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Diffusion models are now the de facto approach to generative modeling across a wide range of data modalities including images, audio, videos, and visuomotor policies. In recent years, there has been a surge of interest in building the mathematical...
A sequence of sparse weighted graphs G_N indexed by the number of vertices
N is said to be left-convergent if all (suitably weighted) subgraph counts
converge to a limit as N diverges. This notion generalizes in a
natural way Benjamini-Schramm's definition of local-weak convergence.
A broad research agenda aims at determining which `global' graph properties
determined by left or local-weak convergence (in other words, which
of these properties are in fact local).
We prove that, under a spectral condition on the weighted adjacency matrices
A_N of graph G_N, the free energy density of the Ising model on
this weighted graph is continuous in the left convergence topology
(hence is a local function.)
We use stochastic dynamics as a proof device.
[Based on joint work with Michael Ren]
We present an algorithmic framework for high-accuracy sampling using unbiased stochastic queries, with two main applications:
Algorithms for diffusion model sampling which obtain $\delta$-error in $\mathrm{polylog}(1/\delta)$ steps, given access to $\widetilde O(\delta)$-accurate score estimates in $L^2$. This is an exponential improvement over all previous results. (ICML 2026, https://arxiv.org/abs/2602.01338)
High-accuracy guarantees for log-concave sampling -- that is, iteration and query complexities which scale as $\mathrm{polylog}(1/\delta)$, where $\delta$ is the desired target accuracy -- are achievable using stochastic gradients with subexponential tails. Notably, this exhibits a separation with the problem of convex optimization, where stochasticity (even additive Gaussian noise) in the gradient oracle incurs $\mathrm{poly}(1/\delta)$ queries. (CoLT 2026, https://arxiv.org/abs/2602.14342)
Reinforcement learning has become a central tool for post-training generative models, especially large language models. Recently, there has been growing interest in applying RL to diffusion models. However, directly applying LLM-style RL along the denoising trajectory is inefficient and memory-intensive. In this talk, I will discuss a systematic study of the RL design space for diffusion models, including policy optimization objectives, likelihood estimation, and rollout schemes. Our key finding is that replacing trajectory-based likelihood estimation with an ELBO-based estimator induced by the linear forward process is crucial for efficiency. This also decouples rollout from training, enabling fast ODE samplers. Based on these insights, we develop RL algorithms for diffusion fine-tuning that are an order of magnitude faster than trajectory-based methods. Time permitting, I will also discuss a related stochastic-control approach based on first-order optimization.