Daily Papers

Reinforcement Learning with Verifiable Rewards (RLVR) offers a promising framework for optimizing large language models in reasoning tasks. However, existing RLVR algorithms focus on different granularities, and each has complementary strengths and limitations. Group Relative Policy Optimization (GRPO) updates the policy with token-level importance ratios, which preserves fine-grained credit assignment but often suffers from high variance and instability. In contrast, Group Sequence Policy Optimization (GSPO) applies single sequence-level importance ratios across all tokens in a response that better matches sequence-level rewards, but sacrifices token-wise credit assignment. In this paper, we propose Dynamic Hybrid Policy Optimization (DHPO) to bridge GRPO and GSPO within a single clipped surrogate objective. DHPO combines token-level and sequence-level importance ratios using weighting mechanisms. We explore two variants of the mixing mechanism, including an averaged mixing and an entropy-guided mixing. To further stabilize training, we employ a branch-specific clipping strategy that constrains token-level and sequence-level ratios within separate trust regions before mixing, preventing outliers in either branch from dominating the update. Across seven challenging mathematical reasoning benchmarks, experiments on both dense and MoE models from the Qwen3 series show that DHPO consistently outperforms GRPO and GSPO. We will release our code upon acceptance of this paper.

Masked Diffusion Models (MDMs) offer flexible, non-autoregressive generation, but this freedom introduces a challenge: final output quality is highly sensitive to the decoding order. We are the first to formalize this issue, attributing the variability in output quality to the cumulative predictive uncertainty along a generative path. To quantify this uncertainty, we introduce Denoising Entropy, a computable metric that serves as an internal signal for evaluating generative process. Leveraging this metric, we propose two algorithms designed to optimize the decoding path: a post-hoc selection method and a real-time guidance strategy. Experiments demonstrate that our entropy-guided methods significantly improve generation quality, consistently boosting accuracy on challenging reasoning, planning, and code benchmarks. Our work establishes Denoising Entropy as a principled tool for understanding and controlling generation, effectively turning the uncertainty in MDMs from a liability into a key advantage for discovering high-quality solutions.

This work presents mixed variational flows (MixFlows), a new variational family that consists of a mixture of repeated applications of a map to an initial reference distribution. First, we provide efficient algorithms for i.i.d. sampling, density evaluation, and unbiased ELBO estimation. We then show that MixFlows have MCMC-like convergence guarantees when the flow map is ergodic and measure-preserving, and provide bounds on the accumulation of error for practical implementations where the flow map is approximated. Finally, we develop an implementation of MixFlows based on uncorrected discretized Hamiltonian dynamics combined with deterministic momentum refreshment. Simulated and real data experiments show that MixFlows can provide more reliable posterior approximations than several black-box normalizing flows, as well as samples of comparable quality to those obtained from state-of-the-art MCMC methods.

Language model performance depends on identifying the optimal mixture of data groups to train on (e.g., law, code, math). Prior work has proposed a diverse set of methods to efficiently learn mixture proportions, ranging from fitting regression models over training runs to dynamically updating proportions throughout training. Surprisingly, we find that no existing method consistently outperforms a simple stratified sampling baseline in terms of average test perplexity. To understand this inconsistency, we unify existing methods into a standard framework, showing they are equivalent to solving a common optimization problem: minimize average loss subject to a method-specific mixing law -- an implicit assumption on the relationship between loss and mixture proportions. This framework suggests that measuring the fidelity of a method's mixing law can offer insights into its performance. Empirically, we find that existing methods set their mixing law parameters inaccurately, resulting in the inconsistent mixing performance we observe. Using this insight, we derive a new online method named Aioli, which directly estimates the mixing law parameters throughout training and uses them to dynamically adjust proportions. Aioli outperforms stratified sampling on 6 out of 6 datasets by an average of 0.27 test perplexity points, whereas existing methods fail to consistently beat stratified sampling, doing up to 6.9 points worse. Moreover, in a practical setting where proportions are learned on shorter runs due to computational constraints, Aioli can dynamically adjust these proportions over the full training run, consistently improving performance over existing methods by up to 12.012 test perplexity points.

We present an open-source Python library for simulating two-dimensional incompressible Kelvin-Helmholtz instabilities in stratified shear flows. The solver employs a fractional-step projection method with spectral Poisson solution via Fast Sine Transform, achieving second-order spatial accuracy. Implementation leverages NumPy, SciPy, and Numba JIT compilation for efficient computation. Four canonical test cases explore Reynolds numbers 1000--5000 and Richardson numbers 0.1--0.3: classical shear layer, double shear configuration, rotating flow, and forced turbulence. Statistical analysis using Shannon entropy and complexity indices reveals that double shear layers achieve 2.8times higher mixing rates than forced turbulence despite lower Reynolds numbers. The solver runs efficiently on standard desktop hardware, with 384times192 grid simulations completing in approximately 31 minutes. Results demonstrate that mixing efficiency depends on instability generation pathways rather than intensity measures alone, challenging Richardson number-based parameterizations and suggesting refinements for subgrid-scale representation in climate models.

While Reinforcement Learning with Verifiable Rewards (RLVR) can enhance LLM reasoning, its training process poses a critical risk: entropy collapse. This phenomenon is a rapid loss of policy diversity, stemming from the exploration-exploitation imbalance and leading to a lack of generalization. Recent entropy-intervention methods aim to prevent entropy collapse, yet their underlying mechanisms remain unclear. In this paper, we conduct a quantitative analysis to reveal token-level entropy changes and how existing entropy intervention methods help avoid entropy collapse. Our findings point out a fundamental limitation of existing methods: they attempt to control entropy dynamics indirectly. By only affecting related factors, such as the advantage signal and generation probability, their effectiveness is inherently limited and could potentially fail. To address this limitation, we introduce an entropy-change-aware reweighting scheme, namely Stabilizing Token-level Entropy-changE via Reweighting (STEER), that adaptively stabilizes entropy dynamics through fine-grained token-level adjustments. Our approach mitigates over-exploitation while fostering robust exploration. Extensive experiments demonstrate that STEER significantly mitigates entropy collapse, stabilizes entropy dynamics, and achieves stronger downstream performance across various mathematical reasoning benchmarks \footnote{Our code is available at https://github.com/zz-haooo/STEER.

The pervasiveness of proprietary language models has raised critical privacy concerns, necessitating advancements in private inference (PI), where computations are performed directly on encrypted data without revealing users' sensitive information. While PI offers a promising solution, its practical deployment is hindered by substantial communication and latency overheads, primarily stemming from nonlinear operations. To address this, we introduce an information-theoretic framework to characterize the role of nonlinearities in decoder-only language models, laying a principled foundation for optimizing transformer-architectures tailored to the demands of PI. By leveraging Shannon's entropy as a quantitative measure, we uncover the previously unexplored dual significance of nonlinearities: beyond ensuring training stability, they are crucial for maintaining attention head diversity. Specifically, we find that their removal triggers two critical failure modes: {\em entropy collapse} in deeper layers that destabilizes training, and {\em entropic overload} in earlier layers that leads to under-utilization of Multi-Head Attention's (MHA) representational capacity. We propose an entropy-guided attention mechanism paired with a novel entropy regularization technique to mitigate entropic overload. Additionally, we explore PI-friendly alternatives to layer normalization for preventing entropy collapse and stabilizing the training of LLMs with reduced-nonlinearities. Our study bridges the gap between information theory and architectural design, establishing entropy dynamics as a principled guide for developing efficient PI architectures. The code and implementation are available at https://github.com/Nandan91/entropy-guided-attention-llm{entropy-guided-llm}.

As foundational models reshape scientific discovery, a bottleneck persists in dynamical system reconstruction (DSR): the ability to learn across system hierarchies. Many meta-learning approaches have been applied successfully to single systems, but falter when confronted with sparse, loosely related datasets requiring multiple hierarchies to be learned. Mixture of Experts (MoE) offers a natural paradigm to address these challenges. Despite their potential, we demonstrate that naive MoEs are inadequate for the nuanced demands of hierarchical DSR, largely due to their gradient descent-based gating update mechanism which leads to slow updates and conflicted routing during training. To overcome this limitation, we introduce MixER: Mixture of Expert Reconstructors, a novel sparse top-1 MoE layer employing a custom gating update algorithm based on K-means and least squares. Extensive experiments validate MixER's capabilities, demonstrating efficient training and scalability to systems of up to ten parametric ordinary differential equations. However, our layer underperforms state-of-the-art meta-learners in high-data regimes, particularly when each expert is constrained to process only a fraction of a dataset composed of highly related data points. Further analysis with synthetic and neuroscientific time series suggests that the quality of the contextual representations generated by MixER is closely linked to the presence of hierarchical structure in the data.

User data confidentiality protection is becoming a rising challenge in the present deep learning research. Without access to data, conventional data-driven model compression faces a higher risk of performance degradation. Recently, some works propose to generate images from a specific pretrained model to serve as training data. However, the inversion process only utilizes biased feature statistics stored in one model and is from low-dimension to high-dimension. As a consequence, it inevitably encounters the difficulties of generalizability and inexact inversion, which leads to unsatisfactory performance. To address these problems, we propose MixMix based on two simple yet effective techniques: (1) Feature Mixing: utilizes various models to construct a universal feature space for generalized inversion; (2) Data Mixing: mixes the synthesized images and labels to generate exact label information. We prove the effectiveness of MixMix from both theoretical and empirical perspectives. Extensive experiments show that MixMix outperforms existing methods on the mainstream compression tasks, including quantization, knowledge distillation, and pruning. Specifically, MixMix achieves up to 4% and 20% accuracy uplift on quantization and pruning, respectively, compared to existing data-free compression work.

Normalizing flows are a promising tool for modeling probability distributions in physical systems. While state-of-the-art flows accurately approximate distributions and energies, applications in physics additionally require smooth energies to compute forces and higher-order derivatives. Furthermore, such densities are often defined on non-trivial topologies. A recent example are Boltzmann Generators for generating 3D-structures of peptides and small proteins. These generative models leverage the space of internal coordinates (dihedrals, angles, and bonds), which is a product of hypertori and compact intervals. In this work, we introduce a class of smooth mixture transformations working on both compact intervals and hypertori. Mixture transformations employ root-finding methods to invert them in practice, which has so far prevented bi-directional flow training. To this end, we show that parameter gradients and forces of such inverses can be computed from forward evaluations via the inverse function theorem. We demonstrate two advantages of such smooth flows: they allow training by force matching to simulation data and can be used as potentials in molecular dynamics simulations.

Pretraining data of large language models composes multiple domains (e.g., web texts, academic papers, codes), whose mixture proportions crucially impact the competence of outcome models. While existing endeavors rely on heuristics or qualitative strategies to tune the proportions, we discover the quantitative predictability of model performance regarding the mixture proportions in function forms, which we refer to as the data mixing laws. Fitting such functions on sample mixtures unveils model performance on unseen mixtures before actual runs, thus guiding the selection of an ideal data mixture. Furthermore, we propose nested use of the scaling laws of training steps, model sizes, and our data mixing law to enable predicting the performance of large models trained on massive data under various mixtures with only small-scale training. Moreover, experimental results verify that our method effectively optimizes the training mixture of a 1B model trained for 100B tokens in RedPajama, reaching a performance comparable to the one trained for 48% more steps on the default mixture. Extending the application of data mixing laws to continual training accurately predicts the critical mixture proportion that avoids catastrophic forgetting and outlooks the potential for dynamic data schedules

Long-term training of large language models (LLMs) requires maintaining stable exploration to prevent the model from collapsing into sub-optimal behaviors. Entropy is crucial in this context, as it controls exploration and helps avoid premature convergence to sub-optimal solutions. However, existing reinforcement learning methods struggle to maintain an appropriate level of entropy, as the training process involves a mix of positive and negative samples, each affecting entropy in different ways across steps. To address this, we propose Entropy stablilization via Proportional-Integral Control (EntroPIC), a novel method that adaptively adjusts the influence of positive and negative samples by dynamically tuning their loss coefficients. This approach stabilizes entropy throughout training, ensuring efficient exploration and steady progress. We provide a comprehensive theoretical analysis for both on-policy and off-policy learning settings, demonstrating that EntroPIC is effective at controlling entropy in large-scale LLM training. Experimental results show that our method successfully maintains desired entropy levels, enabling stable and optimal RL training for LLMs.

The sampling of probability distributions specified up to a normalization constant is an important problem in both machine learning and statistical mechanics. While classical stochastic sampling methods such as Markov Chain Monte Carlo (MCMC) or Langevin Dynamics (LD) can suffer from slow mixing times there is a growing interest in using normalizing flows in order to learn the transformation of a simple prior distribution to the given target distribution. Here we propose a generalized and combined approach to sample target densities: Stochastic Normalizing Flows (SNF) -- an arbitrary sequence of deterministic invertible functions and stochastic sampling blocks. We show that stochasticity overcomes expressivity limitations of normalizing flows resulting from the invertibility constraint, whereas trainable transformations between sampling steps improve efficiency of pure MCMC/LD along the flow. By invoking ideas from non-equilibrium statistical mechanics we derive an efficient training procedure by which both the sampler's and the flow's parameters can be optimized end-to-end, and by which we can compute exact importance weights without having to marginalize out the randomness of the stochastic blocks. We illustrate the representational power, sampling efficiency and asymptotic correctness of SNFs on several benchmarks including applications to sampling molecular systems in equilibrium.

Data mixing -- determining the ratios of data from different domains -- is a first-order concern for training language models (LMs). While existing mixing methods show promise, they fall short when applied during real-world LM development. We present Olmix, a framework that addresses two such challenges. First, the configuration space for developing a mixing method is not well understood -- design choices across existing methods lack justification or consensus and overlook practical issues like data constraints. We conduct a comprehensive empirical study of this space, identifying which design choices lead to a strong mixing method. Second, in practice, the domain set evolves throughout LM development as datasets are added, removed, partitioned, and revised -- a problem setting largely unaddressed by existing works, which assume fixed domains. We study how to efficiently recompute the mixture after the domain set is updated, leveraging information from past mixtures. We introduce mixture reuse, a mechanism that reuses existing ratios and recomputes ratios only for domains affected by the update. Over a sequence of five domain-set updates mirroring real-world LM development, mixture reuse matches the performance of fully recomputing the mix after each update with 74% less compute and improves over training without mixing by 11.6% on downstream tasks.

Large language models exhibit exceptional generalization capabilities, primarily attributed to the utilization of diversely sourced data. However, conventional practices in integrating this diverse data heavily rely on heuristic schemes, lacking theoretical guidance. This research tackles these limitations by investigating strategies based on low-cost proxies for data mixtures, with the aim of streamlining data curation to enhance training efficiency. Specifically, we propose a unified scaling law, termed BiMix, which accurately models the bivariate scaling behaviors of both data quantity and mixing proportions. We conduct systematic experiments and provide empirical evidence for the predictive power and fundamental principles of BiMix. Notably, our findings reveal that entropy-driven training-free data mixtures can achieve comparable or even better performance than more resource-intensive methods. We hope that our quantitative insights can shed light on further judicious research and development in cost-effective language modeling.

