# Convergence rates for optimised adaptive importance samplers

## TL;DR

This paper introduces optimised adaptive importance samplers (OAIS) that use convex optimisation to improve convergence rates in estimating expectations, providing non-asymptotic error bounds and analyzing their performance over iterations.

## Contribution

The paper develops a new class of adaptive importance samplers based on convex optimisation that achieve explicit convergence rates and error bounds, especially when the target is in the exponential family.

## Key findings

- Convergence rate of $\mathcal{O}(1/\sqrt{N})$ for $L_2$ errors when target is in exponential family.
- Explicit non-asymptotic error bounds depending on iterations and samples.
- Asymptotic error increases by a factor related to the $\chi^2$-divergence when target is outside exponential family.

## Abstract

Adaptive importance samplers are adaptive Monte Carlo algorithms to estimate expectations with respect to some target distribution which \textit{adapt} themselves to obtain better estimators over a sequence of iterations. Although it is straightforward to show that they have the same $\mathcal{O}(1/\sqrt{N})$ convergence rate as standard importance samplers, where $N$ is the number of Monte Carlo samples, the behaviour of adaptive importance samplers over the number of iterations has been left relatively unexplored. In this work, we investigate an adaptation strategy based on convex optimisation which leads to a class of adaptive importance samplers termed \textit{optimised adaptive importance samplers} (OAIS). These samplers rely on the iterative minimisation of the $\chi^2$-divergence between an exponential-family proposal and the target. The analysed algorithms are closely related to the class of adaptive importance samplers which minimise the variance of the weight function. We first prove non-asymptotic error bounds for the mean squared errors (MSEs) of these algorithms, which explicitly depend on the number of iterations and the number of samples together. The non-asymptotic bounds derived in this paper imply that when the target belongs to the exponential family, the $L_2$ errors of the optimised samplers converge to the optimal rate of $\mathcal{O}(1/\sqrt{N})$ and the rate of convergence in the number of iterations are explicitly provided. When the target does not belong to the exponential family, the rate of convergence is the same but the asymptotic $L_2$ error increases by a factor $\sqrt{\rho^\star} > 1$, where $\rho^\star - 1$ is the minimum $\chi^2$-divergence between the target and an exponential-family proposal.