High-accuracy log-concave sampling with stochastic queries

TL;DR

This paper demonstrates that high-accuracy log-concave sampling can be achieved efficiently using stochastic gradients with subexponential tails, surpassing previous limitations in query complexity.

Contribution

It introduces a novel framework for high-accuracy log-concave sampling with stochastic gradients, showing a separation from convex optimization and establishing necessary tail conditions.

Findings

High-accuracy sampling scales as poly log(1/δ) with stochastic gradients.

Light-tailed stochastic gradients are necessary for high accuracy.

Provides improved complexity for finite-sum potential sampling.

Abstract

We show that high-accuracy guarantees for log-concave sampling -- that is, iteration and query complexities which scale as $\mathrm{poly}\log(1/\delta)$, 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. We also give an information-theoretic argument that light-tailed stochastic gradients are necessary for high accuracy: for example, in the bounded variance case, we show that the minimax-optimal query complexity scales as $\Theta(1/\delta)$. Our framework also provides similar high accuracy guarantees under stochastic zeroth order (value) queries, and an improved complexity result for sampling from finite-sum potentials.