Knowledge distillation (KD) is a general neural network training approach that uses a teacher model to guide the student model. Existing works mainly study KD from the network output side (e.g., trying to design a better KD loss function), while few have attempted to understand it from the input side. Especially, its interplay with data augmentation (DA) has not been well understood. In this paper, we ask: Why do some DA schemes (e.g., CutMix) inherently perform much better than others in KD? What makes a "good" DA in KD? Our investigation from a statistical perspective suggests that a good DA scheme should reduce the covariance of the teacher-student cross-entropy. A practical metric, the stddev of teacher's mean probability (T. stddev), is further presented and well justified empirically. Besides the theoretical understanding, we also introduce a new entropy-based data-mixing DA scheme, CutMixPick, to further enhance CutMix. Extensive empirical studies support our claims and demonstrate how we can harvest considerable performance gains simply by using a better DA scheme in knowledge distillation.

We establish a fundamental connection between score-based diffusion models and non-equilibrium thermodynamics by deriving performance limits based on entropy rates. Our main theoretical contribution is a lower bound on the negative log-likelihood of the data that relates model performance to entropy rates of diffusion processes. We numerically validate this bound on a synthetic dataset and investigate its tightness. By building a bridge to entropy rates - system, intrinsic, and exchange entropy - we provide new insights into the thermodynamic operation of these models, drawing parallels to Maxwell's demon and implications for thermodynamic computing hardware. Our framework connects generative modeling performance to fundamental physical principles through stochastic thermodynamics.

Temporal causal representation learning methods assume that causal mechanisms switch instantaneously between discrete domains, yet real-world systems often exhibit continuous mechanism transitions. For example, a vehicle's dynamics evolve gradually through a turning maneuver, and human gait shifts smoothly from walking to running. We formalize this setting by modeling transitional mechanisms as convex combinations of finitely many atomic mechanisms, governed by time-varying mixing coefficients. Our theoretical contributions establish that both the latent causal variables and the continuous mixing trajectory are jointly identifiable. We further propose TRACE, a Mixture-of-Experts framework where each expert learns one atomic mechanism during training, enabling recovery of mechanism trajectories at test time. This formulation generalizes to intermediate mechanism states never observed during training. Experiments on synthetic and real-world data demonstrate that TRACE recovers mixing trajectories with up to 0.99 correlation, substantially outperforming discrete-switching baselines.

A primary challenge in using neural networks to approximate nonlinear dynamical systems governed by partial differential equations (PDEs) is transforming these systems into a suitable format, especially when dealing with non-linearizable dynamics or the need for infinite-dimensional spaces for linearization. This paper introduces DyMixOp, a novel neural operator framework for PDEs that integrates insights from complex dynamical systems to address this challenge. Grounded in inertial manifold theory, DyMixOp transforms infinite-dimensional nonlinear PDE dynamics into a finite-dimensional latent space, establishing a structured foundation that maintains essential nonlinear interactions and enhances physical interpretability. A key innovation is the Local-Global-Mixing (LGM) transformation, inspired by convection dynamics in turbulence. This transformation effectively captures both fine-scale details and nonlinear interactions, while mitigating spectral bias commonly found in existing neural operators. The framework is further strengthened by a dynamics-informed architecture that connects multiple LGM layers to approximate linear and nonlinear dynamics, reflecting the temporal evolution of dynamical systems. Experimental results across diverse PDE benchmarks demonstrate that DyMixOp achieves state-of-the-art performance, significantly reducing prediction errors, particularly in convection-dominated scenarios reaching up to 86.7\%, while maintaining computational efficiency and scalability.

Data mixing augmentation have proved to be effective in improving the generalization ability of deep neural networks. While early methods mix samples by hand-crafted policies (e.g., linear interpolation), recent methods utilize saliency information to match the mixed samples and labels via complex offline optimization. However, there arises a trade-off between precise mixing policies and optimization complexity. To address this challenge, we propose a novel automatic mixup (AutoMix) framework, where the mixup policy is parameterized and serves the ultimate classification goal directly. Specifically, AutoMix reformulates the mixup classification into two sub-tasks (i.e., mixed sample generation and mixup classification) with corresponding sub-networks and solves them in a bi-level optimization framework. For the generation, a learnable lightweight mixup generator, Mix Block, is designed to generate mixed samples by modeling patch-wise relationships under the direct supervision of the corresponding mixed labels. To prevent the degradation and instability of bi-level optimization, we further introduce a momentum pipeline to train AutoMix in an end-to-end manner. Extensive experiments on nine image benchmarks prove the superiority of AutoMix compared with state-of-the-art in various classification scenarios and downstream tasks.

Autoregressive language models are trained by minimizing the cross-entropy of the model distribution Q relative to the data distribution P -- that is, minimizing the forward cross-entropy, which is equivalent to maximum likelihood estimation (MLE). We have observed that models trained in this way may "over-generalize", in the sense that they produce non-human-like text. Moreover, we believe that reverse cross-entropy, i.e., the cross-entropy of P relative to Q, is a better reflection of how a human would evaluate text generated by a model. Hence, we propose learning with MixCE, an objective that mixes the forward and reverse cross-entropies. We evaluate models trained with this objective on synthetic data settings (where P is known) and real data, and show that the resulting models yield better generated text without complex decoding strategies. Our code and models are publicly available at https://github.com/bloomberg/mixce-acl2023

The data used to pretrain large language models has a decisive impact on a model's downstream performance, which has led to a large body of work on data selection methods that aim to automatically determine the most suitable data to use for pretraining. Existing data selection methods suffer from slow and computationally expensive processes, a problem amplified by the increasing size of models and of pretraining datasets. Data mixing, on the other hand, reduces the complexity of data selection by grouping data points together and determining sampling probabilities across entire groups. However, data mixing proportions are typically fixed before training and therefore cannot adapt to changing training dynamics. To address these limitations, we develop an efficient algorithm for Online Data Mixing (ODM) that combines elements from both data selection and data mixing. Based on multi-armed bandit algorithms, our online approach optimizes the data mixing proportions during training. Remarkably, our method trains a model that reaches the final perplexity of the next best method with 19\% fewer training iterations, and improves performance on the 5-shot MMLU benchmark by 1.9% relative accuracy, while adding negligible wall-clock time during pretraining.

Reinforcement learning with verifiable rewards (RLVR) has demonstrated superior performance in enhancing the reasoning capability of large language models (LLMs). However, this accuracy-oriented learning paradigm often suffers from entropy collapse, which reduces policy exploration and limits reasoning capabilities. To address this challenge, we propose an efficient reinforcement learning framework that leverages entropy signals at both the semantic and token levels to improve reasoning. From the data perspective, we introduce semantic entropy-guided curriculum learning, organizing training data from low to high semantic entropy to guide progressive optimization from easier to more challenging tasks. For the algorithmic design, we adopt non-uniform token treatment by imposing KL regularization on low-entropy tokens that critically impact policy exploration and applying stronger constraints on high-covariance portions within these tokens. By jointly optimizing data organization and algorithmic design, our method effectively mitigates entropy collapse and enhances LLM reasoning. Experimental results across 6 benchmarks with 3 different parameter-scale base models demonstrate that our method outperforms other entropy-based approaches in improving reasoning.

Training LLM agents in multi-turn environments with sparse rewards, where completing a single task requires 30+ turns of interaction within an episode, presents a fundamental challenge for reinforcement learning. We identify a critical failure mode unique to this setting: the exploration-exploitation cascade failure. This cascade begins with early-stage policy premature convergence, where sparse feedback causes agents to commit to flawed, low-entropy strategies. Subsequently, agents enter late-stage policy collapse, where conventional entropy regularization becomes counterproductive, promoting chaotic exploration that destabilizes training. We propose Entropy-regularized Policy Optimization (EPO), a general framework that breaks this failure cycle through three synergistic mechanisms: (1) adopting entropy regularization in multi-turn settings to enhance exploration, (2) an entropy smoothing regularizer that bounds policy entropy within historical averages to prevent abrupt fluctuations, and (3) adaptive phase-based weighting that balances exploration and exploitation across training. Our analysis justifies that EPO guarantees monotonically decreasing entropy variance while maintaining convergence. EPO achieves up to 152% performance improvement on ScienceWorld and up to 19.8% on ALFWorld. Our work demonstrates that multi-turn sparse-reward settings require fundamentally different entropy control than traditional RL, with broad implications for LLM agent training.

We derive a minimalist but powerful deterministic denoising-diffusion model. While denoising diffusion has shown great success in many domains, its underlying theory remains largely inaccessible to non-expert users. Indeed, an understanding of graduate-level concepts such as Langevin dynamics or score matching appears to be required to grasp how it works. We propose an alternative approach that requires no more than undergrad calculus and probability. We consider two densities and observe what happens when random samples from these densities are blended (linearly interpolated). We show that iteratively blending and deblending samples produces random paths between the two densities that converge toward a deterministic mapping. This mapping can be evaluated with a neural network trained to deblend samples. We obtain a model that behaves like deterministic denoising diffusion: it iteratively maps samples from one density (e.g., Gaussian noise) to another (e.g., cat images). However, compared to the state-of-the-art alternative, our model is simpler to derive, simpler to implement, more numerically stable, achieves higher quality results in our experiments, and has interesting connections to computer graphics.

In long-horizon tasks, recent agents based on Large Language Models (LLMs) face a significant challenge that sparse, outcome-based rewards make it difficult to assign credit to intermediate steps. Previous methods mainly focus on creating dense reward signals to guide learning, either through traditional reinforcement learning techniques like inverse reinforcement learning or by using Process Reward Models for step-by-step feedback. In this paper, we identify a fundamental problem in the learning dynamics of LLMs: the magnitude of policy gradients is inherently coupled with the entropy, which leads to inefficient small updates for confident correct actions and potentially destabilizes large updates for uncertain ones. To resolve this, we propose Entropy-Modulated Policy Gradients (EMPG), a framework that re-calibrates the learning signal based on step-wise uncertainty and the final task outcome. EMPG amplifies updates for confident correct actions, penalizes confident errors, and attenuates updates from uncertain steps to stabilize exploration. We further introduce a bonus term for future clarity that encourages agents to find more predictable solution paths. Through comprehensive experiments on three challenging agent tasks, WebShop, ALFWorld, and Deep Search, we demonstrate that EMPG achieves substantial performance gains and significantly outperforms strong policy gradient baselines. Project page is at https://empgseed-seed.github.io/

Recent reinforcement learning has enhanced the flow matching models on human preference alignment. While stochastic sampling enables the exploration of denoising directions, existing methods which optimize over multiple denoising steps suffer from sparse and ambiguous reward signals. We observe that the high entropy steps enable more efficient and effective exploration while the low entropy steps result in undistinguished roll-outs. To this end, we propose E-GRPO, an entropy aware Group Relative Policy Optimization to increase the entropy of SDE sampling steps. Since the integration of stochastic differential equations suffer from ambiguous reward signals due to stochasticity from multiple steps, we specifically merge consecutive low entropy steps to formulate one high entropy step for SDE sampling, while applying ODE sampling on other steps. Building upon this, we introduce multi-step group normalized advantage, which computes group-relative advantages within samples sharing the same consolidated SDE denoising step. Experimental results on different reward settings have demonstrated the effectiveness of our methods.

Guided sampling is a vital approach for applying diffusion models in real-world tasks that embeds human-defined guidance during the sampling procedure. This paper considers a general setting where the guidance is defined by an (unnormalized) energy function. The main challenge for this setting is that the intermediate guidance during the diffusion sampling procedure, which is jointly defined by the sampling distribution and the energy function, is unknown and is hard to estimate. To address this challenge, we propose an exact formulation of the intermediate guidance as well as a novel training objective named contrastive energy prediction (CEP) to learn the exact guidance. Our method is guaranteed to converge to the exact guidance under unlimited model capacity and data samples, while previous methods can not. We demonstrate the effectiveness of our method by applying it to offline reinforcement learning (RL). Extensive experiments on D4RL benchmarks demonstrate that our method outperforms existing state-of-the-art algorithms. We also provide some examples of applying CEP for image synthesis to demonstrate the scalability of CEP on high-dimensional data.

Energy-based models (EBMs) are known in the Machine Learning community for decades. Since the seminal works devoted to EBMs dating back to the noughties, there have been a lot of efficient methods which solve the generative modelling problem by means of energy potentials (unnormalized likelihood functions). In contrast, the realm of Optimal Transport (OT) and, in particular, neural OT solvers is much less explored and limited by few recent works (excluding WGAN-based approaches which utilize OT as a loss function and do not model OT maps themselves). In our work, we bridge the gap between EBMs and Entropy-regularized OT. We present a novel methodology which allows utilizing the recent developments and technical improvements of the former in order to enrich the latter. From the theoretical perspective, we prove generalization bounds for our technique. In practice, we validate its applicability in toy 2D and image domains. To showcase the scalability, we empower our method with a pre-trained StyleGAN and apply it to high-res AFHQ 512times 512 unpaired I2I translation. For simplicity, we choose simple short- and long-run EBMs as a backbone of our Energy-guided Entropic OT approach, leaving the application of more sophisticated EBMs for future research. Our code is available at: https://github.com/PetrMokrov/Energy-guided-Entropic-OT

Large Language Models (LLMs) and agent-based systems often struggle with compositional generalization due to a data bottleneck in which complex skill combinations follow a long-tailed, power-law distribution, limiting both instruction-following performance and generalization in agent-centric tasks. To address this challenge, we propose STEPS, a Skill Taxonomy guided Entropy-based Post-training data Synthesis framework for generating compositionally challenging data. STEPS explicitly targets compositional generalization by uncovering latent relationships among skills and organizing them into an interpretable, hierarchical skill taxonomy using structural information theory. Building on this taxonomy, we formulate data synthesis as a constrained information maximization problem, selecting skill combinations that maximize marginal structural information within the hierarchy while preserving semantic coherence. Experiments on challenging instruction-following benchmarks show that STEPS outperforms existing data synthesis baselines, while also yielding improved compositional generalization in downstream agent-based evaluations.

Training-free guided generation is a widely used and powerful technique that allows the end user to exert further control over the generative process of flow/diffusion models. Generally speaking, two families of techniques have emerged for solving this problem for gradient-based guidance: namely, posterior guidance (i.e., guidance via projecting the current sample to the target distribution via the target prediction model) and end-to-end guidance (i.e., guidance by performing backpropagation throughout the entire ODE solve). In this work, we show that these two seemingly separate families can actually be unified by looking at posterior guidance as a greedy strategy of end-to-end guidance. We explore the theoretical connections between these two families and provide an in-depth theoretical of these two techniques relative to the continuous ideal gradients. Motivated by this analysis we then show a method for interpolating between these two families enabling a trade-off between compute and accuracy of the guidance gradients. We then validate this work on several inverse image problems and property-guided molecular generation.

