Entropic Mirror Monte Carlo

Importance sampling is a Monte Carlo method which designs estimators of expectations under a target distribution using weighted samples from a proposal distribution. When the target distribution is complex, such as multimodal distributions in high-dimensional spaces, the efficiency of importance sampling critically depends on the choice of the proposal distribution. In this paper, we propose a novel adaptive scheme for the construction of efficient proposal distributions. Our algorithm promotes efficient exploration of the target distribution by combining global sampling mechanisms with a delayed weighting procedure. The proposed weighting mechanism plays a key role by enabling rapid resampling in regions where the proposal distribution is poorly adapted to the target. Our sampling algorithm is shown to be geometrically convergent under mild assumptions and is illustrated through various numerical experiments.

Sampling from complex probability distributions is a core challenge in modern machine learning. Many statistical problems require estimating averages with respect to such distributions, but these quantities are often too difficult to compute exactly. A common strategy is then to approximate them using random samples that are averaged, but this can become inefficient when important regions are rare, far apart, or hidden in high-dimensional spaces. Our work improves a technique called importance sampling, which makes simulations more efficient by drawing samples from a carefully chosen distribution and then correcting for the difference using weights. The main challenge is choosing this sampling distribution well. We propose an adaptive method called Entropic Mirror Monte Carlo that learns where to sample by balancing two complementary ideas: one mechanism helps the algorithm focus on promising regions it has already discovered, while another encourages exploration of regions that may have been missed. This helps the method better navigate complicated probability distributions. We prove that, under mild conditions, an idealized version of our method converges to the target distribution at a geometrically fast rate. Although this idealized version is not directly implementable, we introduce a practical approximation and show through numerical experiments that it performs well in practice. Overall, our work contributes to the development of more reliable sampling tools for machine learning, Bayesian statistics, and scientific computing.