v_rev is the unique safety direction, nor that the dominant manifold is directly observed; rather, we identify a robust, history-defined direction that explains and partially controls early reversion dynamics.

The input side of the Information Bottleneck may contain task-irrelevant variation that still costs rate. We identify this cost exactly. Let T be a source and C a relevance variable. Suppose the deterministic statistic Z = phi(T) satisfies C-Z-T. For any encoder p(X|T), its conditional average over the fibres of phi preserves I(X;C) and lowers the rate by I(X;T|Z). The reverse pullback preserves both coordinates. These maps establish equality of the relevance-rate curves and Lagrangian infima on standard Borel spaces for every tradeoff parameter. They also characterise all attained optima. Every full-source minimiser factors through Z, and every reduced minimiser pulls back to T. When C is finite and distortion is logarithmic loss, replacement of T^n by Z^n also preserves the optimal remote distortion at every blocklength and message budget. The operational rate-distortion functions are therefore equal.

Diffusion models are commonly interpreted as learning the score function, i.e., the gradient of the log-density of noisy data. However, this assumption implies that the target of learning is a conservative vector field, which is not enforced by the neural network architectures used in practice. We present numerical evidence that trained diffusion networks violate both integral and differential constraints required of true score functions, demonstrating that the learned vector fields are not conservative. Despite this, the models perform remarkably well as generative mechanisms. To explain this apparent paradox, we advocate a new theoretical perspective: diffusion training is better understood as flow matching to the velocity field of a Wasserstein Gradient Flow (WGF), rather than as score learning for a reverse-time stochastic differential equation. Under this view, the "probability flow" arises naturally from the WGF framework, eliminating the need to invoke reverse-time SDE theory and clarifying why generative sampling remains successful even when the neural vector field is not a true score. We further show that non-conservative errors from neural approximation do not necessarily harm density transport. Our results advocate for adopting the WGF perspective as a principled, elegant, and theoretically grounded framework for understanding diffusion generative models.

Knowledge distillation has become a primary mechanism for transferring reasoning and domain expertise from frontier Large Language Models (LLMs) to smaller, deployable students. However, the dominant paradigm remains off-policy: students train on static teacher-generated data and never encounter their own errors during learning. This train--test mismatch, an instance of exposure bias, causes prediction errors to compound autoregressively at inference time. On-Policy Distillation (OPD) addresses this by letting the student generate its own trajectories and receive teacher feedback on these self-generated outputs, grounding distillation in the theory of interactive imitation learning. Despite rapid growth spanning divergence minimization, reward-guided learning, and self-play, the OPD literature remains fragmented with no unified treatment. This survey provides the first comprehensive overview of OPD for LLMs. We introduce a unified f-divergence framework over on-policy samples and organize the landscape along three orthogonal dimensions: feedback signal (logit-based, outcome-based, or self-play), teacher access (white-box, black-box, or teacher-free), and loss granularity (token-level, sequence-level, or hybrid). We systematically analyze representative methods, examine industrial deployments, and identify open problems including distillation scaling laws, uncertainty-aware feedback, and agent-level distillation.

Given alphain(0,1] and pin[1,+infty], we define the space DM^{alpha,p}(mathbb R^n) of L^p vector fields whose alpha-divergence is a finite Radon measure, extending the theory of divergence-measure vector fields to the distributional fractional setting. Our main results concern the absolute continuity properties of the alpha-divergence-measure with respect to the Hausdorff measure and fractional analogues of the Leibniz rule and the Gauss-Green formula. The sharpness of our results is discussed via some explicit examples.

Many theoretical results in deep learning can be traced to symmetry or equivariance of neural networks under parameter transformations. However, existing analyses are typically problem-specific and focus on first-order consequences such as conservation laws, while the implications for second-order structure remain less understood. We develop a general equivariance toolbox that yields coupled first- and second-order constraints on learning dynamics. The framework extends classical Noether-type analyses in three directions: from gradient constraints to Hessian constraints, from symmetry to general equivariance, and from continuous to discrete transformations. At the first order, our framework unifies conservation laws and implicit-bias relations as special cases of a single identity. At the second order, it provides structural predictions about curvature: which directions are flat or sharp, how the gradient aligns with Hessian eigenspaces, and how the loss landscape geometry reflects the underlying transformation structure. We illustrate the framework through several applications, recovering known results while also deriving new characterizations that connect transformation structure to modern empirical observations about optimization geometry.