State-of-the-art language generation models can degenerate when applied to open-ended generation problems such as text completion, story generation, or dialog modeling. This degeneration usually shows up in the form of incoherence, lack of vocabulary diversity, and self-repetition or copying from the context. In this paper, we postulate that ``human-like'' generations usually lie in a narrow and nearly flat entropy band, and violation of these entropy bounds correlates with degenerate behavior. Our experiments show that this stable narrow entropy zone exists across models, tasks, and domains and confirm the hypothesis that violations of this zone correlate with degeneration. We then use this insight to propose an entropy-aware decoding algorithm that respects these entropy bounds resulting in less degenerate, more contextual, and "human-like" language generation in open-ended text generation settings.

We combine the capacity of sparsely gated Mixture-of-Experts (MoE) with the speed and stability of linear, mixing transformations to design the Sparse Mixer encoder model. Sparse Mixer slightly outperforms (<1%) BERT on GLUE and SuperGLUE, but more importantly trains 65% faster and runs inference 61% faster. We also present a faster variant, prosaically named Fast Sparse Mixer, that marginally underperforms BERT on SuperGLUE, but trains and runs nearly twice as fast. We justify the design of these two models by carefully ablating through various mixing mechanisms, MoE configurations and hyperparameters. Sparse Mixer overcomes many of the latency and stability concerns of MoE models and offers the prospect of serving sparse student models, without resorting to distilling them to dense variants.

Modern optimization in experimental chemistry employs algorithmic search through black-box parameter spaces. Here we demonstrate that pre-trained knowledge in large language models (LLMs) fundamentally changes this paradigm. Using six fully enumerated categorical reaction datasets (768 - 5,684 experiments), we benchmark LLM-guided optimization (LLM-GO) against Bayesian optimization (BO) and random sampling. Frontier LLMs consistently match or exceed BO performance across five single-objective datasets, with advantages growing as parameter complexity increases and high-performing conditions become scarce (<5% of space). BO retains superiority only for explicit multi-objective trade-offs. To understand these contrasting behaviors, we introduce a topology-agnostic information theory framework quantifying sampling diversity throughout optimization campaigns. This analysis reveals that LLMs maintain systematically higher exploration entropy than BO across all datasets while achieving superior performance, with advantages most pronounced in solution-scarce parameter spaces where high-entropy exploration typically fails - suggesting that pre-trained domain knowledge enables more effective navigation of chemical parameter space rather than replacing structured exploration strategies. To enable transparent benchmarking and community validation, we release Iron Mind (https://gomes.andrew.cmu.edu/iron-mind), a no-code platform for side-by-side evaluation of human, algorithmic, and LLM optimization campaigns with public leaderboards and complete trajectories. Our findings establish that LLM-GO excels precisely where traditional methods struggle: complex categorical spaces requiring domain understanding rather than mathematical optimization.

The entropic fictitious play (EFP) is a recently proposed algorithm that minimizes the sum of a convex functional and entropy in the space of measures -- such an objective naturally arises in the optimization of a two-layer neural network in the mean-field regime. In this work, we provide a concise primal-dual analysis of EFP in the setting where the learning problem exhibits a finite-sum structure. We establish quantitative global convergence guarantees for both the continuous-time and discrete-time dynamics based on properties of a proximal Gibbs measure introduced in Nitanda et al. (2022). Furthermore, our primal-dual framework entails a memory-efficient particle-based implementation of the EFP update, and also suggests a connection to gradient boosting methods. We illustrate the efficiency of our novel implementation in experiments including neural network optimization and image synthesis.

Entropy serves as a critical metric for measuring the diversity of outputs generated by large language models (LLMs), providing valuable insights into their exploration capabilities. While recent studies increasingly focus on monitoring and adjusting entropy to better balance exploration and exploitation in reinforcement fine-tuning (RFT), a principled understanding of entropy dynamics during this process is yet to be thoroughly investigated. In this paper, we establish a theoretical framework for analyzing the entropy dynamics during the RFT process, which begins with a discriminant expression that quantifies entropy change under a single logit update. This foundation enables the derivation of a first-order expression for entropy change, which can be further extended to the update formula of Group Relative Policy Optimization (GRPO). The corollaries and insights drawn from the theoretical analysis inspire the design of entropy control methods, and also offer a unified lens for interpreting various entropy-based methods in existing studies. We provide empirical evidence to support the main conclusions of our analysis and demonstrate the effectiveness of the derived entropy-discriminator clipping methods. This study yields novel insights into RFT training dynamics, providing theoretical support and practical strategies for optimizing the exploration-exploitation balance during LLM fine-tuning.

Have you ever imagined what a corgi-alike coffee machine or a tiger-alike rabbit would look like? In this work, we attempt to answer these questions by exploring a new task called semantic mixing, aiming at blending two different semantics to create a new concept (e.g., corgi + coffee machine -- > corgi-alike coffee machine). Unlike style transfer, where an image is stylized according to the reference style without changing the image content, semantic blending mixes two different concepts in a semantic manner to synthesize a novel concept while preserving the spatial layout and geometry. To this end, we present MagicMix, a simple yet effective solution based on pre-trained text-conditioned diffusion models. Motivated by the progressive generation property of diffusion models where layout/shape emerges at early denoising steps while semantically meaningful details appear at later steps during the denoising process, our method first obtains a coarse layout (either by corrupting an image or denoising from a pure Gaussian noise given a text prompt), followed by injection of conditional prompt for semantic mixing. Our method does not require any spatial mask or re-training, yet is able to synthesize novel objects with high fidelity. To improve the mixing quality, we further devise two simple strategies to provide better control and flexibility over the synthesized content. With our method, we present our results over diverse downstream applications, including semantic style transfer, novel object synthesis, breed mixing, and concept removal, demonstrating the flexibility of our method. More results can be found on the project page https://magicmix.github.io

Algorithmic efficiency techniques such as distillation (hinton2015distillation) are useful in improving model quality without increasing serving costs, provided a larger teacher model is available for a smaller student model to learn from during training. Standard distillation methods are limited to only forcing the student to match the teacher's outputs. Given the costs associated with training a large model, we believe we should be extracting more useful information from a teacher model than by just making the student match the teacher's outputs. In this paper, we introduce \guide (Guided Initialization and Distillation of Embeddings). \guide can be considered a distillation technique that forces the student to match the teacher in the parameter space. Using \guide we show 25-26\% reduction in the teacher-student quality gap when using large student models (400M - 1B parameters) trained on approx 20B tokens. We also present a thorough analysis demonstrating that \guide can be combined with knowledge distillation with near additive improvements. Furthermore, we show that applying \guide alone leads to substantially better model quality than applying knowledge distillation by itself. Most importantly, \guide introduces no training or inference overhead and hence any model quality gains from our method are virtually free.

In this paper, we point out suboptimal noise-data mapping leads to slow training of diffusion models. During diffusion training, current methods diffuse each image across the entire noise space, resulting in a mixture of all images at every point in the noise layer. We emphasize that this random mixture of noise-data mapping complicates the optimization of the denoising function in diffusion models. Drawing inspiration from the immiscible phenomenon in physics, we propose Immiscible Diffusion, a simple and effective method to improve the random mixture of noise-data mapping. In physics, miscibility can vary according to various intermolecular forces. Thus, immiscibility means that the mixing of the molecular sources is distinguishable. Inspired by this, we propose an assignment-then-diffusion training strategy. Specifically, prior to diffusing the image data into noise, we assign diffusion target noise for the image data by minimizing the total image-noise pair distance in a mini-batch. The assignment functions analogously to external forces to separate the diffuse-able areas of images, thus mitigating the inherent difficulties in diffusion training. Our approach is remarkably simple, requiring only one line of code to restrict the diffuse-able area for each image while preserving the Gaussian distribution of noise. This ensures that each image is projected only to nearby noise. To address the high complexity of the assignment algorithm, we employ a quantized-assignment method to reduce the computational overhead to a negligible level. Experiments demonstrate that our method achieve up to 3x faster training for consistency models and DDIM on the CIFAR dataset, and up to 1.3x faster on CelebA datasets for consistency models. Besides, we conduct thorough analysis about the Immiscible Diffusion, which sheds lights on how it improves diffusion training speed while improving the fidelity.

This paper investigates a novel lossy compression framework operating under logarithmic loss, designed to handle situations where the reconstruction distribution diverges from the source distribution. This framework is especially relevant for applications that require joint compression and retrieval, and in scenarios involving distributional shifts due to processing. We show that the proposed formulation extends the classical minimum entropy coupling framework by integrating a bottleneck, allowing for a controlled degree of stochasticity in the coupling. We explore the decomposition of the Minimum Entropy Coupling with Bottleneck (MEC-B) into two distinct optimization problems: Entropy-Bounded Information Maximization (EBIM) for the encoder, and Minimum Entropy Coupling (MEC) for the decoder. Through extensive analysis, we provide a greedy algorithm for EBIM with guaranteed performance, and characterize the optimal solution near functional mappings, yielding significant theoretical insights into the structural complexity of this problem. Furthermore, we illustrate the practical application of MEC-B through experiments in Markov Coding Games (MCGs) under rate limits. These games simulate a communication scenario within a Markov Decision Process, where an agent must transmit a compressed message from a sender to a receiver through its actions. Our experiments highlight the trade-offs between MDP rewards and receiver accuracy across various compression rates, showcasing the efficacy of our method compared to conventional compression baseline.

Can we learn more from data than existed in the generating process itself? Can new and useful information be constructed from merely applying deterministic transformations to existing data? Can the learnable content in data be evaluated without considering a downstream task? On these questions, Shannon information and Kolmogorov complexity come up nearly empty-handed, in part because they assume observers with unlimited computational capacity and fail to target the useful information content. In this work, we identify and exemplify three seeming paradoxes in information theory: (1) information cannot be increased by deterministic transformations; (2) information is independent of the order of data; (3) likelihood modeling is merely distribution matching. To shed light on the tension between these results and modern practice, and to quantify the value of data, we introduce epiplexity, a formalization of information capturing what computationally bounded observers can learn from data. Epiplexity captures the structural content in data while excluding time-bounded entropy, the random unpredictable content exemplified by pseudorandom number generators and chaotic dynamical systems. With these concepts, we demonstrate how information can be created with computation, how it depends on the ordering of the data, and how likelihood modeling can produce more complex programs than present in the data generating process itself. We also present practical procedures to estimate epiplexity which we show capture differences across data sources, track with downstream performance, and highlight dataset interventions that improve out-of-distribution generalization. In contrast to principles of model selection, epiplexity provides a theoretical foundation for data selection, guiding how to select, generate, or transform data for learning systems.

The data mixture used in the pre-training of a language model is a cornerstone of its final performance. However, a static mixing strategy is suboptimal, as the model's learning preferences for various data domains shift dynamically throughout training. Crucially, observing these evolving preferences in a computationally efficient manner remains a significant challenge. To address this, we propose TiKMiX, a method that dynamically adjusts the data mixture according to the model's evolving preferences. TiKMiX introduces Group Influence, an efficient metric for evaluating the impact of data domains on the model. This metric enables the formulation of the data mixing problem as a search for an optimal, influence-maximizing distribution. We solve this via two approaches: TiKMiX-D for direct optimization, and TiKMiX-M, which uses a regression model to predict a superior mixture. We trained models with different numbers of parameters, on up to 1 trillion tokens. TiKMiX-D exceeds the performance of state-of-the-art methods like REGMIX while using just 20% of the computational resources. TiKMiX-M leads to an average performance gain of 2% across 9 downstream benchmarks. Our experiments reveal that a model's data preferences evolve with training progress and scale, and we demonstrate that dynamically adjusting the data mixture based on Group Influence, a direct measure of these preferences, significantly improves performance by mitigating the underdigestion of data seen with static ratios.

Recent advances in Reinforcement Learning with Verified Reward (RLVR) have driven the emergence of more sophisticated cognitive behaviors in large language models (LLMs), thereby enhancing their reasoning capabilities. However, in prior RLVR pipelines, the repeated use of static initial-state sampling drawn exactly from the dataset distribution during each sampling phase produced overly deterministic, low diversity model behavior, which manifested as rapid entropy collapse and hindered sustained performance gains during prolonged training. To address this issue, we introduce CURE (Critical-token-gUided Re concatenation for Entropy-collapse prevention), a two-stage framework that balances exploration and exploitation. Specifically, in the first stage, to deliberately steer the model toward novel yet coherent contexts, we re-generate at high-entropy critical tokens and jointly optimize the original and the branched trajectories. The further comparison with vanilla DAPO shows that the regeneration process achieves a better performance on math reasoning tasks while sustaining a high-level entropy degree for exploration. In the second stage, we continue training with static initial-state sampling by DAPO, intentionally placing the model in a familiar state to gradually strengthen exploitation. Extensive experiments on Qwen-2.5-Math-7B show that, compared to other RLVR methods, CURE achieves a 5% performance gain across six math benchmarks, establishing state-of-the-art performance in both entropy and accuracy. A series of experiments further validate the effectiveness of our approach. Code is available at https://github.com/bytedance/CURE.

Conditional flow matching (CFM) has emerged as a powerful framework for training continuous normalizing flows due to its computational efficiency and effectiveness. However, standard CFM often produces paths that deviate significantly from straight-line interpolations between prior and target distributions, making generation slower and less accurate due to the need for fine discretization at inference. Recent methods enhance CFM performance by inducing shorter and straighter trajectories but typically rely on computationally expensive mini-batch optimal transport (OT). Drawing insights from entropic optimal transport (EOT), we propose Weighted Conditional Flow Matching (W-CFM), a novel approach that modifies the classical CFM loss by weighting each training pair (x, y) with a Gibbs kernel. We show that this weighting recovers the entropic OT coupling up to some bias in the marginals, and we provide the conditions under which the marginals remain nearly unchanged. Moreover, we establish an equivalence between W-CFM and the minibatch OT method in the large-batch limit, showing how our method overcomes computational and performance bottlenecks linked to batch size. Empirically, we test our method on unconditional generation on various synthetic and real datasets, confirming that W-CFM achieves comparable or superior sample quality, fidelity, and diversity to other alternative baselines while maintaining the computational efficiency of vanilla CFM.

We propose a new method called the N-particle underdamped Langevin algorithm for optimizing a special class of non-linear functionals defined over the space of probability measures. Examples of problems with this formulation include training mean-field neural networks, maximum mean discrepancy minimization and kernel Stein discrepancy minimization. Our algorithm is based on a novel spacetime discretization of the mean-field underdamped Langevin dynamics, for which we provide a new, fast mixing guarantee. In addition, we demonstrate that our algorithm converges globally in total variation distance, bridging the theoretical gap between the dynamics and its practical implementation.

