Adaptive Importance Sampling meets Mirror Descent: a Bias-variance tradeoff

TL;DR

This paper introduces a regularization strategy for adaptive importance sampling that balances bias and variance by raising importance weights to a power, connecting it to mirror descent, and demonstrating its effectiveness through theoretical analysis and empirical results.

Contribution

It proposes a novel regularization method for importance sampling weights, linking it to mirror descent and providing convergence guarantees and practical parameter selection strategies.

Findings

Regularization reduces variance of importance weights.

The method achieves uniform convergence under mild conditions.

Empirical results show improved accuracy in Monte Carlo estimates.

Abstract

Adaptive importance sampling is a widely spread Monte Carlo technique that uses a re-weighting strategy to iteratively estimate the so-called target distribution. A major drawback of adaptive importance sampling is the large variance of the weights which is known to badly impact the accuracy of the estimates. This paper investigates a regularization strategy whose basic principle is to raise the importance weights at a certain power. This regularization parameter, that might evolve between zero and one during the algorithm, is shown (i) to balance between the bias and the variance and (ii) to be connected to the mirror descent framework. Using a kernel density estimate to build the sampling policy, the uniform convergence is established under mild conditions. Finally, several practical ways to choose the regularization parameter are discussed and the benefits of the proposed approach are illustrated empirically.