We characterize the statistical efficiency of knowledge transfer through n samples from a teacher to a probabilistic student classifier with input space mathcal S over labels mathcal A. We show that privileged information at three progressive levels accelerates the transfer. At the first level, only samples with hard labels are known, via which the maximum likelihood estimator attains the minimax rate {|{mathcal S||{mathcal A}|}/{n}}. The second level has the teacher probabilities of sampled labels available in addition, which turns out to boost the convergence rate lower bound to {{|{mathcal S}||{mathcal A}|}/{n}}. However, under this second data acquisition protocol, minimizing a naive adaptation of the cross-entropy loss results in an asymptotically biased student. We overcome this limitation and achieve the fundamental limit by using a novel empirical variant of the squared error logit loss. The third level further equips the student with the soft labels (complete logits) on {mathcal A} given every sampled input, thereby provably enables the student to enjoy a rate {|{mathcal S}|}/{n} free of |{mathcal A}|. We find any Kullback-Leibler divergence minimizer to be optimal in the last case. Numerical simulations distinguish the four learners and corroborate our theory.

On-policy distillation (OPD) provides dense token-level supervision on student rollouts, but this supervision can become unreliable when the student and teacher diverge, either because of a large initial policy gap or compounding errors in long-horizon reasoning and tool use. We introduce Divergence-Triggered Dynamic Distillation (DTDD), an adaptive OPD framework that uses segment-level student-teacher divergence to trigger teacher takeover during training. When divergence triggers an intervention during reasoning, the student prefix continues to receive standard OPD supervision. The teacher then generates a recovery suffix, which the student learns through behavior cloning, using cross-entropy on the teacher-generated tokens. We evaluate DTDD in three regimes: long-horizon reasoning, distillation from a base initialization without offline supervised fine-tuning (SFT), and multi-turn agents. We also compare alternative trigger designs and teacher-intervention rates. DTDD reduces the post-warmup peak gradient norm by 8.6times and, from a base initialization, reaches 21.3% held-out math accuracy in 2.6times fewer optimizer updates than OPD; both methods finish near 22%. It improves best-observed performance by 4.8 and 4.1 percentage points on ALFWorld with LoRA and agentic code, respectively. Ablations show that segment-averaged triggers outperform token-spike triggers. Behavior cloning is also more effective than the tested OPD surrogate for training teacher-generated recoveries. Across 23 experimental arms, takeover tends to help when the teacher substantially outperforms the OPD-trained student, provided that the teacher can recover from student-visited states, the student can learn those recoveries, and teacher intervention is neither excessive nor insufficient.

We rigorously quantify the improvement in the sample complexity of variational divergence estimations for group-invariant distributions. In the cases of the Wasserstein-1 metric and the Lipschitz-regularized alpha-divergences, the reduction of sample complexity is proportional to an ambient-dimension-dependent power of the group size. For the maximum mean discrepancy (MMD), the improvement of sample complexity is more nuanced, as it depends on not only the group size but also the choice of kernel. Numerical simulations verify our theories.

Policy-shift distillation reuses the difference between a teacher's log probabilities before and after reinforcement learning (RL) as supervision on student-generated trajectories. However, this endpoint policy shift cannot reveal whether local action preferences evolved consistently during teacher training. The same endpoint shift can result from early and late shifts that reinforce or partially cancel each other. We investigate whether agreement between early and late teacher policy shifts can guide selective transfer. We introduce Waypoint-guided On-Policy Distillation (Waypoint-OPD), which uses an intermediate teacher checkpoint from the same RL run to compare early and late shifts in relative action preferences at student-visited prefixes. We gate the endpoint shift according to the directional agreement between the early and late shifts, suppressing it when they point in opposing directions. Compared with endpoint-only transfer, our method requires only one additional frozen-teacher scoring pass. Using RL-trained Qwen3-4B teachers, we evaluate our method with Qwen3-8B, Qwen3-14B, and Qwen3-30B-A3B as student models in mathematical and code reasoning. In all six domain--student settings, Waypoint-OPD achieves the highest domain-average score among the compared distillation methods. Ablations with the Qwen3-8B student show that removing or shuffling gates reduces mathematical reasoning performance. These results support the use of directional agreement between early and late teacher policy shifts to weight the endpoint shift during student training.

Diffusion models are generative models that have recently demonstrated impressive performances in terms of sampling quality and density estimation in high dimensions. They rely on a forward continuous diffusion process and a backward continuous denoising process, which can be described by a time-dependent vector field and is used as a generative model. In the original formulation of the diffusion model, this vector field is assumed to be the score function (i.e. it is the gradient of the log-probability at a given time in the diffusion process). Curiously, on the practical side, most studies on diffusion models implement this vector field as a neural network function and do not constrain it be the gradient of some energy function (that is, most studies do not constrain the vector field to be conservative). Even though some studies investigated empirically whether such a constraint will lead to a performance gain, they lead to contradicting results and failed to provide analytical results. Here, we provide three analytical results regarding the extent of the modeling freedom of this vector field. {Firstly, we propose a novel decomposition of vector fields into a conservative component and an orthogonal component which satisfies a given (gauge) freedom. Secondly, from this orthogonal decomposition, we show that exact density estimation and exact sampling is achieved when the conservative component is exactly equals to the true score and therefore conservativity is neither necessary nor sufficient to obtain exact density estimation and exact sampling. Finally, we show that when it comes to inferring local information of the data manifold, constraining the vector field to be conservative is desirable.

