Multi-fidelity No-U-Turn Sampling

TL;DR

This paper introduces a multi-fidelity Gaussian Process surrogate to efficiently approximate derivatives in No-U-Turn Sampling, significantly reducing computational costs while maintaining ergodicity for high-dimensional Bayesian inference.

Contribution

It proposes a novel multi-fidelity surrogate approach for NUTS that leverages hierarchies of models to accelerate sampling in computationally expensive scenarios.

Findings

Outperforms existing methods on benchmark tests

Reduces computational cost significantly

Maintains ergodicity with high fidelity models

Abstract

Markov Chain Monte Carlo (MCMC) methods often take many iterations to converge for highly correlated or high-dimensional target density functions. Methods such as Hamiltonian Monte Carlo (HMC) or No-U-Turn Sampling (NUTS) use the first-order derivative of the density function to tackle the aforementioned issues. However, the calculation of the derivative represents a bottleneck for computationally expensive models. We propose to first build a multi-fidelity Gaussian Process (GP) surrogate. The building block of the multi-fidelity surrogate is a hierarchy of models of decreasing approximation error and increasing computational cost. Then the generated multi-fidelity surrogate is used to approximate the derivative. The majority of the computation is assigned to the cheap models thereby reducing the overall computational cost. The derivative of the multi-fidelity method is used to explore the target density function and generate proposals. We select or reject the proposals using the Metropolis Hasting criterion using the highest fidelity model which ensures that the proposed method is ergodic with respect to the highest fidelity density function. We apply the proposed method to three test cases including some well-known benchmarks to compare it with existing methods and show that multi-fidelity No-U-turn sampling outperforms other methods.