Multi-fidelity Hamiltonian Monte Carlo

TL;DR

This work tackles the high computational cost of gradient-based Hamiltonian Monte Carlo in Bayesian inverse problems with high-dimensional parameters and/or black-box forward models. It introduces Multi-fidelity HMC (MFHMC), a two-stage approach that uses a cheap differentiable surrogate to drive proposals and a high-fidelity solver to correct and produce accurate samples, preserving detailed balance. Across synthetic and real-data experiments, MFHMC demonstrates orders-of-magnitude improvements in both computational and statistical efficiency, capabilities with Gaussian and GAN-based priors, and compatibility with various surrogate types (TSVD, DNN, CNN). The method broadens the practical applicability of HMC to black-box simulators and complex priors, offering a flexible framework for scalable uncertainty quantification in diverse inverse problems.

Abstract

Numerous applications in biology, statistics, science, and engineering require generating samples from high-dimensional probability distributions. In recent years, the Hamiltonian Monte Carlo (HMC) method has emerged as a state-of-the-art Markov chain Monte Carlo technique, exploiting the shape of such high-dimensional target distributions to efficiently generate samples. Despite its impressive empirical success and increasing popularity, its wide-scale adoption remains limited due to the high computational cost of gradient calculation. Moreover, applying this method is impossible when the gradient of the posterior cannot be computed (for example, with black-box simulators). To overcome these challenges, we propose a novel two-stage Hamiltonian Monte Carlo algorithm with a surrogate model. In this multi-fidelity algorithm, the acceptance probability is computed in the first stage via a standard HMC proposal using an inexpensive differentiable surrogate model, and if the proposal is accepted, the posterior is evaluated in the second stage using the high-fidelity (HF) numerical solver. Splitting the standard HMC algorithm into these two stages allows for approximating the gradient of the posterior efficiently, while producing accurate posterior samples by using HF numerical solvers in the second stage. We demonstrate the effectiveness of this algorithm for a range of problems, including linear and nonlinear Bayesian inverse problems with in-silico data and experimental data. The proposed algorithm is shown to seamlessly integrate with various low-fidelity and HF models, priors, and datasets. Remarkably, our proposed method outperforms the traditional HMC algorithm in both computational and statistical efficiency by several orders of magnitude, all while retaining or improving the accuracy in computed posterior statistics.

Figures (19)

• Figure 1: Outline of the proposed two-stage MFHMC algorithm: In the first stage of the algorithm a surrogate forward model (with an easy-to-compute gradient) is used in the standard HMC step. If a given sample is accepted by this first stage, then it is passed to the second stage, where a high-fidelity numerical solver is used in the Metropolis-Hastings step (for which only a forward model evaluation is required with no gradient requirement) to produce accurate posterior samples. If a sample is rejected in either stages, we stay at the current Markov chain state.
• Figure 2: Comparison of the number of accepted moves per HF evaluation for HMC and MFHMC at different computation budgets (as defined by the number of HF target distribution evaluations ($n_{hf}$)). To minimize the influence of randomness, each experiment was performed with five different random seeds. Each data point in the figure represents the average value of these five experiments.
• Figure 3: Comparison of effective sample size (ESS) per HF evaluation for HMC and MFHMC at different computation budgets (as defined by the number of HF target distribution evaluations ($n_{hf}$)). To minimize the influence of randomness, each experiment was performed with five different random seeds. Each data point in the figure represents the average value of these five experiments.
• Figure 4: Comparison of relative error in covariance (in %) for HMC and MFHMC at different computation budgets (as defined by the number of HF target distribution evaluations ($n_{hf}$)). To minimize the influence of randomness, each experiment was performed with five different random seeds. Each data point in the figure represents the average value of these five experiments.
• Figure 5: True initial condition ($\bm{x}$).