Aligned language models gate behaviors such as refusal and language routing through sparse feed forward neurons, yet no theory predicts when a single neuron intervention controls a behavior coherently rather than collapsing the output. We develop a budget normalized control window framework for single neuron steering. A dose along one write direction reduces to one control coordinate: the alignment between the residual stream and the write, driven along a universal saturation curve in units of a coherence budget set by the residual norm divided by the write norm. Coherent control exists when a behavior trigger lies below the collapse ceiling. The same coordinate governs benign mode switches and refusal; the ceiling follows from weights and one generic forward pass, while triggers are measured at rollout. On fifteen held out neurons, the predicted ceiling has mean absolute error 0.14, about 0.07 in bulk layers, and the committed open or closed verdict holds on eleven against a ten of fifteen majority baseline. Closed cases expose three failure modes rather than violations: collapse before trigger, too little depth to propagate, or a normalization that caps how far one neuron can push. The law explains why local gradient attribution anti predicts control: true controllers write off the readout axis and carry a near zero first order gradient. A forward only contrastive screen made precise by the window recovers controllers that attribution misses. On refusal, the hardest case, intervention success is typed, not scalar: coherent bypass and strict actionable reach separate, so a neuron can flip refusal in fluent, on task text with no actionable content, and genuine actionable reach appears only for three of six audited Llama pivots and only at later rollout horizons. Single neuron steering is therefore a budgeted, typed audit of controllability rather than a fixed dose anecdote.

In reinforcement learning and imitation learning, an object of central importance is the state distribution induced by the policy. It plays a crucial role in the policy gradient theorem, and references to it--along with the related state-action distribution--can be found all across the literature. Despite its importance, the state distribution is mostly discussed indirectly and theoretically, rather than being modeled explicitly. The reason being an absence of appropriate density estimation tools. In this work, we investigate applications of a normalizing flow-based model for the aforementioned distributions. In particular, we use a pair of flows coupled through the optimality point of the Donsker-Varadhan representation of the Kullback-Leibler (KL) divergence, for distribution matching based imitation learning. Our algorithm, Coupled Flow Imitation Learning (CFIL), achieves state-of-the-art performance on benchmark tasks with a single expert trajectory and extends naturally to a variety of other settings, including the subsampled and state-only regimes.

As data propagates through a Transformer, the norm of its hidden states grows by orders of magnitude with depth, a phenomenon framed as 'curse of depth' and nearly universally treated as a pathology to be suppressed. We take the opposite view. Across 16 pre-trained LLMs from 9 families, spanning dense, mixture-of-experts and hybrid architectures and Pre-, Peri- and Post-Norm designs, we find that this growth reflects an emergent depth-positional encoding, carried by the only learned per-layer gain on the residual stream, the normalization weight γ: with depth, γ grows in magnitude and rotates in direction, jointly encoding the layer index. We make this depth-conditioned encoding explicit with LayerRoPE, an implicit analog of RoPE along the depth axis, which replaces all layerwise γ vectors with a single shared vector and depth-conditioned scalars, at a net reduction in parameters and <0.02% change in FLOPs. Across a model ladder scaled up to 100B+ tokens, LayerRoPE consistently outperforms Pre-, Post- and Peri-Norm and Layer-Norm Scaling, reaching Pre-Norm's 1.3B loss with 3.4times less compute; LayerRoPE is the only approach that shows strong convergence and improves near monotonically as depth scales to 512 layers. It improves learning-rate sensitivity by 3-10times, and transfers naively to and consistently improves looped latent models and Vision Transformers. Inspecting its learned schedule inverts the prevailing premise: LayerRoPE does not shrink the residual stream but widens it, damping what each block reads while amplifying what it writes. Depth stability, our results suggest, calls not for suppressing the residual stream, but for depth-conditioned regulation of the computational blocks it feeds.

The strong generalization performance of on-policy post-training paradigms has motivated studies of their parameter update behaviors. However, these studies treat the observed behaviors only as byproducts in on-policy training, overlooking their potential to serve as optimization principles for improving the generalization of other paradigms such as supervised fine-tuning (SFT). To address this limitation, we investigate whether there exists a specific on-policy update behavior that can achieve such improvements. First, our analyses reveal that SFT updates parameters along consistent directions, while the on-policy paradigm continuously adjusts the direction during training. This difference inspires us to focus on the cumulative update direction of each parameter as a promising behavior. Then, we evaluate its effectiveness for improving generalization by proposing On-Policy direction-constrained Supervised Fine-Tuning (OPSFT), which constrains SFT updates to the direction identified by on-policy paradigms. The strong performance of OPSFT indicates that the generalization advantage of on-policy paradigms can be transferred to SFT through the parameter update direction. Once such a direction is identified, even SFT can generalize with its updates constrained to this direction. This finding offers two practical benefits by combining the strong generalization of on-policy paradigms with the advantages of SFT, including the high training efficiency and ability to leverage high-quality trajectories. For efficiency, we identify update directions that support strong generalization using a few on-policy training steps, and subsequently apply OPSFT to achieve high training efficiency. For leveraging high-quality trajectories, OPSFT can utilize these trajectories to continue improving a post-trained model along its update direction without disrupting the ability learned from on-policy training.

