Hamiltonian Adaptive Importance Sampling

TL;DR

The paper introduces Hamiltonian Adaptive Importance Sampling (HAIS), a novel method combining Hamiltonian Monte Carlo with adaptive importance sampling to improve high-dimensional integral approximation.

Contribution

It proposes a new HAIS algorithm that integrates parallel HMC chains into an adaptive importance sampling framework, enhancing performance in high-dimensional problems.

Findings

HAIS outperforms state-of-the-art algorithms in high-dimensional tasks.

HAIS effectively adapts proposals using parallel HMC chains.

The method demonstrates significant performance improvements in challenging examples.

Abstract

Importance sampling (IS) is a powerful Monte Carlo (MC) methodology for approximating integrals, for instance in the context of Bayesian inference. In IS, the samples are simulated from the so-called proposal distribution, and the choice of this proposal is key for achieving a high performance. In adaptive IS (AIS) methods, a set of proposals is iteratively improved. AIS is a relevant and timely methodology although many limitations remain yet to be overcome, e.g., the curse of dimensionality in high-dimensional and multi-modal problems. Moreover, the Hamiltonian Monte Carlo (HMC) algorithm has become increasingly popular in machine learning and statistics. HMC has several appealing features such as its exploratory behavior, especially in high-dimensional targets, when other methods suffer. In this paper, we introduce the novel Hamiltonian adaptive importance sampling (HAIS) method. HAIS implements a two-step adaptive process with parallel HMC chains that cooperate at each iteration. The proposed HAIS efficiently adapts a population of proposals, extracting the advantages of HMC. HAIS can be understood as a particular instance of the generic layered AIS family with an additional resampling step. HAIS achieves a significant performance improvement in high-dimensional problems w.r.t. state-of-the-art algorithms. We discuss the statistical properties of HAIS and show its high performance in two challenging examples.