Enhanced convergence rates of Adaptive Importance Sampling with recycling schemes via quasi-Monte Carlo methods

This article investigates the integration of quasi-Monte Carlo (QMC) methods using the Adaptive Multiple Importance Sampling (AMIS). Traditional Importance Sampling (IS) often suffers from poor performance since it heavily relies on the choice of the proposal distributions. The AMIS and the Modified version of AMIS (MAMIS) address this by iteratively refining proposal distributions and reusing all past samples through a recycling strategy. We introduce the RQMC methods into the MAMIS, achieving higher convergence rates compared to the Monte Carlo (MC) methods. Our main contributions include a detailed convergence analysis of the MAMIS estimator under randomized QMC (RQMC) sampling. Specifically, we establish the $L^q$ $(q \geq 2)$ error bound for the RQMC-based estimator using a smoothed projection method, which enables us to apply the H\"older's inequality in the error analysis of the RQMC-based MAMIS estimator. As a result, we prove that the root mean square error of the RQMC-based MAMIS estimator converges at a rate of $\mathcal{O}(\bar{N}_T^{-1+\epsilon})$, where $\bar{N}_T$ is the average number of samples used in each step over $T$ iterations, and $\epsilon > 0$ is arbitrarily small. Numerical experiments validate the effectiveness of our method, including mixtures of Gaussians, a banana-shaped model, and Bayesian Logistic regression.