Distillation enables compact Vision-Language Models (VLMs) to obtain strong reasoning capabilities, yet the prompts driving this process are typically chosen via simple heuristics or aggregated from off-the-shelf datasets. We reveal a critical inefficiency in this approach: up to 69% of the prompts in standard chart / document reasoning datasets are effectively zero-delta, meaning the teacher and student already induce the exact same answer distribution. Training on these prompts provides minimal learning signal, causing student improvement to rapidly saturate regardless of data scale. To escape the zero-delta trap, we return to first principles: distillation fundamentally minimizes distributional divergence, and thus a prompt is valuable only if it exposes a functional capability gap between the teacher and student. We quantify this gap through answer divergence (Δ), demonstrating that non-zero divergence is critical for effective scaling. Building on this insight, we propose a staged synthesis pipeline that repurposes existing datasets as seeds, actively targeting student failure modes to produce better prompts. The result is DeltaPrompts, a diverse dataset of 200k synthetic, high-divergence reasoning problems. We evaluate DeltaPrompts across three distinct settings: on-policy distillation with the target teacher-student pair, transfer to a novel model family without regenerating the data, and off-policy fine-tuning of a non-reasoning model. Across all scenarios, DeltaPrompts drives substantial gains, yielding up to 15% relative improvement even on top of a highly-optimized reasoning model (e.g., Qwen3-VL-8B-Thinking) -- averaged over 10 benchmarks spanning chart, document and perception-centric reasoning.

Adapting reasoning models to new tasks during post-training with only output-level supervision stalls under reinforcement learning from verifiable rewards (RLVR) when the initial success probability p_0 is small. Using the Tsallis q-logarithm, we define a loss family J_Q that interpolates between RLVR (at q{=}0, the exploitation pole) and the log-marginal-likelihood over latent trajectories (at q{=}1, the density-estimation pole). All members share the same per-example gradient direction, differing only by a scalar amplification P_{θ^{-q}} that reweights each instance independently of the learning rate. This amplification is the mechanism that addresses cold-start stalling: under gradient flow, the exploitation pole requires Ω(1{p_0}) time to escape cold start, while the density-estimation pole escapes in Θbig(log(1{p_0})big); intermediate q trades escape speed against noise memorization. Because P_θ is intractable, we derive two Monte Carlo estimators from the two factorizations of the gradient: Gradient-Amplified RL (GARL) samples from the prior and amplifies the RL gradient, and Posterior-Attenuated Fine-Tuning (PAFT) importance-resamples from the posterior and runs standard SFT. Both have bias Obig(q{M P_θ^{q+1}}big); GARL has lower variance, PAFT has semantically coherent gradients. On FinQA, HotPotQA, and MuSiQue, GARL at q{=}0.75 substantially mitigates cold-start stalling, escaping cold start where GRPO fails entirely. In warm start, GARL at low q dominates FinQA where training is stable; on HotPotQA and MuSiQue, GARL destabilizes during training, and PAFT at q{=}0.75 provides stable gradients (best overall on HotPotQA at 47.9 maj@16, +14.4 over GRPO).

Knowledge distillation is a model compression technique in which a compact "student" network is trained to replicate the predictive behavior of a larger "teacher" network. In logit-based knowledge distillation, it has become the de facto approach to augment cross-entropy with a distillation term. Typically, this term is either a KL divergence that matches marginal probabilities or a correlation-based loss that captures intra- and inter-class relationships. In every case, it acts as an additional term to cross-entropy. This term has its own weight, which must be carefully tuned. In this paper, we adopt a choice-theoretic perspective and recast knowledge distillation under the Plackett-Luce model by interpreting teacher logits as "worth" scores. We introduce "Plackett-Luce Distillation (PLD)", a weighted list-wise ranking loss. In PLD, the teacher model transfers knowledge of its full ranking of classes, weighting each ranked choice by its own confidence. PLD directly optimizes a single "teacher-optimal" ranking. The true label is placed first, followed by the remaining classes in descending teacher confidence. This process yields a convex and translation-invariant surrogate that subsumes weighted cross-entropy. Empirically, across CIFAR-100, ImageNet-1K, and MS-COCO, PLD achieves consistent gains across diverse architectures and distillation objectives, including divergence-based, correlation-based, and feature-based methods, in both homogeneous and heterogeneous teacher-student pairs.

