Probabilistic generative models based on measure transport, such as diffusion and flow-based models, are often formulated in the language of Markovian stochastic dynamics, where the choice of the underlying process impacts both algorithmic design choices and theoretical analysis. In this paper, we aim to establish a rigorous mathematical foundation for denoising Markov models, a broad class of generative models that postulate a forward process transitioning from the target distribution to a simple, easy-to-sample distribution, alongside a backward process particularly constructed to enable efficient sampling in the reverse direction. Leveraging deep connections with nonequilibrium statistical mechanics and generalized Doob's $h$-transform, we propose a minimal set of assumptions that ensure: (1) explicit construction of the backward generator, (2) a unified variational objective directly minimizing the measure transport discrepancy, and (3) adaptations of the classical score-matching approach across diverse dynamics. Our framework unifies existing formulations of continuous and discrete diffusion models, identifies the most general form of denoising Markov models under certain regularity assumptions on forward generators, and provides a systematic recipe for designing denoising Markov models driven by arbitrary L\'evy-type processes. We illustrate the versatility and practical effectiveness of our approach through novel denoising Markov models employing geometric Brownian motion and jump processes as forward dynamics, highlighting the framework's potential flexibility and capability in modeling complex distributions.

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

With the emergence of large-scale pre-trained neural networks, methods to adapt such "foundation" models to data-limited downstream tasks have become a necessity. Fine-tuning, preference optimization, and transfer learning have all been successfully employed for these purposes when the target task closely resembles the source task, but a precise theoretical understanding of "task similarity" is still lacking. While conventional wisdom suggests that simple measures of similarity between source and target distributions, such as $\phi$-divergences or integral probability metrics, can directly predict the success of transfer, we prove the surprising fact that, in general, this is not the case. We adopt, instead, a feature-centric viewpoint on transfer learning and establish a number of theoretical results that demonstrate that when the target task is well represented by the feature space of the pre-trained model, transfer learning outperforms training from scratch. We study deep linear networks as a minimal model of transfer learning in which we can analytically characterize the transferability phase diagram as a function of the target dataset size and the feature space overlap. For this model, we establish rigorously that when the feature space overlap between the source and target tasks is sufficiently strong, both linear transfer and fine-tuning improve performance, especially in the low data limit. These results build on an emerging understanding of feature learning dynamics in deep linear networks, and we demonstrate numerically that the rigorous results we derive for the linear case also apply to nonlinear networks.

Discrete diffusion models have gained increasing attention for their ability to model complex distributions with tractable sampling and inference. However, the error analysis for discrete diffusion models remains less well-understood. In this work, we propose a comprehensive framework for the error analysis of discrete diffusion models based on L\'evy-type stochastic integrals. By generalizing the Poisson random measure to that with a time-independent and state-dependent intensity, we rigorously establish a stochastic integral formulation of discrete diffusion models and provide the corresponding change of measure theorems that are intriguingly analogous to It\^o integrals and Girsanov's theorem for their continuous counterparts. Our framework unifies and strengthens the current theoretical results on discrete diffusion models and obtains the first error bound for the $\tau$-leaping scheme in KL divergence. With error sources clearly identified, our analysis gives new insight into the mathematical properties of discrete diffusion models and offers guidance for the design of efficient and accurate algorithms for real-world discrete diffusion model applications.

Diffusion models have become a leading method for generativ e modeling of both image and scientific data. As these models are costly to train and evaluate, reducing the inference cost for diffusion models remains a maj or goal. Inspired by the recent empirical success in accelerating diffusion mod els via the parallel sampling technique [1], we propose to divide the sampling proce ss into O (1) blocks with parallelizable Picard iterations within each block. R igorous theoretical analysis reveals that our algorithm achieves null O (poly log d) overall time complexity, marking the first implementation with provable sub-linear complexi ty w.r .t. the data dimension d. Our analysis is based on a generalized version of Girsanov' s theorem and is compatible with both the SDE and probability fl ow ODE implementations. Our results shed light on the potential of fast a nd efficient sampling of high-dimensional data on fast-evolving modern large-me mory GPU clusters.

