Convergence rates for optimised adaptive importance samplers
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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 theoretical guarantees with importance sampling with respect to the sample size, their behaviour over the number of iterations has been relatively left unexplored despite these adaptive algorithms aim at improving the proposal quality over time. 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 χ^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 O(1/√(N)). We also show that when the target is not from the exponential family, the asymptotic error rate is O(√(ρ^/N)) where ρ^ is the minimum χ^2-divergence between the target and an exponential family proposal.
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