This paper tackles optimization problems whose objective and constraints involve a trained Neural Network (NN), where the goal is to maximize f(Phi(x)) subject to c(Phi(x)) leq 0, with f smooth, c general and non-stringent, and Phi an already trained and possibly nonwhite-box NN. We address two challenges regarding this problem: identifying ascent directions for local search, and ensuring reliable convergence towards relevant local solutions. To this end, we re-purpose the notion of directional NN attacks as efficient optimization subroutines, since directional NN attacks use the neural structure of Phi to compute perturbations of x that steer Phi(x) in prescribed directions. Precisely, we develop an attack operator that computes attacks of Phi at any x along the direction nabla f(Phi(x)). Then, we propose a hybrid algorithm combining the attack operator with derivative-free optimization (DFO) techniques, designed for numerical reliability by remaining oblivious to the structure of the problem. We consider the cDSM algorithm, which offers asymptotic guarantees to converge to a local solution under mild assumptions on the problem. The resulting method alternates between attack-based steps for heuristic yet fast local intensification and cDSM steps for certified convergence and numerical reliability. Experiments on three problems show that this hybrid approach consistently outperforms standard DFO baselines.

One-step generative modeling has emerged as a leading approach for amortizing the inference cost of diffusion and flow-matching models. Among distillation-free methods, MeanFlow training is notoriously unstable, with non-decreasing loss and unbounded gradient variance. In this work, we establish a theory that attributes this pathology to a misuse of the conditional velocity field. We show that the conditional velocity plays two distinct statistical roles in the loss: both as an unbiased regression target and as a Monte Carlo control variate in a Jacobi-vector product, with the original MeanFlow loss assigning the wrong coefficient to the latter. We derive the optimal coefficient in closed form and show that a family of fixes in concurrent works corresponds to different practical realizations of the same optimum. A controlled sweep of this coefficient on two-dimensional benchmarks and on a latent Diffusion Transformer recovers the predicted bias-variance ordering. Our DiT experiment also reveals a quantitative FID-MSE landscape mismatch. Specifically, although the gradient-MSE is minimized at an interior coefficient value near β!=!0.94, the coefficient that minimizes FID prefers to use conditional velocity directly at the unbiased corner. Our analysis therefore explains why MeanFlow is unstable and unifies its concurrent remedies, and shows that the variance-optimal coefficient need not coincide with the quality-optimal one.

A growing literature in computational neuroscience leverages gradient descent and learning algorithms that approximate it to study synaptic plasticity in the brain. However, the vast majority of this work ignores a critical underlying assumption: the choice of distance for synaptic changes - i.e. the geometry of synaptic plasticity. Gradient descent assumes that the distance is Euclidean, but many other distances are possible, and there is no reason that biology necessarily uses Euclidean geometry. Here, using the theoretical tools provided by mirror descent, we show that the distribution of synaptic weights will depend on the geometry of synaptic plasticity. We use these results to show that experimentally-observed log-normal weight distributions found in several brain areas are not consistent with standard gradient descent (i.e. a Euclidean geometry), but rather with non-Euclidean distances. Finally, we show that it should be possible to experimentally test for different synaptic geometries by comparing synaptic weight distributions before and after learning. Overall, our work shows that the current paradigm in theoretical work on synaptic plasticity that assumes Euclidean synaptic geometry may be misguided and that it should be possible to experimentally determine the true geometry of synaptic plasticity in the brain.

Large language models often answer structurally unanswerable questions, such as computing cot(-540°) or evaluating (1).startswith("1"), instead of abstaining. We ask whether this failure reflects missing recognition or failed routing from recognition to abstention. Across instruction-tuned models from 1.7B to 70B parameters, a single linear direction in the hidden state separates answerable from structurally impossible math and code prompts, showing that models represent impossibility before generation. Yet this recognition direction is nearly orthogonal to the canonical safety-refusal direction that mediates trained harmful-content refusal. An in-domain behavior-defined invalidity-aware direction is closer to recognition, but only partially aligned with it, and remains near-orthogonal to safety refusal. Generation-time steering along the recognition direction changes invalidity-aware behavior bidirectionally and dose-responsively on structural math and code cells, while random directions do not. Base/instruct comparisons further show that the low-cosine geometry is already present at the pretraining endpoint. The confident-on-impossible failure is therefore better explained as a routing failure than as an encoding failure: the model has a usable "no admissible answer" signal, but the safety-refusal pathway is not aligned to use it.

Wasserstein Gradient Flows (WGF) with respect to specific functionals have been widely used in the machine learning literature. Recently, neural networks have been adopted to approximate certain intractable parts of the underlying Wasserstein gradient flow and result in efficient inference procedures. In this paper, we introduce the Neural Sinkhorn Gradient Flow (NSGF) model, which parametrizes the time-varying velocity field of the Wasserstein gradient flow w.r.t. the Sinkhorn divergence to the target distribution starting a given source distribution. We utilize the velocity field matching training scheme in NSGF, which only requires samples from the source and target distribution to compute an empirical velocity field approximation. Our theoretical analyses show that as the sample size increases to infinity, the mean-field limit of the empirical approximation converges to the true underlying velocity field. To further enhance model efficiency on high-dimensional tasks, a two-phase NSGF++ model is devised, which first follows the Sinkhorn flow to approach the image manifold quickly (le 5 NFEs) and then refines the samples along a simple straight flow. Numerical experiments with synthetic and real-world benchmark datasets support our theoretical results and demonstrate the effectiveness of the proposed methods.

