Global convergence of optimized adaptive importance samplers

TL;DR

This paper presents a new analysis of optimized adaptive importance samplers (OAIS), providing theoretical guarantees for their convergence and performance in Monte Carlo integration with general proposals.

Contribution

It introduces a novel scheme for globally optimizing the chi-squared divergence in importance sampling, extending previous results to general proposals using nonasymptotic bounds.

Findings

Achieves uniform-in-time theoretical guarantees for AIS schemes.

Develops a scheme utilizing stochastic gradient Langevin dynamics for divergence optimization.

Provides nonasymptotic bounds for mean-squared error in importance sampling.

Abstract

We analyze the optimized adaptive importance sampler (OAIS) for performing Monte Carlo integration with general proposals. We leverage a classical result which shows that the bias and the mean-squared error (MSE) of the importance sampling scales with the $\chi^2$-divergence between the target and the proposal and develop a scheme which performs global optimization of $\chi^2$-divergence. While it is known that this quantity is convex for exponential family proposals, the case of the general proposals has been an open problem. We close this gap by utilizing the nonasymptotic bounds for stochastic gradient Langevin dynamics (SGLD) for the global optimization of $\chi^2$-divergence and derive nonasymptotic bounds for the MSE by leveraging recent results from non-convex optimization literature. The resulting AIS schemes have explicit theoretical guarantees that are uniform-in-time.