This paper proposes a new optimization algorithm called Entropy-SGD for training deep neural networks that is motivated by the local geometry of the energy landscape. Local extrema with low generalization error have a large proportion of almost-zero eigenvalues in the Hessian with very few positive or negative eigenvalues. We leverage upon this observation to construct a local-entropy-based objective function that favors well-generalizable solutions lying in large flat regions of the energy landscape, while avoiding poorly-generalizable solutions located in the sharp valleys. Conceptually, our algorithm resembles two nested loops of SGD where we use Langevin dynamics in the inner loop to compute the gradient of the local entropy before each update of the weights. We show that the new objective has a smoother energy landscape and show improved generalization over SGD using uniform stability, under certain assumptions. Our experiments on convolutional and recurrent networks demonstrate that Entropy-SGD compares favorably to state-of-the-art techniques in terms of generalization error and training time.

Growing efforts to improve knowledge distillation (KD) in large language models (LLMs) replace dense teacher supervision with selective distillation, which uses a subset of token positions, vocabulary classes, or training samples for supervision. However, it remains unclear which importance signals, selection policies, and their interplay are most effective. In this work, we revisit where and how to distill in autoregressive LLMs. We disentangle selective KD along the position, class, and sample axes and systematically compare importance signals and selection policies. Then, guided by this analysis, we identify underexplored opportunities and introduce student-entropy-guided position selection (SE-KD). Across a suite of benchmarks, SE-KD often improves accuracy, downstream task adherence, and memory efficiency over dense distillation. Extending this approach across the class and sample axes (SE-KD 3X) yields complementary efficiency gains that make offline teacher caching feasible. In practice, this reduces wall time by 70% and peak memory by 18%, while cutting storage usage by 80% over prior methods without sacrificing performance.

Wasserstein gradient flow has emerged as a promising approach to solve optimization problems over the space of probability distributions. A recent trend is to use the well-known JKO scheme in combination with input convex neural networks to numerically implement the proximal step. The most challenging step, in this setup, is to evaluate functions involving density explicitly, such as entropy, in terms of samples. This paper builds on the recent works with a slight but crucial difference: we propose to utilize a variational formulation of the objective function formulated as maximization over a parametric class of functions. Theoretically, the proposed variational formulation allows the construction of gradient flows directly for empirical distributions with a well-defined and meaningful objective function. Computationally, this approach replaces the computationally expensive step in existing methods, to handle objective functions involving density, with inner loop updates that only require a small batch of samples and scale well with the dimension. The performance and scalability of the proposed method are illustrated with the aid of several numerical experiments involving high-dimensional synthetic and real datasets.

Many existing reinforcement learning (RL) methods employ stochastic gradient iteration on the back end, whose stability hinges upon a hypothesis that the data-generating process mixes exponentially fast with a rate parameter that appears in the step-size selection. Unfortunately, this assumption is violated for large state spaces or settings with sparse rewards, and the mixing time is unknown, making the step size inoperable. In this work, we propose an RL methodology attuned to the mixing time by employing a multi-level Monte Carlo estimator for the critic, the actor, and the average reward embedded within an actor-critic (AC) algorithm. This method, which we call Multi-level Actor-Critic (MAC), is developed especially for infinite-horizon average-reward settings and neither relies on oracle knowledge of the mixing time in its parameter selection nor assumes its exponential decay; it, therefore, is readily applicable to applications with slower mixing times. Nonetheless, it achieves a convergence rate comparable to the state-of-the-art AC algorithms. We experimentally show that these alleviated restrictions on the technical conditions required for stability translate to superior performance in practice for RL problems with sparse rewards.

Reinforcement fine-tuning (RFT) is essential for enhancing the reasoning capabilities of large language models (LLM), yet the widely adopted Group Relative Policy Optimization (GRPO) suffers from entropy collapse, where entropy monotonically decreases, exploration vanishes, and policies converge prematurely. Existing entropy-regularized methods only partially alleviate this issue while introducing bias and instability, leaving entropy control unresolved and the connection between entropy, exploration, and performance unclear. We propose Arbitrary Entropy Policy Optimization (AEPO), which eliminates entropy collapse by replacing entropy bonuses with REINFORCE policy gradient on temperature-adjusted distributions and stabilizing entropy through temperature regulation. AEPO integrates three key designs: policy gradient as regularization, distribution as regularization, and REINFORCE as regularization, enabling precise entropy control without distorting optimization. Experiments demonstrate three major contributions: AEPO (1) stabilizes entropy at arbitrary target levels, effectively removing collapse in GRPO; (2) reveals a non-monotonic relation where performance first improves then declines with increasing entropy, clarifying the link between entropy, exploration, and reasoning; and (3) generalizes beyond entropy, providing a broader RFT paradigm where superior target distributions can serve as REINFORCE regularizers.

We propose Score-of-Mixture Training (SMT), a novel framework for training one-step generative models by minimizing a class of divergences called the alpha-skew Jensen-Shannon divergence. At its core, SMT estimates the score of mixture distributions between real and fake samples across multiple noise levels. Similar to consistency models, our approach supports both training from scratch (SMT) and distillation using a pretrained diffusion model, which we call Score-of-Mixture Distillation (SMD). It is simple to implement, requires minimal hyperparameter tuning, and ensures stable training. Experiments on CIFAR-10 and ImageNet 64x64 show that SMT/SMD are competitive with and can even outperform existing methods.

Discrete diffusion models have recently gained significant attention due to their ability to process complex discrete structures for language modeling. However, fine-tuning these models with policy gradient methods, as is commonly done in Reinforcement Learning from Human Feedback (RLHF), remains a challenging task. We propose an efficient, broadly applicable, and theoretically justified policy gradient algorithm, called Score Entropy Policy Optimization (SEPO), for fine-tuning discrete diffusion models over non-differentiable rewards. Our numerical experiments across several discrete generative tasks demonstrate the scalability and efficiency of our method. Our code is available at https://github.com/ozekri/SEPO.

Distribution Matching Distillation (DMD) distills score-based generative models into efficient one-step generators, without requiring a one-to-one correspondence with the sampling trajectories of their teachers. However, limited model capacity causes one-step distilled models underperform on complex generative tasks, e.g., synthesizing intricate object motions in text-to-video generation. Directly extending DMD to multi-step distillation increases memory usage and computational depth, leading to instability and reduced efficiency. While prior works propose stochastic gradient truncation as a potential solution, we observe that it substantially reduces the generation diversity of multi-step distilled models, bringing it down to the level of their one-step counterparts. To address these limitations, we propose Phased DMD, a multi-step distillation framework that bridges the idea of phase-wise distillation with Mixture-of-Experts (MoE), reducing learning difficulty while enhancing model capacity. Phased DMD is built upon two key ideas: progressive distribution matching and score matching within subintervals. First, our model divides the SNR range into subintervals, progressively refining the model to higher SNR levels, to better capture complex distributions. Next, to ensure the training objective within each subinterval is accurate, we have conducted rigorous mathematical derivations. We validate Phased DMD by distilling state-of-the-art image and video generation models, including Qwen-Image (20B parameters) and Wan2.2 (28B parameters). Experimental results demonstrate that Phased DMD preserves output diversity better than DMD while retaining key generative capabilities. We will release our code and models.

Neural networks often exhibit emergent behavior, where qualitatively new capabilities arise from scaling up the amount of parameters, training data, or training steps. One approach to understanding emergence is to find continuous progress measures that underlie the seemingly discontinuous qualitative changes. We argue that progress measures can be found via mechanistic interpretability: reverse-engineering learned behaviors into their individual components. As a case study, we investigate the recently-discovered phenomenon of ``grokking'' exhibited by small transformers trained on modular addition tasks. We fully reverse engineer the algorithm learned by these networks, which uses discrete Fourier transforms and trigonometric identities to convert addition to rotation about a circle. We confirm the algorithm by analyzing the activations and weights and by performing ablations in Fourier space. Based on this understanding, we define progress measures that allow us to study the dynamics of training and split training into three continuous phases: memorization, circuit formation, and cleanup. Our results show that grokking, rather than being a sudden shift, arises from the gradual amplification of structured mechanisms encoded in the weights, followed by the later removal of memorizing components.

This paper aims to overcome a major obstacle in scaling RL for reasoning with LLMs, namely the collapse of policy entropy. Such phenomenon is consistently observed across vast RL runs without entropy intervention, where the policy entropy dropped sharply at the early training stage, this diminished exploratory ability is always accompanied with the saturation of policy performance. In practice, we establish a transformation equation R=-a*e^H+b between entropy H and downstream performance R. This empirical law strongly indicates that, the policy performance is traded from policy entropy, thus bottlenecked by its exhaustion, and the ceiling is fully predictable H=0, R=-a+b. Our finding necessitates entropy management for continuous exploration toward scaling compute for RL. To this end, we investigate entropy dynamics both theoretically and empirically. Our derivation highlights that, the change in policy entropy is driven by the covariance between action probability and the change in logits, which is proportional to its advantage when using Policy Gradient-like algorithms. Empirical study shows that, the values of covariance term and entropy differences matched exactly, supporting the theoretical conclusion. Moreover, the covariance term stays mostly positive throughout training, further explaining why policy entropy would decrease monotonically. Through understanding the mechanism behind entropy dynamics, we motivate to control entropy by restricting the update of high-covariance tokens. Specifically, we propose two simple yet effective techniques, namely Clip-Cov and KL-Cov, which clip and apply KL penalty to tokens with high covariances respectively. Experiments show that these methods encourage exploration, thus helping policy escape entropy collapse and achieve better downstream performance.

Sequence and channel mixers, the core mechanism in sequence models, have become the de facto standard in time series analysis (TSA). However, recent studies have questioned the necessity of complex sequence mixers, such as attention mechanisms, demonstrating that simpler architectures can achieve comparable or even superior performance. This suggests that the benefits attributed to complex sequencemixers might instead emerge from other architectural or optimization factors. Based on this observation, we pose a central question: Are common sequence mixers necessary for time-series analysis? Therefore, we propose JustDense, an empirical study that systematically replaces sequence mixers in various well-established TSA models with dense layers. Grounded in the MatrixMixer framework, JustDense treats any sequence mixer as a mixing matrix and replaces it with a dense layer. This substitution isolates the mixing operation, enabling a clear theoretical foundation for understanding its role. Therefore, we conducted extensive experiments on 29 benchmarks covering five representative TSA tasks using seven state-of-the-art TSA models to address our research question. The results show that replacing sequence mixers with dense layers yields comparable or even superior performance. In the cases where dedicated sequence mixers still offer benefits, JustDense challenges the assumption that "deeper and more complex architectures are inherently better" in TSA.

Mixup data augmentation approaches have been applied for various tasks of deep learning to improve the generalization ability of deep neural networks. Some existing approaches CutMix, SaliencyMix, etc. randomly replace a patch in one image with patches from another to generate the mixed image. Similarly, the corresponding labels are linearly combined by a fixed ratio lambda by l. The objects in two images may be overlapped during the mixing process, so some semantic information is corrupted in the mixed samples. In this case, the mixed image does not match the mixed label information. Besides, such a label may mislead the deep learning model training, which results in poor performance. To solve this problem, we proposed a novel approach named SUMix to learn the mixing ratio as well as the uncertainty for the mixed samples during the training process. First, we design a learnable similarity function to compute an accurate mix ratio. Second, an approach is investigated as a regularized term to model the uncertainty of the mixed samples. We conduct experiments on five image benchmarks, and extensive experimental results imply that our method is capable of improving the performance of classifiers with different cutting-based mixup approaches. The source code is available at https://github.com/JinXins/SUMix.

For RL algorithms, appropriate entropy control is crucial to their effectiveness. To control the policy entropy, a commonly used method is entropy regularization, which is adopted in various popular RL algorithms including PPO, SAC and A3C. Although entropy regularization proves effective in robotic and games RL conventionally, studies found that it gives weak to no gains in LLM-RL training. In this work, we study the issues of entropy bonus in LLM-RL setting. Specifically, we first argue that the conventional entropy regularization suffers from the LLM's extremely large response space and the sparsity of the optimal outputs. As a remedy, we propose AEnt, an entropy control method that utilizes a new clamped entropy bonus with an automatically adjusted coefficient. The clamped entropy is evaluated with the re-normalized policy defined on certain smaller token space, which encourages exploration within a more compact response set. In addition, the algorithm automatically adjusts entropy coefficient according to the clamped entropy value, effectively controlling the entropy-induced bias while leveraging the entropy's benefits. AEnt is tested in math-reasoning tasks under different base models and datasets, and it is observed that AEnt outperforms the baselines consistently across multiple benchmarks.

Discrete diffusion or flow models could enable faster and more controllable sequence generation than autoregressive models. We show that na\"ive linear flow matching on the simplex is insufficient toward this goal since it suffers from discontinuities in the training target and further pathologies. To overcome this, we develop Dirichlet flow matching on the simplex based on mixtures of Dirichlet distributions as probability paths. In this framework, we derive a connection between the mixtures' scores and the flow's vector field that allows for classifier and classifier-free guidance. Further, we provide distilled Dirichlet flow matching, which enables one-step sequence generation with minimal performance hits, resulting in O(L) speedups compared to autoregressive models. On complex DNA sequence generation tasks, we demonstrate superior performance compared to all baselines in distributional metrics and in achieving desired design targets for generated sequences. Finally, we show that our classifier-free guidance approach improves unconditional generation and is effective for generating DNA that satisfies design targets. Code is available at https://github.com/HannesStark/dirichlet-flow-matching.

Instruction Tuning involves finetuning a language model on a collection of instruction-formatted datasets in order to enhance the generalizability of the model to unseen tasks. Studies have shown the importance of balancing different task proportions during finetuning, but finding the right balance remains challenging. Unfortunately, there's currently no systematic method beyond manual tuning or relying on practitioners' intuition. In this paper, we introduce SMART (Submodular data Mixture strAtegy for instRuction Tuning) - a novel data mixture strategy which makes use of a submodular function to assign importance scores to tasks which are then used to determine the mixture weights. Given a fine-tuning budget, SMART redistributes the budget among tasks and selects non-redundant samples from each task. Experimental results demonstrate that SMART significantly outperforms traditional methods such as examples proportional mixing and equal mixing. Furthermore, SMART facilitates the creation of data mixtures based on a few representative subsets of tasks alone and through task pruning analysis, we reveal that in a limited budget setting, allocating budget among a subset of representative tasks yields superior performance compared to distributing the budget among all tasks. The code for reproducing our results is open-sourced at https://github.com/kowndinya-renduchintala/SMART.

