Learning Optimal Flows for Non-Equilibrium Importance Sampling

TL;DR

This paper introduces a non-equilibrium importance sampling method that uses flow-based transformations and deep learning to efficiently estimate expectations and normalization constants for complex high-dimensional distributions, achieving significant variance reduction.

Contribution

It develops a novel flow-based importance sampling approach, connecting transport maps with neural network training for zero-variance estimators in high-dimensional settings.

Findings

Variance reduced by up to 6 orders of magnitude.

Applicable to distributions up to 10 dimensions.

Outperforms Neal's annealed importance sampling in benchmarks.

Abstract

Many applications in computational sciences and statistical inference require the computation of expectations with respect to complex high-dimensional distributions with unknown normalization constants, as well as the estimation of these constants. Here we develop a method to perform these calculations based on generating samples from a simple base distribution, transporting them by the flow generated by a velocity field, and performing averages along these flowlines. This non-equilibrium importance sampling (NEIS) strategy is straightforward to implement and can be used for calculations with arbitrary target distributions. On the theory side, we discuss how to tailor the velocity field to the target and establish general conditions under which the proposed estimator is a perfect estimator with zero-variance. We also draw connections between NEIS and approaches based on mapping a base distribution onto a target via a transport map. On the computational side, we show how to use deep learning to represent the velocity field by a neural network and train it towards the zero variance optimum. These results are illustrated numerically on benchmark examples (with dimension up to $10$), where after training the velocity field, the variance of the NEIS estimator is reduced by up to $6$ orders of magnitude than that of a vanilla estimator. We also compare the performances of NEIS with those of Neal's annealed importance sampling (AIS).