Convergence rates for optimised adaptive importance samplers

Adaptive importance samplers are adaptive Monte Carlo algorithms to estimate expectations with respect to some target distribution which adapt themselves to obtain better estimators over iterations. Although it is straightforward to show that they have the same $\mathcal{O}(1/\sqrt{N})$ convergence rate as the importance sampling 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 despite these adaptive algorithms aim at improving the proposal quality iteratively. In this work, we explore an adaptation strategy based on convex optimisation which leads to a class of adaptive importance samplers, termed optimised adaptive importance samplers (OAIS). These samplers rely on an adaptation idea based on minimizing the $\chi^2$-divergence between an exponential family proposal and the target. The analysed algorithms are closely related to the 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 particles together. The non-asymptotic bounds derived in this paper imply that when the target is from the exponential family, the $L_2$ errors of the optimised samplers converge to the perfect Monte Carlo sampling error $\mathcal{O}(1/\sqrt{N})$. We also show that when the target is not from the exponential family, the asymptotic error rate is $\mathcal{O}(\sqrt{\rho^\star/N})$ where $\rho^\star$ is the minimum $\chi^2$-divergence between the target and an exponential family proposal.