While advancements in the reasoning abilities of LLMs have significantly enhanced their performance in solving mathematical problems, coding tasks, and general puzzles, their effectiveness in accurately adhering to instructions remains inconsistent, particularly with more complex directives. Our investigation identifies lazy reasoning during the thinking stage as the primary factor contributing to poor instruction adherence. To mitigate this issue, we propose a comprehensive framework designed to enable rigorous reasoning processes involving preview and self-checking, essential for satisfying strict instruction constraints. Specifically, we first generate instructions with complex constraints and apply a filtering process to obtain valid prompts, resulting in three distinct prompt datasets categorized as hard, easy, and pass. Then, we employ rejection sampling on the pass prompts to curate a small yet high-quality dataset, enabling a cold-start initialization of the model and facilitating its adaptation to effective reasoning patterns. Subsequently, we employ an entropy-preserving supervised fine-tuning (Entropy-SFT) strategy coupled with token-wise entropy-adaptive (TEA-RL) reinforcement learning guided by rule-based dense rewards. This approach encourages the model to transform its reasoning mechanism, ultimately fostering generalizable reasoning abilities that encompass preview and self-checking. Extensive experiments conducted on instruction-following benchmarks demonstrate remarkable performance improvements across various model scales. Notably, our Light-IF-32B model surpasses both larger open-source models such as DeepSeek-R1 and closed-source models like Doubao-1.6.

Boltzmann generators approach the sampling problem in many-body physics by combining a normalizing flow and a statistical reweighting method to generate samples in thermodynamic equilibrium. The equilibrium distribution is usually defined by an energy function and a thermodynamic state. Here we propose temperature-steerable flows (TSF) which are able to generate a family of probability densities parametrized by a choosable temperature parameter. TSFs can be embedded in generalized ensemble sampling frameworks to sample a physical system across multiple thermodynamic states.

Although GRPO substantially enhances flow matching models in human preference alignment of image generation, methods such as FlowGRPO still exhibit inefficiency due to the necessity of sampling and optimizing over all denoising steps specified by the Markov Decision Process (MDP). In this paper, we propose MixGRPO, a novel framework that leverages the flexibility of mixed sampling strategies through the integration of stochastic differential equations (SDE) and ordinary differential equations (ODE). This streamlines the optimization process within the MDP to improve efficiency and boost performance. Specifically, MixGRPO introduces a sliding window mechanism, using SDE sampling and GRPO-guided optimization only within the window, while applying ODE sampling outside. This design confines sampling randomness to the time-steps within the window, thereby reducing the optimization overhead, and allowing for more focused gradient updates to accelerate convergence. Additionally, as time-steps beyond the sliding window are not involved in optimization, higher-order solvers are supported for sampling. So we present a faster variant, termed MixGRPO-Flash, which further improves training efficiency while achieving comparable performance. MixGRPO exhibits substantial gains across multiple dimensions of human preference alignment, outperforming DanceGRPO in both effectiveness and efficiency, with nearly 50% lower training time. Notably, MixGRPO-Flash further reduces training time by 71%. Codes and models are available at https://github.com/Tencent-Hunyuan/MixGRPO{MixGRPO}.

Data mixing strategies (e.g., CutMix) have shown the ability to greatly improve the performance of convolutional neural networks (CNNs). They mix two images as inputs for training and assign them with a mixed label with the same ratio. While they are shown effective for vision transformers (ViTs), we identify a token fluctuation phenomenon that has suppressed the potential of data mixing strategies. We empirically observe that the contributions of input tokens fluctuate as forward propagating, which might induce a different mixing ratio in the output tokens. The training target computed by the original data mixing strategy can thus be inaccurate, resulting in less effective training. To address this, we propose a token-label alignment (TL-Align) method to trace the correspondence between transformed tokens and the original tokens to maintain a label for each token. We reuse the computed attention at each layer for efficient token-label alignment, introducing only negligible additional training costs. Extensive experiments demonstrate that our method improves the performance of ViTs on image classification, semantic segmentation, objective detection, and transfer learning tasks. Code is available at: https://github.com/Euphoria16/TL-Align.

Iterative generative policies, such as diffusion models and flow matching, offer superior expressivity for continuous control but complicate Maximum Entropy Reinforcement Learning because their action log-densities are not directly accessible. To address this, we propose Field Least-Energy Actor-Critic (FLAC), a likelihood-free framework that regulates policy stochasticity by penalizing the kinetic energy of the velocity field. Our key insight is to formulate policy optimization as a Generalized Schrödinger Bridge (GSB) problem relative to a high-entropy reference process (e.g., uniform). Under this view, the maximum-entropy principle emerges naturally as staying close to a high-entropy reference while optimizing return, without requiring explicit action densities. In this framework, kinetic energy serves as a physically grounded proxy for divergence from the reference: minimizing path-space energy bounds the deviation of the induced terminal action distribution. Building on this view, we derive an energy-regularized policy iteration scheme and a practical off-policy algorithm that automatically tunes the kinetic energy via a Lagrangian dual mechanism. Empirically, FLAC achieves superior or comparable performance on high-dimensional benchmarks relative to strong baselines, while avoiding explicit density estimation.

Recent advancements in learned image compression (LIC) have yielded impressive performance gains. Notably, the learned image compression models with multi-reference entropy models (MLIC series) have significantly outperformed existing traditional image codecs such as the Versatile Video Coding (VVC) Intra. In this paper, we present MLICv2 and MLICv2^+, enhanced versions of the MLIC series, featuring improved transform techniques, entropy modeling, and instance adaptability. For better transform, we introduce a simple token mixing transform block inspired by the meta transformer architecture, addressing the performance degradation at high bit-rates observed in previous MLIC series while maintaining computational efficiency. To enhance entropy modeling, we propose a hyperprior-guided global correlation prediction, enabling the capture of global contexts in the initial slice of the latent representation. We also develop a channel reweighting module to dynamically prioritize important channels within each context. Additionally, advanced positional embedding for context modeling and selective compression with guided optimization are investigated. To boost instance adaptability, we employ stochastic Gumbel annealing to iteratively refine the latent representation according to the rate-distortion optimization of a specific input image. This approach further enhances performance without impacting decoding speed. Experimental results demonstrate that our MLICv2 and MLICv2^+ achieve state-of-the-art performance, reducing Bjontegaard-Delta rate (BD-rate) by 16.54%, 21.61%, 16.05% and 20.46%, 24.35%, 19.14% respectively, compared to VTM-17.0 Intra on the Kodak, Tecnick, CLIC Pro Val dataset, respectively.

Streaming video generation, as one fundamental component in interactive world models and neural game engines, aims to generate high-quality, low-latency, and temporally coherent long video streams. However, most existing work suffers from severe error accumulation that often significantly degrades the generated stream videos over long horizons. We design Rolling Forcing, a novel video generation technique that enables streaming long videos with minimal error accumulation. Rolling Forcing comes with three novel designs. First, instead of iteratively sampling individual frames, which accelerates error propagation, we design a joint denoising scheme that simultaneously denoises multiple frames with progressively increasing noise levels. This design relaxes the strict causality across adjacent frames, effectively suppressing error growth. Second, we introduce the attention sink mechanism into the long-horizon stream video generation task, which allows the model to keep key value states of initial frames as a global context anchor and thereby enhances long-term global consistency. Third, we design an efficient training algorithm that enables few-step distillation over largely extended denoising windows. This algorithm operates on non-overlapping windows and mitigates exposure bias conditioned on self-generated histories. Extensive experiments show that Rolling Forcing enables real-time streaming generation of multi-minute videos on a single GPU, with substantially reduced error accumulation.

We introduce a differentiable estimator of range-partition entropy, a recent concept from computational geometry that enables algorithms to adapt to the "sortedness" of their input. While range-partition entropy provides strong guarantees in algorithm design, it has not yet been made accessible to deep learning. In this work, we (i) propose the first differentiable approximation of range-partition entropy, enabling its use as a trainable loss or regularizer; (ii) design EntropyNet, a neural module that restructures data into low-entropy forms to accelerate downstream instance-optimal algorithms; and (iii) extend this principle beyond geometry by applying entropy regularization directly to Transformer attention. Across tasks, we demonstrate that differentiable entropy improves efficiency without degrading correctness: in geometry, our method achieves up to 4.1times runtime speedups with negligible error (<0.2%); in deep learning, it induces structured attention patterns that yield 6% higher accuracy at 80% sparsity compared to L1 baselines. Our theoretical analysis provides approximation bounds for the estimator, and extensive ablations validate design choices. These results suggest that entropy-bounded computation is not only theoretically elegant but also a practical mechanism for adaptive learning, efficiency, and structured representation.

Activation steering methods were shown to be effective in conditioning language model generation by additively intervening over models' intermediate representations. However, the evaluation of these techniques has so far been limited to single conditioning properties and synthetic settings. In this work, we conduct a comprehensive evaluation of various activation steering strategies, highlighting the property-dependent nature of optimal parameters to ensure a robust effect throughout generation. To address this issue, we propose Dynamic Activation Composition, an information-theoretic approach to modulate the steering intensity of one or more properties throughout generation. Our experiments on multi-property steering show that our method successfully maintains high conditioning while minimizing the impact of conditioning on generation fluency.

Learning PDE dynamics with neural solvers can significantly improve wall-clock efficiency and accuracy compared with classical numerical solvers. In recent years, foundation models for PDEs have largely adopted multi-scale windowed self-attention, with the scOT backbone in Poseidon serving as a representative example. However, because of their locality, truly globally consistent spectral coupling can only be propagated gradually through deep stacking and window shifting. This weakens global coupling and leads to error accumulation and drift during closed-loop rollouts. To address this, we propose DRIFT-Net. It employs a dual-branch design comprising a spectral branch and an image branch. The spectral branch is responsible for capturing global, large-scale low-frequency information, whereas the image branch focuses on local details and nonstationary structures. Specifically, we first perform controlled, lightweight mixing within the low-frequency range. Then we fuse the spectral and image paths at each layer via bandwise weighting, which avoids the width inflation and training instability caused by naive concatenation. The fused result is transformed back into the spatial domain and added to the image branch, thereby preserving both global structure and high-frequency details across scales. Compared with strong attention-based baselines, DRIFT-Net achieves lower error and higher throughput with fewer parameters under identical training settings and budget. On Navier--Stokes benchmarks, the relative L_{1} error is reduced by 7\%--54\%, the parameter count decreases by about 15\%, and the throughput remains higher than scOT. Ablation studies and theoretical analyses further demonstrate the stability and effectiveness of this design. The code is available at https://github.com/cruiseresearchgroup/DRIFT-Net.

The substantial training cost of diffusion models hinders their deployment. Immiscible Diffusion recently showed that reducing diffusion trajectory mixing in the noise space via linear assignment accelerates training by simplifying denoising. To extend immiscible diffusion beyond the inefficient linear assignment under high batch sizes and high dimensions, we refine this concept to a broader miscibility reduction at any layer and by any implementation. Specifically, we empirically demonstrate the bijective nature of the denoising process with respect to immiscible diffusion, ensuring its preservation of generative diversity. Moreover, we provide thorough analysis and show step-by-step how immiscibility eases denoising and improves efficiency. Extending beyond linear assignment, we propose a family of implementations including K-nearest neighbor (KNN) noise selection and image scaling to reduce miscibility, achieving up to >4x faster training across diverse models and tasks including unconditional/conditional generation, image editing, and robotics planning. Furthermore, our analysis of immiscibility offers a novel perspective on how optimal transport (OT) enhances diffusion training. By identifying trajectory miscibility as a fundamental bottleneck, we believe this work establishes a potentially new direction for future research into high-efficiency diffusion training. The code is available at https://github.com/yhli123/Immiscible-Diffusion.

Generating new molecules is fundamental to advancing critical applications such as drug discovery and material synthesis. Flows can generate molecules effectively by inverting the encoding process, however, existing flow models either require artifactual dequantization or specific node/edge orderings, lack desiderata such as permutation invariance, or induce discrepancy between the encoding and the decoding steps that necessitates post hoc validity correction. We circumvent these issues with novel continuous normalizing E(3)-equivariant flows, based on a system of node ODEs coupled as a graph PDE, that repeatedly reconcile locally toward globally aligned densities. Our models can be cast as message-passing temporal networks, and result in superlative performance on the tasks of density estimation and molecular generation. In particular, our generated samples achieve state-of-the-art on both the standard QM9 and ZINC250K benchmarks.

Fast and high-quality language generation is the holy grail that people pursue in the age of AI. In this work, we introduce Discrete Diffusion Divergence Instruct (DiDi-Instruct), a training-based method that initializes from a pre-trained (masked) discrete diffusion language model (dLLM) and distills a few-step student for fast generation. The resulting DiDi-Instruct model achieves comparable or superior performance to its dLLM teacher and the GPT-2 baseline while enabling up to 64times acceleration. The theoretical foundation of DiDi-Instruct is a novel framework based on integral KL-divergence minimization, which yields a practical training algorithm. We further introduce grouped reward normalization, intermediate-state matching, and the reward-guided ancestral sampler that significantly improve training stability, model coverage, and inference quality. On OpenWebText, DiDi-Instruct achieves perplexity from 62.2 (8 NFEs) to 18.4 (128 NFEs), which outperforms prior accelerated dLLMs and GPT-2 baseline. These gains come with a negligible entropy loss (around 1%) and reduce additional training wall-clock time by more than 20times compared to competing dLLM distillation methods. We further validate the robustness and effectiveness of DiDi-Instruct through extensive ablation studies, model scaling, and the generation of discrete protein sequences. In conclusion, DiDi-Instruct is an efficient yet effective distillation method, enabling language generation in the blink of an eye. We will release both code and models at github.com/haoyangzheng-ai/didi-instruct.

Recent advances in diffusion models have shown remarkable potential in the conditional generation of novel molecules. These models can be guided in two ways: (i) explicitly, through additional features representing the condition, or (ii) implicitly, using a property predictor. However, training property predictors or conditional diffusion models requires an abundance of labeled data and is inherently challenging in real-world applications. We propose a novel approach that attenuates the limitations of acquiring large labeled datasets by leveraging domain knowledge from quantum chemistry as a non-differentiable oracle to guide an unconditional diffusion model. Instead of relying on neural networks, the oracle provides accurate guidance in the form of estimated gradients, allowing the diffusion process to sample from a conditional distribution specified by quantum chemistry. We show that this results in more precise conditional generation of novel and stable molecular structures. Our experiments demonstrate that our method: (1) significantly reduces atomic forces, enhancing the validity of generated molecules when used for stability optimization; (2) is compatible with both explicit and implicit guidance in diffusion models, enabling joint optimization of molecular properties and stability; and (3) generalizes effectively to molecular optimization tasks beyond stability optimization.