Most single-channel separators estimate a source by applying a real gain to the mixture in each time-frequency bin. The optimum of that format, which the oracle masks used as bounds do not attain, is the orthogonal projection of the source onto the line spanned by the mixture, its residual set by the angle between them. Locating an estimator reduces to the block structure of a real-linear operator on stacked spectra, giving a chain of four nested classes whose three larger terms match three assumptions on the prior: zero means, circularity and absence of inter-frequency coupling. Held fixed the chain is a cascade of four orthogonal projections; refitted per frame it collapses onto its first term, attributing the whole residual to one missing real parameter per bin, the phase. When the phase posterior is symmetric about the mixture direction, the minimum mean-square estimate falls back onto the line, with gain the posterior mean of the oracle gain and excess error its variance. On MUSDB18 a posterior mean under a non-circular Gaussian-mixture prior leaves the class yet stays 11.44 dB under the per-frame ceiling, which four times as many components and 7.5x the data do not close; a closed-form gate attributes some 70% of it, in decibels, to the predicted variance. The widest fixed class stays 6.70 dB under the same ceiling. Leaving the class and minimising squared error are conflicting requests: the barrier lies in the criterion rather than in the prior.

A selector that allocates work across opaque components must evaluate them without being told how they performed. We study the least communication that suffices. Each parent maintains an allocation vector over its children and updates it from outcomes by proportional redistribution. Each child reads the sign of its own allocation change, one bit per round it is selected, so no evaluation message crosses the parent-child boundary. (1) The update preserves the simplex with strict positivity. (2) Conditional on being selected, a child's sign is exactly the root outcome at every depth, a sufficient statistic. The conditioning is necessary. (3) Under interiority a single selector has a unique interior equilibrium, exact and closed-form for N=2, with almost sure convergence under decreasing steps. For general N an equi-ratio condition gives an explicit affine equilibrium. (4) The linearised mean flow has real, strictly negative tangent eigenvalues with explicit bounds for all N >= 2. (5) The mean flow converges globally. An explicit potential, up to normalisation the KL divergence between the failure and slack distributions, decreases at a rate equal to the variance of the equi-ratio statistic. Without interiority the rule drops exactly the children below the surviving support's threshold. (6) The update law composes along a single active path, since ancestors select a node's active rounds but do not alter a realised signal. No joint convergence theorem for a hierarchy whose levels adapt simultaneously is claimed. (7) The one-bit interface admits exactly four deterministic memoryless distortions, and at a single selector passing the bit on unchanged delivers strictly higher mean quality upward than negation or constant transmission. Illustrations on synthetic hierarchies up to 16,384 leaves and three real datasets (3.4 billion comparisons, no mismatches) are consistent with the theory.

Prior work argues that refusal in large language models is mediated by a single activation-space direction, enabling effective steering and ablation. We show that this account is incomplete. Across eleven categories of refusal and non-compliance, including safety, incomplete or unsupported requests, anthropomorphization, and over-refusal, we find that these refusal behaviors correspond to geometrically distinct directions in activation space. Yet despite this diversity, linear steering along any refusal-related direction produces nearly identical refusal to over-refusal trade-offs, acting as a shared one-dimensional control knob. The primary effect of different directions is not whether the model refuses, but how it refuses.

Discrete diffusion models have emerged as a powerful generative modeling framework for discrete data with successful applications spanning from text generation to image synthesis. However, their deployment faces challenges due to the high dimensionality of the state space, necessitating the development of efficient inference algorithms. Current inference approaches mainly fall into two categories: exact simulation and approximate methods such as tau-leaping. While exact methods suffer from unpredictable inference time and redundant function evaluations, tau-leaping is limited by its first-order accuracy. In this work, we advance the latter category by tailoring the first extension of high-order numerical inference schemes to discrete diffusion models, enabling larger step sizes while reducing error. We rigorously analyze the proposed schemes and establish the second-order accuracy of the theta-trapezoidal method in KL divergence. Empirical evaluations on GPT-2 level text and ImageNet-level image generation tasks demonstrate that our method achieves superior sample quality compared to existing approaches under equivalent computational constraints.

Variational inference (VI) seeks to approximate a target distribution pi by an element of a tractable family of distributions. Of key interest in statistics and machine learning is Gaussian VI, which approximates pi by minimizing the Kullback-Leibler (KL) divergence to pi over the space of Gaussians. In this work, we develop the (Stochastic) Forward-Backward Gaussian Variational Inference (FB-GVI) algorithm to solve Gaussian VI. Our approach exploits the composite structure of the KL divergence, which can be written as the sum of a smooth term (the potential) and a non-smooth term (the entropy) over the Bures-Wasserstein (BW) space of Gaussians endowed with the Wasserstein distance. For our proposed algorithm, we obtain state-of-the-art convergence guarantees when pi is log-smooth and log-concave, as well as the first convergence guarantees to first-order stationary solutions when pi is only log-smooth.

