Matthew S. Zhang

Yunbum Kook, Matthew S. Zhang

Uniform sampling over a convex body is a fundamental algorithmic problem, yet the convergence in KL or Rényi divergence of most samplers remains poorly understood. In this work, we propose a constrained proximal sampler, a principled and simple algorithm that possesses elegant convergence guarantees. Leveraging the uniform ergodicity of this sampler, we show that it converges in the Rényi-infinity divergence ($\mathcal R_\infty$) with no query complexity overhead when starting from a warm start. This is the strongest of commonly considered performance metrics, implying rates in $\{\mathcal R_q, \mathsf{KL}\}$ convergence as special cases. By applying this sampler within an annealing scheme, we propose an algorithm which can approximately sample $\varepsilon$-close to the uniform distribution on convex bodies in $\mathcal R_\infty$-divergence with $\widetilde{\mathcal{O}}(d^3\, \text{polylog} \frac{1}{\varepsilon})$ query complexity. This improves on all prior results in $\{\mathcal R_q, \mathsf{KL}\}$-divergences, without resorting to any algorithmic modifications or post-processing of the sample. It also matches the prior best known complexity in total variation distance.

Yunbum Kook, Santosh S. Vempala, Matthew S. Zhang

We present a new random walk for uniformly sampling high-dimensional convex bodies. It achieves state-of-the-art runtime complexity with stronger guarantees on the output than previously known, namely in Rényi divergence (which implies TV, $\mathcal{W}_2$, KL, $χ^2$). The proof departs from known approaches for polytime algorithms for the problem -- we utilize a stochastic diffusion perspective to show contraction to the target distribution with the rate of convergence determined by functional isoperimetric constants of the target distribution.

Yunbum Kook, Matthew S. Zhang, Sinho Chewi et al. · princeton, utoronto

We study the complexity of sampling from the stationary distribution of a mean-field SDE, or equivalently, the complexity of minimizing a functional over the space of probability measures which includes an interaction term. Our main insight is to decouple the two key aspects of this problem: (1) approximation of the mean-field SDE via a finite-particle system, via uniform-in-time propagation of chaos, and (2) sampling from the finite-particle stationary distribution, via standard log-concave samplers. Our approach is conceptually simpler and its flexibility allows for incorporating the state-of-the-art for both algorithms and theory. This leads to improved guarantees in numerous settings, including better guarantees for optimizing certain two-layer neural networks in the mean-field regime. A key technical contribution is to establish a new uniform-in-$N$ log-Sobolev inequality for the stationary distribution of the mean-field Langevin dynamics.

Yunbum Kook, Matthew S. Zhang

We investigate the complexity of covariance matrix estimation for Gibbs distributions based on dependent samples from a Markov chain. We show that when $π$ satisfies a Poincaré inequality and the chain possesses a spectral gap, we can achieve similar sample complexity using MCMC as compared to an estimator constructed using i.i.d. samples, with potentially much better query complexity. As an application of our methods, we show improvements for the query complexity in both constrained and unconstrained settings for concrete instances of MCMC. In particular, we provide guarantees regarding isotropic rounding procedures for sampling uniformly on convex bodies.

Matthew S. Zhang, Murat A. Erdogdu, Animesh Garg

Policy gradient methods have been frequently applied to problems in control and reinforcement learning with great success, yet existing convergence analysis still relies on non-intuitive, impractical and often opaque conditions. In particular, existing rates are achieved in limited settings, under strict regularity conditions. In this work, we establish explicit convergence rates of policy gradient methods, extending the convergence regime to weakly smooth policy classes with $L_2$ integrable gradient. We provide intuitive examples to illustrate the insight behind these new conditions. Notably, our analysis also shows that convergence rates are achievable for both the standard policy gradient and the natural policy gradient algorithms under these assumptions. Lastly we provide performance guarantees for the converged policies.

Murat A. Erdogdu, Rasa Hosseinzadeh, Matthew S. Zhang

We study sampling from a target distribution $ν_* = e^{-f}$ using the unadjusted Langevin Monte Carlo (LMC) algorithm when the potential $f$ satisfies a strong dissipativity condition and it is first-order smooth with a Lipschitz gradient. We prove that, initialized with a Gaussian random vector that has sufficiently small variance, iterating the LMC algorithm for $\widetilde{\mathcal{O}}(λ^2 dε^{-1})$ steps is sufficient to reach $ε$-neighborhood of the target in both Chi-squared and Renyi divergence, where $λ$ is the logarithmic Sobolev constant of $ν_*$. Our results do not require warm-start to deal with the exponential dimension dependency in Chi-squared divergence at initialization. In particular, for strongly convex and first-order smooth potentials, we show that the LMC algorithm achieves the rate estimate $\widetilde{\mathcal{O}}(dε^{-1})$ which improves the previously known rates in both of these metrics, under the same assumptions. Translating this rate to other metrics, our results also recover the state-of-the-art rate estimates in KL divergence, total variation and $2$-Wasserstein distance in the same setup. Finally, as we rely on the logarithmic Sobolev inequality, our framework covers a range of non-convex potentials that are first-order smooth and exhibit strong convexity outside of a compact region.