Diffusion probabilistic models (DPMs) have achieved impressive success in high-resolution image synthesis, especially in recent large-scale text-to-image generation applications. An essential technique for improving the sample quality of DPMs is guided sampling, which usually needs a large guidance scale to obtain the best sample quality. The commonly-used fast sampler for guided sampling is DDIM, a first-order diffusion ODE solver that generally needs 100 to 250 steps for high-quality samples. Although recent works propose dedicated high-order solvers and achieve a further speedup for sampling without guidance, their effectiveness for guided sampling has not been well-tested before. In this work, we demonstrate that previous high-order fast samplers suffer from instability issues, and they even become slower than DDIM when the guidance scale grows large. To further speed up guided sampling, we propose DPM-Solver++, a high-order solver for the guided sampling of DPMs. DPM-Solver++ solves the diffusion ODE with the data prediction model and adopts thresholding methods to keep the solution matches training data distribution. We further propose a multistep variant of DPM-Solver++ to address the instability issue by reducing the effective step size. Experiments show that DPM-Solver++ can generate high-quality samples within only 15 to 20 steps for guided sampling by pixel-space and latent-space DPMs.

Discrete diffusion models (DDMs) have shown powerful generation ability for discrete data modalities like text and molecules. However, their practical application is hindered by inefficient sampling, requiring a large number of sampling steps. Accelerating DDMs by using larger step sizes typically introduces significant problems in generation quality, as it amplifies the impact of both the compounding decoding error due to factorized predictions and discretization error from numerical approximations, leading to a significant decrease in sampling quality. To address these challenges, we propose learnable sampler distillation (LSD), a novel approach to train fast and high-fidelity samplers for DDMs. LSD employs a distillation approach where a student sampler with a few steps learns to align its intermediate score trajectory with that of a high-quality teacher sampler with numerous steps. This alignment is achieved by optimizing learnable sampler coefficients that adaptively adjust sampling dynamics. Additionally, we further propose LSD+, which also learns time schedules that allocate steps non-uniformly. Experiments across text generation, image generation, and synthetic tasks demonstrate that our proposed approaches outperform existing samplers for DDMs, achieving substantially higher sampling quality with significantly fewer sampling steps. Our code is available at https://github.com/feiyangfu/LSD{https://github.com/feiyangfu/LSD}.

Recent advancements in diffusion and flow-matching models have demonstrated remarkable capabilities in high-fidelity image synthesis. A prominent line of research involves reward-guided guidance, which steers the generation process during inference to align with specific objectives. However, leveraging this reward-guided approach to the task of image editing, which requires preserving the semantic content of the source image while enhancing a target reward, is largely unexplored. In this work, we introduce a novel framework for training-free, reward-guided image editing. We formulate the editing process as a trajectory optimal control problem where the reverse process of a diffusion model is treated as a controllable trajectory originating from the source image, and the adjoint states are iteratively updated to steer the editing process. Through extensive experiments across distinct editing tasks, we demonstrate that our approach significantly outperforms existing inversion-based training-free guidance baselines, achieving a superior balance between reward maximization and fidelity to the source image without reward hacking.

Classifier-free guidance is a key component for enhancing the performance of conditional generative models across diverse tasks. While it has previously demonstrated remarkable improvements for the sample quality, it has only been exclusively employed for diffusion models. In this paper, we integrate classifier-free guidance into Flow Matching (FM) models, an alternative simulation-free approach that trains Continuous Normalizing Flows (CNFs) based on regressing vector fields. We explore the usage of Guided Flows for a variety of downstream applications. We show that Guided Flows significantly improves the sample quality in conditional image generation and zero-shot text-to-speech synthesis, boasting state-of-the-art performance. Notably, we are the first to apply flow models for plan generation in the offline reinforcement learning setting, showcasing a 10x speedup in computation compared to diffusion models while maintaining comparable performance.

The Koopman operator provides a powerful framework for modeling dynamical systems and has attracted growing interest from the machine learning community. However, its infinite-dimensional nature makes identifying suitable finite-dimensional subspaces challenging, especially for deep architectures. We argue that these difficulties come from suboptimal representation learning, where latent variables fail to balance expressivity and simplicity. This tension is closely related to the information bottleneck (IB) dilemma: constructing compressed representations that are both compact and predictive. Rethinking Koopman learning through this lens, we demonstrate that latent mutual information promotes simplicity, yet an overemphasis on simplicity may cause latent space to collapse onto a few dominant modes. In contrast, expressiveness is sustained by the von Neumann entropy, which prevents such collapse and encourages mode diversity. This insight leads us to propose an information-theoretic Lagrangian formulation that explicitly balances this tradeoff. Furthermore, we propose a new algorithm based on the Lagrangian formulation that encourages both simplicity and expressiveness, leading to a stable and interpretable Koopman representation. Beyond quantitative evaluations, we further visualize the learned manifolds under our representations, observing empirical results consistent with our theoretical predictions. Finally, we validate our approach across a diverse range of dynamical systems, demonstrating improved performance over existing Koopman learning methods. The implementation is publicly available at https://github.com/Wenxuan52/InformationKoopman.

We present a probabilistic model for point cloud generation, which is fundamental for various 3D vision tasks such as shape completion, upsampling, synthesis and data augmentation. Inspired by the diffusion process in non-equilibrium thermodynamics, we view points in point clouds as particles in a thermodynamic system in contact with a heat bath, which diffuse from the original distribution to a noise distribution. Point cloud generation thus amounts to learning the reverse diffusion process that transforms the noise distribution to the distribution of a desired shape. Specifically, we propose to model the reverse diffusion process for point clouds as a Markov chain conditioned on certain shape latent. We derive the variational bound in closed form for training and provide implementations of the model. Experimental results demonstrate that our model achieves competitive performance in point cloud generation and auto-encoding. The code is available at https://github.com/luost26/diffusion-point-cloud.

We study the problem of symbolic music generation (e.g., generating piano rolls), with a technical focus on non-differentiable rule guidance. Musical rules are often expressed in symbolic form on note characteristics, such as note density or chord progression, many of which are non-differentiable which pose a challenge when using them for guided diffusion. We propose Stochastic Control Guidance (SCG), a novel guidance method that only requires forward evaluation of rule functions that can work with pre-trained diffusion models in a plug-and-play way, thus achieving training-free guidance for non-differentiable rules for the first time. Additionally, we introduce a latent diffusion architecture for symbolic music generation with high time resolution, which can be composed with SCG in a plug-and-play fashion. Compared to standard strong baselines in symbolic music generation, this framework demonstrates marked advancements in music quality and rule-based controllability, outperforming current state-of-the-art generators in a variety of settings. For detailed demonstrations, code and model checkpoints, please visit our project website: https://scg-rule-guided-music.github.io/.

We introduce Interleaved Gibbs Diffusion (IGD), a novel generative modeling framework for mixed continuous-discrete data, focusing on constrained generation problems. Prior works on discrete and continuous-discrete diffusion models assume factorized denoising distribution for fast generation, which can hinder the modeling of strong dependencies between random variables encountered in constrained generation. IGD moves beyond this by interleaving continuous and discrete denoising algorithms via a discrete time Gibbs sampling type Markov chain. IGD provides flexibility in the choice of denoisers, allows conditional generation via state-space doubling and inference time scaling via the ReDeNoise method. Empirical evaluations on three challenging tasks-solving 3-SAT, generating molecule structures, and generating layouts-demonstrate state-of-the-art performance. Notably, IGD achieves a 7% improvement on 3-SAT out of the box and achieves state-of-the-art results in molecule generation without relying on equivariant diffusion or domain-specific architectures. We explore a wide range of modeling, and interleaving strategies along with hyperparameters in each of these problems.

We present an open-source Python implementation of an idealized high-order pseudo-spectral solver for the one-dimensional nonlinear Schr\"odinger equation (NLSE). The solver combines Fourier spectral spatial discretization with an adaptive eighth-order Dormand-Prince time integration scheme to achieve machine-precision conservation of mass and near-perfect preservation of momentum and energy for smooth solutions. The implementation accurately reproduces fundamental NLSE phenomena including soliton collisions with analytically predicted phase shifts, Akhmediev breather dynamics, and the development of modulation instability from noisy initial conditions. Four canonical test cases validate the numerical scheme: single soliton propagation, two-soliton elastic collision, breather evolution, and noise-seeded modulation instability. The solver employs a 2/3 dealiasing rule with exponential filtering to prevent aliasing errors from the cubic nonlinearity. Statistical analysis using Shannon, R\'enyi, and Tsallis entropies quantifies the spatio-temporal complexity of solutions, while phase space representations reveal the underlying coherence structure. The implementation prioritizes code transparency and educational accessibility over computational performance, providing a valuable pedagogical tool for exploring nonlinear wave dynamics. Complete source code, documentation, and example configurations are freely available, enabling reproducible computational experiments across diverse physical contexts where the NLSE governs wave evolution, including nonlinear optics, Bose-Einstein condensates, and ocean surface waves.

Reinforcement learning (RL) is the dominant paradigm for sharpening strategic tool use capabilities of LLMs on long-horizon, sparsely-rewarded agent tasks, yet it faces a fundamental challenge of exploration-exploitation trade-off. Existing studies stimulate exploration through the lens of policy entropy, but such mechanical entropy maximization is prone to RL training instability due to the multi-turn distribution shifting. In this paper, we target the progressive exploration-exploitation balance under the guidance of the agent own experiences without succumbing to either entropy collapsing or runaway divergence. We propose SPEAR, a curriculum-based self-imitation learning (SIL) recipe for training agentic LLMs. It extends the vanilla SIL framework, where a replay buffer stores self-generated promising trajectories for off-policy update, by gradually steering the policy evolution within a well-balanced range of entropy across stages. Specifically, our approach incorporates a curriculum to manage the exploration process, utilizing intrinsic rewards to foster skill-level exploration and facilitating action-level exploration through SIL. At first, the auxiliary tool call reward plays a critical role in the accumulation of tool-use skills, enabling broad exposure to the unfamiliar distributions of the environment feedback with an upward entropy trend. As training progresses, self-imitation gets strengthened to exploit existing successful patterns from replayed experiences for comparative action-level exploration, accelerating solution iteration without unbounded entropy growth. To further stabilize training, we recalibrate the advantages of experiences in the replay buffer to address the potential policy drift. Reugularizations such as the clipping of tokens with high covariance between probability and advantage are introduced to the trajectory-level entropy control to curb over-confidence.

Guidance of generative models is typically achieved by modifying the probability flow vector field through the addition of a guidance field. In this paper, we instead propose the Source-Guided Flow Matching (SGFM) framework, which modifies the source distribution directly while keeping the pre-trained vector field intact. This reduces the guidance problem to a well-defined problem of sampling from the source distribution. We theoretically show that SGFM recovers the desired target distribution exactly. Furthermore, we provide bounds on the Wasserstein error for the generated distribution when using an approximate sampler of the source distribution and an approximate vector field. The key benefit of our approach is that it allows the user to flexibly choose the sampling method depending on their specific problem. To illustrate this, we systematically compare different sampling methods and discuss conditions for asymptotically exact guidance. Moreover, our framework integrates well with optimal flow matching models since the straight transport map generated by the vector field is preserved. Experimental results on synthetic 2D benchmarks, physics-informed generative tasks, and imaging inverse problems demonstrate the effectiveness and flexibility of the proposed framework.

Popular guidance for denoising diffusion probabilistic model (DDPM) linearly combines distinct conditional models together to provide enhanced control over samples. However, this approach overlooks nonlinear effects that become significant when guidance scale is large. To address this issue, we propose characteristic guidance, a guidance method that provides first-principle non-linear correction for classifier-free guidance. Such correction forces the guided DDPMs to respect the Fokker-Planck (FP) equation of diffusion process, in a way that is training-free and compatible with existing sampling methods. Experiments show that characteristic guidance enhances semantic characteristics of prompts and mitigate irregularities in image generation, proving effective in diverse applications ranging from simulating magnet phase transitions to latent space sampling.

Backpropagation, which uses the chain rule, is the de-facto standard algorithm for optimizing neural networks nowadays. Recently, Hinton (2022) proposed the forward-forward algorithm, a promising alternative that optimizes neural nets layer-by-layer, without propagating gradients throughout the network. Although such an approach has several advantages over back-propagation and shows promising results, the fact that each layer is being trained independently limits the optimization process. Specifically, it prevents the network's layers from collaborating to learn complex and rich features. In this work, we study layer collaboration in the forward-forward algorithm. We show that the current version of the forward-forward algorithm is suboptimal when considering information flow in the network, resulting in a lack of collaboration between layers of the network. We propose an improved version that supports layer collaboration to better utilize the network structure, while not requiring any additional assumptions or computations. We empirically demonstrate the efficacy of the proposed version when considering both information flow and objective metrics. Additionally, we provide a theoretical motivation for the proposed method, inspired by functional entropy theory.

Deep neural networks tend to memorize noisy labels, severely degrading their generalization performance. Although Mixup has demonstrated effectiveness in improving generalization and robustness, existing Mixup-based methods typically perform indiscriminate mixing without principled guidance on sample selection and mixing strategy, inadvertently propagating noisy supervision. To overcome these limitations, we propose SelectMix, a confidence-guided mixing framework explicitly tailored for noisy labels. SelectMix first identifies potentially noisy or ambiguous samples through confidence based mismatch analysis using K-fold cross-validation, then selectively blends identified uncertain samples with confidently predicted peers from their potential classes. Furthermore, SelectMix employs soft labels derived from all classes involved in the mixing process, ensuring the labels accurately represent the composition of the mixed samples, thus aligning supervision signals closely with the actual mixed inputs. Through extensive theoretical analysis and empirical evaluations on multiple synthetic (MNIST, Fashion-MNIST, CIFAR-10, CIFAR-100) and real-world benchmark datasets (CIFAR-N, MNIST and Clothing1M), we demonstrate that SelectMix consistently outperforms strong baseline methods, validating its effectiveness and robustness in learning with noisy labels.

The mechanisms behind the success of multi-view self-supervised learning (MVSSL) are not yet fully understood. Contrastive MVSSL methods have been studied through the lens of InfoNCE, a lower bound of the Mutual Information (MI). However, the relation between other MVSSL methods and MI remains unclear. We consider a different lower bound on the MI consisting of an entropy and a reconstruction term (ER), and analyze the main MVSSL families through its lens. Through this ER bound, we show that clustering-based methods such as DeepCluster and SwAV maximize the MI. We also re-interpret the mechanisms of distillation-based approaches such as BYOL and DINO, showing that they explicitly maximize the reconstruction term and implicitly encourage a stable entropy, and we confirm this empirically. We show that replacing the objectives of common MVSSL methods with this ER bound achieves competitive performance, while making them stable when training with smaller batch sizes or smaller exponential moving average (EMA) coefficients. Github repo: https://github.com/apple/ml-entropy-reconstruction.