Our world is full of asymmetries. Gravity and wind can make reaching a place easier than coming back. Social artifacts such as genealogy charts and citation graphs are inherently directed. In reinforcement learning and control, optimal goal-reaching strategies are rarely reversible (symmetrical). Distance functions supported on these asymmetrical structures are called quasimetrics. Despite their common appearance, little research has been done on the learning of quasimetrics. Our theoretical analysis reveals that a common class of learning algorithms, including unconstrained multilayer perceptrons (MLPs), provably fails to learn a quasimetric consistent with training data. In contrast, our proposed Poisson Quasimetric Embedding (PQE) is the first quasimetric learning formulation that both is learnable with gradient-based optimization and enjoys strong performance guarantees. Experiments on random graphs, social graphs, and offline Q-learning demonstrate its effectiveness over many common baselines.

Knowledge distillation aims to transfer the factual knowledge of large language models to smaller models for efficient deployment. Yet a teacher may recall a relation in one direction while failing to generate the answer in the reverse direction. Distillation from its generated answers can therefore propagate this directional limitation to the student. The same teacher can nevertheless recognize such an answer by scoring the relation in the direction it knows. We introduce directional label distillation, in which frozen teachers score candidate answers in that known direction and the best-scoring candidate becomes the student's training target. On facts about parents and their children, known-direction scoring yields more accurate labels than scoring the requested direction, even after tuned corrections for name priors. With prior-corrected scores, the better direction depends on the facts rather than the template, and reverses on mined facts whose notable entity is the parent rather than the child. With the evaluated children's forward facts withheld, students trained on known-direction labels improve open-ended accuracy on their trained queries by 13 to 15 points over students trained on prior-corrected reverse labels. After generated answers are matched to a fixed name list by lexical similarity, students reproduce nearly all selected labels. Their accuracy largely follows label quality. The label advantage holds on unscreened queries and when candidates are retrieved without inserting correct answers. Our findings show that directional verification mitigates the transfer of errors from teacher-generated answers to students by providing more accurate training targets. Code is available at https://github.com/js-lee-AI/directional-verification.

In nature, symmetry governs regularities, while symmetry breaking brings texture. In artificial neural networks, symmetry has been a central design principle to efficiently capture regularities in the world, but the role of symmetry breaking is not well understood. Here, we develop a theoretical framework to study the "geometry of learning dynamics" in neural networks, and reveal a key mechanism of explicit symmetry breaking behind the efficiency and stability of modern neural networks. To build this understanding, we model the discrete learning dynamics of gradient descent using a continuous-time Lagrangian formulation, in which the learning rule corresponds to the kinetic energy and the loss function corresponds to the potential energy. Then, we identify "kinetic symmetry breaking" (KSB), the condition when the kinetic energy explicitly breaks the symmetry of the potential function. We generalize Noether's theorem known in physics to take into account KSB and derive the resulting motion of the Noether charge: "Noether's Learning Dynamics" (NLD). Finally, we apply NLD to neural networks with normalization layers and reveal how KSB introduces a mechanism of "implicit adaptive optimization", establishing an analogy between learning dynamics induced by normalization layers and RMSProp. Overall, through the lens of Lagrangian mechanics, we have established a theoretical foundation to discover geometric design principles for the learning dynamics of neural networks.

Vision-language-action (VLA) models adapted through supervised fine-tuning (SFT) inherit a structural asymmetry: expert demonstrations teach the policy where success behavior lies, but provide no signal about where it ceases to be reliable. We argue that robust VLA adaptation should therefore be viewed not as further demonstration fitting, but as **Failure-Boundary Learning**---the problem of *Discovering*, *Localizing*, and *Shaping* the boundary between recoverable deviations and task failure. To instantiate this view, we propose **DLS**: built on a **real-grounded behavioral prior** from few real demonstrations and simulated co-training, DLS *discovers* failure boundaries at scale through on-policy digital twin rollouts. Rather than reducing each rollout to a binary label, **semantic progress localization** uses privileged simulator states to assign progress-aware signals that capture *where* the failure boundary is crossed, not merely *whether*. These signals drive **directional boundary shaping** in the flow dynamics---reinforcing success-producing denoising directions and suppressing failure-producing ones, without action likelihoods or auxiliary critics. Across real-robot manipulation tasks, DLS improves robustness over SFT and online RL baselines, especially under randomized initial states and unseen visual conditions.

