Accelerating Multilevel Markov Chain Monte Carlo Using Machine Learning Models
Abstract
This work presents an efficient approach for accelerating multilevel Markov Chain Monte Carlo (MCMC) sampling for largescale problems using lowfidelity machine learning models. While conventional techniques for largescale Bayesian inference often substitute computationally expensive highfidelity models with machine learning models, thereby introducing approximation errors, our approach offers a computationally efficient alternative by augmenting highfidelity models with lowfidelity ones within a hierarchical framework. The multilevel approach utilizes the lowfidelity machine learning model (MLM) for inexpensive evaluation of proposed samples thereby improving the acceptance of samples by the highfidelity model. The hierarchy in our multilevel algorithm is derived from geometric multigrid hierarchy. We utilize an MLM to acclerate the coarse level sampling. Training machine learning model for the coarsest level significantly reduces the computational cost associated with generating training data and training the model. We present an MCMC algorithm to accelerate the coarsest level sampling using MLM and account for the approximation error introduced. We provide theoretical proofs of detailed balance and demonstrate that our multilevel approach constitutes a consistent MCMC algorithm. Additionally, we derive conditions on the accuracy of the machine learning model to facilitate more efficient hierarchical sampling. Our technique is demonstrated on a standard benchmark inference problem in groundwater flow, where we estimate the probability density of a quantity of interest using a fourlevel MCMC algorithm. Our proposed algorithm accelerates multilevel sampling by a factor of two while achieving similar accuracy compared to sampling using the standard multilevel algorithm.
 Publication:

arXiv eprints
 Pub Date:
 May 2024
 DOI:
 10.48550/arXiv.2405.11179
 arXiv:
 arXiv:2405.11179
 Bibcode:
 2024arXiv240511179R
 Keywords:

 Statistics  Machine Learning;
 Computer Science  Machine Learning;
 Mathematics  Numerical Analysis;
 Mathematics  Probability