Recent advances in deep learning have mainly relied on Transformers due to their data dependency and ability to learn at scale. The attention module in these architectures, however, exhibits quadratic time and space in input size, limiting their scalability for long-sequence modeling. Despite recent attempts to design efficient and effective architecture backbone for multi-dimensional data, such as images and multivariate time series, existing models are either data independent, or fail to allow inter- and intra-dimension communication. Recently, State Space Models (SSMs), and more specifically Selective State Space Models, with efficient hardware-aware implementation, have shown promising potential for long sequence modeling. Motivated by the success of SSMs, we present MambaMixer, a new architecture with data-dependent weights that uses a dual selection mechanism across tokens and channels, called Selective Token and Channel Mixer. MambaMixer connects selective mixers using a weighted averaging mechanism, allowing layers to have direct access to early features. As a proof of concept, we design Vision MambaMixer (ViM2) and Time Series MambaMixer (TSM2) architectures based on the MambaMixer block and explore their performance in various vision and time series forecasting tasks. Our results underline the importance of selective mixing across both tokens and channels. In ImageNet classification, object detection, and semantic segmentation tasks, ViM2 achieves competitive performance with well-established vision models and outperforms SSM-based vision models. In time series forecasting, TSM2 achieves outstanding performance compared to state-of-the-art methods while demonstrating significantly improved computational cost. These results show that while Transformers, cross-channel attention, and MLPs are sufficient for good performance in time series forecasting, neither is necessary.

Generalist robot policies trained on large-scale, visually homogeneous datasets can be susceptible to shortcut learning, which impairs their out-of-distribution (OOD) generalization. While generative data augmentation is a common approach to introduce diversity, it presents a subtle challenge: data composition. Naively mixing real and synthetic data can corrupt the learning signal, as this process often prioritizes visual diversity at the expense of information fidelity. This paper suggests that robust generalization depends on principled, fidelity-aware data composition. We introduce Coherent Information Fidelity Tuning (CIFT), a framework that treats data composition as an optimization problem. CIFT uses a practical proxy for Information Fidelity based on the feature-space geometry of a dataset. This enables the identification of a phase transition, termed the Decoherence Point, where training stability degrades. The framework includes a generative engine, Multi-View Video Augmentation (MVAug), to synthesize a causally disentangled data spectrum for this tuning process. Applying CIFT to policy architectures such as pi_0 and Diffusion Policy improves OOD success rates by over 54\%. These results indicate that fidelity-aware composition, beyond data synthesis alone, is an important component for developing robust, general-purpose robots.

Designing sequences that satisfy multiple, often conflicting, objectives is a central challenge in therapeutic and biomolecular engineering. Existing generative frameworks largely operate in continuous spaces with single-objective guidance, while discrete approaches lack guarantees for multi-objective Pareto optimality. We introduce AReUReDi (Annealed Rectified Updates for Refining Discrete Flows), a discrete optimization algorithm with theoretical guarantees of convergence to the Pareto front. Building on Rectified Discrete Flows (ReDi), AReUReDi combines Tchebycheff scalarization, locally balanced proposals, and annealed Metropolis-Hastings updates to bias sampling toward Pareto-optimal states while preserving distributional invariance. Applied to peptide and SMILES sequence design, AReUReDi simultaneously optimizes up to five therapeutic properties (including affinity, solubility, hemolysis, half-life, and non-fouling) and outperforms both evolutionary and diffusion-based baselines. These results establish AReUReDi as a powerful, sequence-based framework for multi-property biomolecule generation.

We introduce Entropy-Guided Sequence Weighting (EGSW), a novel approach that enhances the exploration-exploitation tradeoff by dynamically assigning weights to generated outputs based on their advantage and entropy for Reinforcement Learning-based Large Language Model fine-tuning. EGSW integrates entropy regularization with advantage-based weighting to balance policy updates, enabling efficient exploration in high-dimensional state spaces. By employing temperature-scaled softmax weighting over sequences, EGSW prioritizing high-reward, high-uncertainty steps while maintaining training stability. Although originally developed to improve Group Relative Policy Optimization (GRPO) during large language model (LLM) fine-tuning, EGSW is generalizable to other reinforcement learning (RL) algorithms and can be implemented in both step-wise and trajectory-wise settings. Empirical evaluations demonstrate that EGSW enhances GRPO reasoning ability, yielding improvements in sample efficiency. Future work will explore the application of EGSW to advanced RL methodologies.

Given a set of K probability densities, we consider the multimarginal generative modeling problem of learning a joint distribution that recovers these densities as marginals. The structure of this joint distribution should identify multi-way correspondences among the prescribed marginals. We formalize an approach to this task within a generalization of the stochastic interpolant framework, leading to efficient learning algorithms built upon dynamical transport of measure. Our generative models are defined by velocity and score fields that can be characterized as the minimizers of simple quadratic objectives, and they are defined on a simplex that generalizes the time variable in the usual dynamical transport framework. The resulting transport on the simplex is influenced by all marginals, and we show that multi-way correspondences can be extracted. The identification of such correspondences has applications to style transfer, algorithmic fairness, and data decorruption. In addition, the multimarginal perspective enables an efficient algorithm for reducing the dynamical transport cost in the ordinary two-marginal setting. We demonstrate these capacities with several numerical examples.

Reinforcement Learning with Verifiable Rewards (RLVR) has emerged as a critical method for enhancing the reasoning capabilities of Large Language Models (LLMs). However, continuous training often leads to policy entropy collapse, characterized by a rapid decay in entropy that results in premature overconfidence, reduced output diversity, and vanishing gradient norms that inhibit learning. Gradient-Preserving Clipping is a primary factor influencing these dynamics, but existing mitigation strategies are largely static and lack a framework connecting clipping mechanisms to precise entropy control. This paper proposes reshaping entropy control in RL from the perspective of Gradient-Preserving Clipping. We first theoretically and empirically verify the contributions of specific importance sampling ratio regions to entropy growth and reduction. Leveraging these findings, we introduce a novel regulation mechanism using dynamic clipping threshold to precisely manage entropy. Furthermore, we design and evaluate dynamic entropy control strategies, including increase-then-decrease, decrease-increase-decrease, and oscillatory decay. Experimental results demonstrate that these strategies effectively mitigate entropy collapse, and achieve superior performance across multiple benchmarks.

In this work, we explore uncertainty estimation as a proxy for correctness in LLM-generated code. To this end, we adapt two state-of-the-art techniques from natural language generation -- one based on entropy and another on mutual information -- to the domain of code generation. Given the distinct semantic properties of code, we introduce modifications, including a semantic equivalence check based on symbolic execution. Our findings indicate a strong correlation between the uncertainty computed through these techniques and correctness, highlighting the potential of uncertainty estimation for quality assessment. Additionally, we propose a simplified version of the entropy-based method that assumes a uniform distribution over the LLM's responses, demonstrating comparable effectiveness. Using these techniques, we develop an abstention policy that prevents the model from making predictions when uncertainty is high, reducing incorrect outputs to near zero. Our evaluation on the LiveCodeBench shows that our approach significantly outperforms a baseline relying solely on LLM-reported log-probabilities.

Multi-layer perceptron (MLP) is a fundamental component of deep learning that has been extensively employed for various problems. However, recent empirical successes in MLP-based architectures, particularly the progress of the MLP-Mixer, have revealed that there is still hidden potential in improving MLPs to achieve better performance. In this study, we reveal that the MLP-Mixer works effectively as a wide MLP with certain sparse weights. Initially, we clarify that the mixing layer of the Mixer has an effective expression as a wider MLP whose weights are sparse and represented by the Kronecker product. This expression naturally defines a permuted-Kronecker (PK) family, which can be regarded as a general class of mixing layers and is also regarded as an approximation of Monarch matrices. Subsequently, because the PK family effectively constitutes a wide MLP with sparse weights, one can apply the hypothesis proposed by Golubeva, Neyshabur and Gur-Ari (2021) that the prediction performance improves as the width (sparsity) increases when the number of weights is fixed. We empirically verify this hypothesis by maximizing the effective width of the MLP-Mixer, which enables us to determine the appropriate size of the mixing layers quantitatively.

People enjoy food photography because they appreciate food. Behind each meal there is a story described in a complex recipe and, unfortunately, by simply looking at a food image we do not have access to its preparation process. Therefore, in this paper we introduce an inverse cooking system that recreates cooking recipes given food images. Our system predicts ingredients as sets by means of a novel architecture, modeling their dependencies without imposing any order, and then generates cooking instructions by attending to both image and its inferred ingredients simultaneously. We extensively evaluate the whole system on the large-scale Recipe1M dataset and show that (1) we improve performance w.r.t. previous baselines for ingredient prediction; (2) we are able to obtain high quality recipes by leveraging both image and ingredients; (3) our system is able to produce more compelling recipes than retrieval-based approaches according to human judgment. We make code and models publicly available.

Normalizing flows can transform a simple prior probability distribution into a more complex target distribution. Here, we evaluate the ability and efficiency of generative machine learning methods to sample the Boltzmann distribution of an atomistic model for glass-forming liquids. This is a notoriously difficult task, as it amounts to ergodically exploring the complex free energy landscape of a disordered and frustrated many-body system. We optimize a normalizing flow model to successfully transform high-temperature configurations of a dense liquid into low-temperature ones, near the glass transition. We perform a detailed comparative analysis with established enhanced sampling techniques developed in the physics literature to assess and rank the performance of normalizing flows against state-of-the-art algorithms. We demonstrate that machine learning methods are very promising, showing a large speedup over conventional molecular dynamics. Normalizing flows show performances comparable to parallel tempering and population annealing, while still falling far behind the swap Monte Carlo algorithm. Our study highlights the potential of generative machine learning models in scientific computing for complex systems, but also points to some of its current limitations and the need for further improvement.

Generative models inspired by dynamical transport of measure -- such as flows and diffusions -- construct a continuous-time map between two probability densities. Conventionally, one of these is the target density, only accessible through samples, while the other is taken as a simple base density that is data-agnostic. In this work, using the framework of stochastic interpolants, we formalize how to couple the base and the target densities. This enables us to incorporate information about class labels or continuous embeddings to construct dynamical transport maps that serve as conditional generative models. We show that these transport maps can be learned by solving a simple square loss regression problem analogous to the standard independent setting. We demonstrate the usefulness of constructing dependent couplings in practice through experiments in super-resolution and in-painting.

High-performance learned image compression codecs require flexible probability models to fit latent representations. Gaussian Mixture Models (GMMs) were proposed to satisfy this demand, but suffer from a significant runtime performance bottleneck due to the large Cumulative Distribution Function (CDF) tables that must be built for rANS coding. This paper introduces a fast coding algorithm that entirely eliminates this bottleneck. By leveraging the CDF's monotonic property, our decoder performs a dynamic binary search to find the correct symbol, eliminating the need for costly table construction and lookup. Aided by SIMD optimizations and numerical approximations, our approach accelerates the GMM entropy coding process by up to approximately 90x without compromising rate-distortion performance, significantly improving the practicality of GMM-based codecs. The implementation will be made publicly available at https://github.com/tokkiwa/FlashGMM.

Reinforcement learning (RL) is a pivotal task for enhancing Large Language Model (LLM) reasoning. Conventional algorithms, however, typically adhere to a coarse-grained credit assignment paradigm, applying a uniform reward to all tokens in a sequence, a critical flaw in long-chain reasoning tasks. In this paper, we address this challenge and propose Dynamic Entropy Weighting, a novel mechanism that facilitates fine-grained rewards through two new algorithms: Group Token Policy Optimization (GTPO), which assigns an entropy-weighted reward to each token, and the analogous algorithm Sequence-Level GRPO (GRPO-S). Our approach is founded on the hypothesis that high policy entropy within a reasoning path is a powerful heuristic for cognitive effort at pivotal junctures, which can be repurposed into a learning signal. By repurposing policy entropy for reward shaping, we achieve true per-token credit assignment. Experimental results across challenging reasoning benchmarks validate the superiority of our approach, showing our methods significantly outperform a strong DAPO baseline and confirming our entropy-weighting mechanism as the key driver of this performance boost.

Recently, Agentic Reinforcement Learning (Agentic RL) has made significant progress in incentivizing the multi-turn, long-horizon tool-use capabilities of web agents. While mainstream agentic RL algorithms autonomously explore high-uncertainty tool-call steps under the guidance of entropy, excessive reliance on entropy signals can impose further constraints, leading to the training collapse. In this paper, we delve into the challenges caused by entropy and propose the Agentic Entropy-Balanced Policy Optimization (AEPO), an agentic RL algorithm designed to balance entropy in both the rollout and policy update phases. AEPO comprises two core components: (1) a dynamic entropy-balanced rollout mechanism that adaptively allocate global and branch sampling budget through entropy pre-monitoring, while imposing a branch penalty on consecutive high-entropy tool-call steps to prevent over-branching issues; and (2) Entropy-Balanced Policy Optimization that inserts a stop-gradient operation into the high-entropy clipping term to preserve and properly rescale gradients on high-entropy tokens, while incorporating entropy-aware advantage estimation to prioritize learning on high-uncertainty tokens. Results across 14 challenging datasets show that AEPO consistently outperforms 7 mainstream RL algorithms. With just 1K RL samples, Qwen3-14B with AEPO achieves impressive results: 47.6% on GAIA, 11.2% on Humanity's Last Exam, and 43.0% on WebWalker for Pass@1; 65.0% on GAIA, 26.0% on Humanity's Last Exam, and 70.0% on WebWalker for Pass@5. Further analysis reveals that AEPO improves rollout sampling diversity while maintaining stable policy entropy, facilitating scalable web agent training.

MLP-Mixer has newly appeared as a new challenger against the realm of CNNs and transformer. Despite its simplicity compared to transformer, the concept of channel-mixing MLPs and token-mixing MLPs achieves noticeable performance in visual recognition tasks. Unlike images, point clouds are inherently sparse, unordered and irregular, which limits the direct use of MLP-Mixer for point cloud understanding. In this paper, we propose PointMixer, a universal point set operator that facilitates information sharing among unstructured 3D points. By simply replacing token-mixing MLPs with a softmax function, PointMixer can "mix" features within/between point sets. By doing so, PointMixer can be broadly used in the network as inter-set mixing, intra-set mixing, and pyramid mixing. Extensive experiments show the competitive or superior performance of PointMixer in semantic segmentation, classification, and point reconstruction against transformer-based methods.

Learning the distribution of data on Riemannian manifolds is crucial for modeling data from non-Euclidean space, which is required by many applications in diverse scientific fields. Yet, existing generative models on manifolds suffer from expensive divergence computation or rely on approximations of heat kernel. These limitations restrict their applicability to simple geometries and hinder scalability to high dimensions. In this work, we introduce the Riemannian Diffusion Mixture, a principled framework for building a generative diffusion process on manifolds. Instead of following the denoising approach of previous diffusion models, we construct a diffusion process using a mixture of bridge processes derived on general manifolds without requiring heat kernel estimations. We develop a geometric understanding of the mixture process, deriving the drift as a weighted mean of tangent directions to the data points that guides the process toward the data distribution. We further propose a scalable training objective for learning the mixture process that readily applies to general manifolds. Our method achieves superior performance on diverse manifolds with dramatically reduced number of in-training simulation steps for general manifolds.