Knowledge distillation (KD) has been widely adopted to compress large language models (LLMs). Existing KD methods investigate various divergence measures including the Kullback-Leibler (KL), reverse Kullback-Leibler (RKL), and Jensen-Shannon (JS) divergences. However, due to limitations inherent in their assumptions and definitions, these measures fail to deliver effective supervision when few distribution overlap exists between the teacher and the student. In this paper, we show that the aforementioned KL, RKL, and JS divergences respectively suffer from issues of mode-averaging, mode-collapsing, and mode-underestimation, which deteriorates logits-based KD for diverse NLP tasks. We propose the Sinkhorn Knowledge Distillation (SinKD) that exploits the Sinkhorn distance to ensure a nuanced and precise assessment of the disparity between teacher and student distributions. Besides, profit by properties of the Sinkhorn metric, we can get rid of sample-wise KD that restricts the perception of divergence in each teacher-student sample pair. Instead, we propose a batch-wise reformulation to capture geometric intricacies of distributions across samples in the high-dimensional space. Comprehensive evaluation on GLUE and SuperGLUE, in terms of comparability, validity, and generalizability, highlights our superiority over state-of-the-art methods on all kinds of LLMs with encoder-only, encoder-decoder, and decoder-only architectures.

The loss landscapes of neural networks contain minima and saddle points that may be connected in flat regions or appear in isolation. We identify and characterize a special structure in the loss landscape: channels along which the loss decreases extremely slowly, while the output weights of at least two neurons, a_i and a_j, diverge to pminfinity, and their input weight vectors, w_i and w_j, become equal to each other. At convergence, the two neurons implement a gated linear unit: a_iσ(w_i cdot x) + a_jσ(w_j cdot x) rightarrow σ(w cdot x) + (v cdot x) σ'(w cdot x). Geometrically, these channels to infinity are asymptotically parallel to symmetry-induced lines of critical points. Gradient flow solvers, and related optimization methods like SGD or ADAM, reach the channels with high probability in diverse regression settings, but without careful inspection they look like flat local minima with finite parameter values. Our characterization provides a comprehensive picture of these quasi-flat regions in terms of gradient dynamics, geometry, and functional interpretation. The emergence of gated linear units at the end of the channels highlights a surprising aspect of the computational capabilities of fully connected layers.

Steering aligns large language models (LLMs) by injecting a bias into selected activations at inference time, offering a far cheaper alternative to weight-update methods such as supervised fine-tuning or reinforcement learning. However, most existing training-free steering methods are input-independent: a single direction is fitted once and shared across all inputs. This is fundamentally limiting as different inputs occupy different regions of the activation space and admit different optimal steering directions toward the same target concept, much as the gradient with respect to a fixed loss varies from input to input. We close this gap with IDEEA (Input-Dependent stEEring via Activation cluster matching), a training-free framework for input-dependent steering. IDEEA clusters the positive and negative activation supports per attention head, and solves an optimal-matching problem to construct a set of cluster-conditional directions, all about the target concept. At inference time, it picks from this pool of directions and uses the one that best matches the input's own activation for steering. IDEEA aligns the model toward the target concept while preserving the input's original representation, evidence that activations encoding a concept occupy several distinct sub-regions of the representation space rather than a single one. IDEEA improves the truth times info rate in TruthfulQA by an average of 9.9% (up to 23.5%) over the best input-independent baseline.

This paper extends the classic theory of convex optimization to the minimization of functions that are equal to the negated logarithm of what we term as a sum-log-concave function, i.e., a sum of log-concave functions. In particular, we show that such functions are in general not convex but still satisfy generalized convexity inequalities. These inequalities unveil the key importance of a certain vector that we call the cross-gradient and that is, in general, distinct from the usual gradient. Thus, we propose the Cross Gradient Descent (XGD) algorithm moving in the opposite direction of the cross-gradient and derive a convergence analysis. As an application of our sum-log-concave framework, we introduce the so-called checkered regression method relying on a sum-log-concave function. This classifier extends (multiclass) logistic regression to non-linearly separable problems since it is capable of tessellating the feature space by using any given number of hyperplanes, creating a checkerboard-like pattern of decision regions.

Despite the fact that multilingual agreement (MA) has shown its importance for multilingual neural machine translation (MNMT), current methodologies in the field have two shortages: (i) require parallel data between multiple language pairs, which is not always realistic and (ii) optimize the agreement in an ambiguous direction, which hampers the translation performance. We present Bidirectional Multilingual Agreement via Switched Back-translation (BMA-SBT), a novel and universal multilingual agreement framework for fine-tuning pre-trained MNMT models, which (i) exempts the need for aforementioned parallel data by using a novel method called switched BT that creates synthetic text written in another source language using the translation target and (ii) optimizes the agreement bidirectionally with the Kullback-Leibler Divergence loss. Experiments indicate that BMA-SBT clearly improves the strong baselines on the task of MNMT with three benchmarks: TED Talks, News, and Europarl. In-depth analyzes indicate that BMA-SBT brings additive improvements to the conventional BT method.