Searching through chemical space is an exceptionally challenging problem because the number of possible molecules grows combinatorially with the number of atoms. Large, autoregressive models trained on databases of chemical compounds have yielded powerful generators, but we still lack robust strategies for generating molecules with desired properties. This molecular search problem closely resembles the "alignment" problem for large language models, though for many chemical tasks we have a specific and easily evaluable reward function. Here, we introduce an algorithm called energy rank alignment (ERA) that leverages an explicit reward function to produce a gradient-based objective that we use to optimize autoregressive policies. We show theoretically that this algorithm is closely related to proximal policy optimization (PPO) and direct preference optimization (DPO), but has a minimizer that converges to an ideal Gibbs-Boltzmann distribution with the reward playing the role of an energy function. Furthermore, this algorithm is highly scalable, does not require reinforcement learning, and performs well relative to DPO when the number of preference observations per pairing is small. We deploy this approach to align molecular transformers to generate molecules with externally specified properties and find that it does so robustly, searching through diverse parts of chemical space. While our focus here is on chemical search, we also obtain excellent results on an AI supervised task for LLM alignment, showing that the method is scalable and general.

Inferring a diffusion equation from discretely-observed measurements is a statistical challenge of significant importance in a variety of fields, from single-molecule tracking in biophysical systems to modeling financial instruments. Assuming that the underlying dynamical process obeys a $d$-dimensional stochastic differential equation of the form $$\mathrm{d}\boldsymbol{x}_t=\boldsymbol{b}(\boldsymbol{x}_t)\mathrm{d} t+\Sigma(\boldsymbol{x}_t)\mathrm{d}\boldsymbol{w}_t,$$ we propose neural network-based estimators of both the drift $\boldsymbol{b}$ and the spatially-inhomogeneous diffusion tensor $D = \Sigma\Sigma^{T}$ and provide statistical convergence guarantees when $\boldsymbol{b}$ and $D$ are $s$-H\"older continuous. Notably, our bound aligns with the minimax optimal rate $N^{-\frac{2s}{2s+d}}$ for nonparametric function estimation even in the presence of correlation within observational data, which necessitates careful handling when establishing fast-rate generalization bounds. Our theoretical results are bolstered by numerical experiments demonstrating accurate inference of spatially-inhomogeneous diffusion tensors.

Experimental advances enabling high-resolution external control create new opportunities to produce materials with exotic properties. In this work, we investigate how a multi-agent reinforcement learning approach can be used to design external control protocols for self-assembly. We find that a fully decentralized approach performs remarkably well even with a "coarse" level of external control. More importantly, we see that a partially decentralized approach, where we include information about the local environment allows us to better control our system towards some target distribution. We explain this by analyzing our approach as a partially-observed Markov decision process. With a partially decentralized approach, the agent is able to act more presciently, both by preventing the formation of undesirable structures and by better stabilizing target structures as compared to a fully decentralized approach.

Markov Chain Monte Carlo (MCMC) algorithms (Liu, Since no data set from the target posterior distribution 2008) are nowadays the methods of choice to sample complex is available beforehand, the flow is typically posterior distributions. MCMC methods generate a trained using the reverse Kullback-Leibler (KL) sequence of configurations over which the time average of divergence that only requires samples from a base any suitable observable converges towards its ensemble average distribution. This strategy may perform poorly over some target distribution, here the posterior. This when the posterior is complicated and hard to is achieved by proposing new samples from a proposal density sample with an untrained normalizing flow. Here that is easy to sample, then accepting or rejecting them we explore a distinct training strategy, using the using a criterion that guarantees that the transition kernel of direct KL divergence as loss, in which samples the chain is in detailed balance with respect to the posterior from the posterior are generated by (i) assisting density: a popular choice is Metropolis-Hastings criterion.

Recent theoretical work has characterized the dynamics of wide shallow neural networks trained via gradient descent in an asymptotic regime called the mean-field limit as the number of parameters tends towards infinity. At initialization, the randomly sampled parameters lead to a deviation from the mean-field limit that is dictated by the classical Central Limit Theorem (CLT). However, the dynamics of training introduces correlations among the parameters, raising the question of how the fluctuations evolve during training. Here, we analyze the mean-field dynamics as a Wasserstein gradient flow and prove that the deviations from the mean-field limit scaled by the width, in the width-asymptotic limit, remain bounded throughout training. In particular, they eventually vanish in the CLT scaling if the mean-field dynamics converges to a measure that interpolates the training data. This observation has implications for both the approximation rate and the generalization: the upper bound we obtain is given by a Monte-Carlo type resampling error, which does not depend explicitly on the dimension. This bound motivates a regularizaton term on the 2-norm of the underlying measure, which is also connected to generalization via the variation-norm function spaces.