Subsampling is commonly used to mitigate costs associated with data acquisition, such as time or energy requirements, motivating the development of algorithms for estimating the fully-sampled signal of interest x from partially observed measurements y. In maximum-entropy sampling, one selects measurement locations that are expected to have the highest entropy, so as to minimize uncertainty about x. This approach relies on an accurate model of the posterior distribution over future measurements, given the measurements observed so far. Recently, diffusion models have been shown to produce high-quality posterior samples of high-dimensional signals using guided diffusion. In this work, we propose Active Diffusion Subsampling (ADS), a method for performing active subsampling using guided diffusion in which the model tracks a distribution of beliefs over the true state of x throughout the reverse diffusion process, progressively decreasing its uncertainty by choosing to acquire measurements with maximum expected entropy, and ultimately generating the posterior distribution p(x | y). ADS can be applied using pre-trained diffusion models for any subsampling rate, and does not require task-specific retraining - just the specification of a measurement model. Furthermore, the maximum entropy sampling policy employed by ADS is interpretable, enhancing transparency relative to existing methods using black-box policies. Experimentally, we show that ADS outperforms fixed sampling strategies, and study an application of ADS in Magnetic Resonance Imaging acceleration using the fastMRI dataset, finding that ADS performs competitively with supervised methods. Code available at https://active-diffusion-subsampling.github.io/.

This monograph presents the core principles that have guided the development of diffusion models, tracing their origins and showing how diverse formulations arise from shared mathematical ideas. Diffusion modeling starts by defining a forward process that gradually corrupts data into noise, linking the data distribution to a simple prior through a continuum of intermediate distributions. The goal is to learn a reverse process that transforms noise back into data while recovering the same intermediates. We describe three complementary views. The variational view, inspired by variational autoencoders, sees diffusion as learning to remove noise step by step. The score-based view, rooted in energy-based modeling, learns the gradient of the evolving data distribution, indicating how to nudge samples toward more likely regions. The flow-based view, related to normalizing flows, treats generation as following a smooth path that moves samples from noise to data under a learned velocity field. These perspectives share a common backbone: a time-dependent velocity field whose flow transports a simple prior to the data. Sampling then amounts to solving a differential equation that evolves noise into data along a continuous trajectory. On this foundation, the monograph discusses guidance for controllable generation, efficient numerical solvers, and diffusion-motivated flow-map models that learn direct mappings between arbitrary times. It provides a conceptual and mathematically grounded understanding of diffusion models for readers with basic deep-learning knowledge.

Taming the generation outcome of state of the art Diffusion and Flow-Matching (FM) models without having to re-train a task-specific model unlocks a powerful tool for solving inverse problems, conditional generation, and controlled generation in general. In this work we introduce D-Flow, a simple framework for controlling the generation process by differentiating through the flow, optimizing for the source (noise) point. We motivate this framework by our key observation stating that for Diffusion/FM models trained with Gaussian probability paths, differentiating through the generation process projects gradient on the data manifold, implicitly injecting the prior into the optimization process. We validate our framework on linear and non-linear controlled generation problems including: image and audio inverse problems and conditional molecule generation reaching state of the art performance across all.

Hybrid training methods for large language models combine supervised fine tuning (SFT) on expert demonstrations with reinforcement learning (RL) on model rollouts, typically at the sample level. We propose Entropy Gated Selective Policy Optimization (EGSPO), a three stage framework that extends sample level mixing with token level gradient modulation. Stage 1, SFT expert learning, establishes a reliable warm up policy using expert demonstrations with a pure SFT loss. Stage 2, RL rollout generation, samples trajectories from the current policy and computes per token predictive entropy. Stage 3, the EGSPO mechanism, applies entropy gated gradient allocation: a predictive entropy module routes high entropy tokens to full PPO updates to encourage exploration, and low entropy tokens to attenuated PPO updates to reduce variance and preserve knowledge. Critically, both branches incorporate the advantage function A_t, ensuring that incorrect trajectories receive consistent negative learning signals and preventing reinforcement of confident errors. EGSPO achieves consistent improvements on mathematical reasoning benchmarks, with gains of 3.8 percent on AIME and 2.9 percent on MATH over the CHORD phi baseline, while incurring only 3.4 percent additional computational overhead.

Existing pretraining data mixing methods for large language models (LLMs) typically follow a domain-wise methodology, a top-down process that first determines domain weights and then performs uniform data sampling across each domain. However, these approaches neglect significant inter-domain overlaps and commonalities, failing to control the global diversity of the constructed training dataset. Further, uniform sampling within domains ignores fine-grained sample-specific features, potentially leading to suboptimal data distribution. To address these shortcomings, we propose a novel sample-wise data mixture approach based on a bottom-up paradigm. This method performs global cross-domain sampling by systematically evaluating the quality and diversity of each sample, thereby dynamically determining the optimal domain distribution. Comprehensive experiments across multiple downstream tasks and perplexity assessments demonstrate that SampleMix surpasses existing domain-based methods. Meanwhile, SampleMix requires 1.4x to 2.1x training steps to achieves the baselines' performance, highlighting the substantial potential of SampleMix to optimize pre-training data.

Diffusion models have achieved remarkable generation quality, but they suffer from significant inference cost due to their reliance on multiple sequential denoising steps, motivating recent efforts to distill this inference process into a few-step regime. However, existing distillation methods typically approximate the teacher trajectory by using linear shortcuts, which makes it difficult to match its constantly changing tangent directions as velocities evolve across timesteps, thereby leading to quality degradation. To address this limitation, we propose ArcFlow, a few-step distillation framework that explicitly employs non-linear flow trajectories to approximate pre-trained teacher trajectories. Concretely, ArcFlow parameterizes the velocity field underlying the inference trajectory as a mixture of continuous momentum processes. This enables ArcFlow to capture velocity evolution and extrapolate coherent velocities to form a continuous non-linear trajectory within each denoising step. Importantly, this parameterization admits an analytical integration of this non-linear trajectory, which circumvents numerical discretization errors and results in high-precision approximation of the teacher trajectory. To train this parameterization into a few-step generator, we implement ArcFlow via trajectory distillation on pre-trained teacher models using lightweight adapters. This strategy ensures fast, stable convergence while preserving generative diversity and quality. Built on large-scale models (Qwen-Image-20B and FLUX.1-dev), ArcFlow only fine-tunes on less than 5% of original parameters and achieves a 40x speedup with 2 NFEs over the original multi-step teachers without significant quality degradation. Experiments on benchmarks show the effectiveness of ArcFlow both qualitatively and quantitatively.

Time series forecasting is widely used in extensive applications, such as traffic planning and weather forecasting. However, real-world time series usually present intricate temporal variations, making forecasting extremely challenging. Going beyond the mainstream paradigms of plain decomposition and multiperiodicity analysis, we analyze temporal variations in a novel view of multiscale-mixing, which is based on an intuitive but important observation that time series present distinct patterns in different sampling scales. The microscopic and the macroscopic information are reflected in fine and coarse scales respectively, and thereby complex variations can be inherently disentangled. Based on this observation, we propose TimeMixer as a fully MLP-based architecture with Past-Decomposable-Mixing (PDM) and Future-Multipredictor-Mixing (FMM) blocks to take full advantage of disentangled multiscale series in both past extraction and future prediction phases. Concretely, PDM applies the decomposition to multiscale series and further mixes the decomposed seasonal and trend components in fine-to-coarse and coarse-to-fine directions separately, which successively aggregates the microscopic seasonal and macroscopic trend information. FMM further ensembles multiple predictors to utilize complementary forecasting capabilities in multiscale observations. Consequently, TimeMixer is able to achieve consistent state-of-the-art performances in both long-term and short-term forecasting tasks with favorable run-time efficiency.

Recently, Large Language Models (LLMs) have demonstrated outstanding performance across a wide range of downstream language tasks. Temperature sampling is a commonly used decoding strategy for LLMs' generation process. However, a fixed temperature parameter is used in most cases, which may not always be an optimal choice for balancing generation quality and diversity. In this paper, we propose an effective Entropy-based Dynamic Temperature (EDT) Sampling method, to achieve a more balanced performance in terms of both generation quality and diversity by dynamically selecting the temperature parameter. Additionally, we also show model performance and comprehensive analyses for 4 different generation benchmarks. Our experiments show that EDT significantly outperforms the existing strategies across different tasks.

Doubly-stochastic matrices (DSM) are increasingly utilized in structure-preserving deep architectures -- such as Optimal Transport layers and Sinkhorn-based attention -- to enforce numerical stability and probabilistic interpretability. In this work, we identify a critical spectral degradation phenomenon inherent to these constraints, termed the Homogeneity Trap. We demonstrate that the maximum-entropy bias, typical of Sinkhorn-based projections, drives the mixing operator towards the uniform barycenter, thereby suppressing the subdominant singular value σ_2 and filtering out high-frequency feature components. We derive a spectral bound linking σ_2 to the network's effective depth, showing that high-entropy constraints restrict feature transformation to a shallow effective receptive field. Furthermore, we formally demonstrate that Layer Normalization fails to mitigate this collapse in noise-dominated regimes; specifically, when spectral filtering degrades the Signal-to-Noise Ratio (SNR) below a critical threshold, geometric structure is irreversibly lost to noise-induced orthogonal collapse. Our findings highlight a fundamental trade-off between entropic stability and spectral expressivity in DSM-constrained networks.

Diffusion models excel in high-quality generation but suffer from slow inference due to iterative sampling. While recent methods have successfully transformed diffusion models into one-step generators, they neglect model size reduction, limiting their applicability in compute-constrained scenarios. This paper aims to develop small, efficient one-step diffusion models based on the powerful rectified flow framework, by exploring joint compression of inference steps and model size. The rectified flow framework trains one-step generative models using two operations, reflow and distillation. Compared with the original framework, squeezing the model size brings two new challenges: (1) the initialization mismatch between large teachers and small students during reflow; (2) the underperformance of naive distillation on small student models. To overcome these issues, we propose Annealing Reflow and Flow-Guided Distillation, which together comprise our SlimFlow framework. With our novel framework, we train a one-step diffusion model with an FID of 5.02 and 15.7M parameters, outperforming the previous state-of-the-art one-step diffusion model (FID=6.47, 19.4M parameters) on CIFAR10. On ImageNet 64times64 and FFHQ 64times64, our method yields small one-step diffusion models that are comparable to larger models, showcasing the effectiveness of our method in creating compact, efficient one-step diffusion models.

The rapid development of generative models has significantly advanced image and video applications. Among these, video creation, aimed at generating videos under various conditions, has gained substantial attention. However, existing video creation models either focus solely on a few specific conditions or suffer from excessively long generation times due to complex model inference, making them impractical for real-world applications. To mitigate these issues, we propose an efficient unified video creation model, named VDOT. Concretely, we model the training process with the distribution matching distillation (DMD) paradigm. Instead of using the Kullback-Leibler (KL) minimization, we additionally employ a novel computational optimal transport (OT) technique to optimize the discrepancy between the real and fake score distributions. The OT distance inherently imposes geometric constraints, mitigating potential zero-forcing or gradient collapse issues that may arise during KL-based distillation within the few-step generation scenario, and thus, enhances the efficiency and stability of the distillation process. Further, we integrate a discriminator to enable the model to perceive real video data, thereby enhancing the quality of generated videos. To support training unified video creation models, we propose a fully automated pipeline for video data annotation and filtering that accommodates multiple video creation tasks. Meanwhile, we curate a unified testing benchmark, UVCBench, to standardize evaluation. Experiments demonstrate that our 4-step VDOT outperforms or matches other baselines with 100 denoising steps.

Distribution Matching Distillation (DMD) distills a pre-trained multi-step diffusion model to a few-step one to improve inference efficiency. However, the performance of the latter is often capped by the former. To circumvent this dilemma, we propose DMDR, a novel framework that combines Reinforcement Learning (RL) techniques into the distillation process. We show that for the RL of the few-step generator, the DMD loss itself is a more effective regularization compared to the traditional ones. In turn, RL can help to guide the mode coverage process in DMD more effectively. These allow us to unlock the capacity of the few-step generator by conducting distillation and RL simultaneously. Meanwhile, we design the dynamic distribution guidance and dynamic renoise sampling training strategies to improve the initial distillation process. The experiments demonstrate that DMDR can achieve leading visual quality, prompt coherence among few-step methods, and even exhibit performance that exceeds the multi-step teacher.

We propose using Normalizing Flows as a trainable kernel within the molecular dynamics update of Hamiltonian Monte Carlo (HMC). By learning (invertible) transformations that simplify our dynamics, we can outperform traditional methods at generating independent configurations. We show that, using a carefully constructed network architecture, our approach can be easily scaled to large lattice volumes with minimal retraining effort. The source code for our implementation is publicly available online at https://github.com/nftqcd/fthmc.

We present an algorithm for learning mixtures of Markov chains and Markov decision processes (MDPs) from short unlabeled trajectories. Specifically, our method handles mixtures of Markov chains with optional control input by going through a multi-step process, involving (1) a subspace estimation step, (2) spectral clustering of trajectories using "pairwise distance estimators," along with refinement using the EM algorithm, (3) a model estimation step, and (4) a classification step for predicting labels of new trajectories. We provide end-to-end performance guarantees, where we only explicitly require the length of trajectories to be linear in the number of states and the number of trajectories to be linear in a mixing time parameter. Experimental results support these guarantees, where we attain 96.6% average accuracy on a mixture of two MDPs in gridworld, outperforming the EM algorithm with random initialization (73.2% average accuracy).

Training-free guided sampling in diffusion models leverages off-the-shelf pre-trained networks, such as an aesthetic evaluation model, to guide the generation process. Current training-free guided sampling algorithms obtain the guidance energy function based on a one-step estimate of the clean image. However, since the off-the-shelf pre-trained networks are trained on clean images, the one-step estimation procedure of the clean image may be inaccurate, especially in the early stages of the generation process in diffusion models. This causes the guidance in the early time steps to be inaccurate. To overcome this problem, we propose Symplectic Adjoint Guidance (SAG), which calculates the gradient guidance in two inner stages. Firstly, SAG estimates the clean image via n function calls, where n serves as a flexible hyperparameter that can be tailored to meet specific image quality requirements. Secondly, SAG uses the symplectic adjoint method to obtain the gradients accurately and efficiently in terms of the memory requirements. Extensive experiments demonstrate that SAG generates images with higher qualities compared to the baselines in both guided image and video generation tasks.