Local-Global MCMC kernels: the best of both worlds
This work addresses sampling inefficiencies in machine learning and statistics, particularly for high-dimensional and tail-heavy distributions, offering a hybrid approach that is incremental but with proven theoretical guarantees.
The paper tackles the challenge of sampling from complex distributions by proposing an Explore-Exploit MCMC strategy (Ex^2MCMC) that combines local and global samplers, achieving improved efficiency on classical benchmarks and high-dimensional distributions like those from GANs, with explicit bounds on mixing rates provided.
Recent works leveraging learning to enhance sampling have shown promising results, in particular by designing effective non-local moves and global proposals. However, learning accuracy is inevitably limited in regions where little data is available such as in the tails of distributions as well as in high-dimensional problems. In the present paper we study an Explore-Exploit Markov chain Monte Carlo strategy ($Ex^2MCMC$) that combines local and global samplers showing that it enjoys the advantages of both approaches. We prove $V$-uniform geometric ergodicity of $Ex^2MCMC$ without requiring a uniform adaptation of the global sampler to the target distribution. We also compute explicit bounds on the mixing rate of the Explore-Exploit strategy under realistic conditions. Moreover, we also analyze an adaptive version of the strategy ($FlEx^2MCMC$) where a normalizing flow is trained while sampling to serve as a proposal for global moves. We illustrate the efficiency of $Ex^2MCMC$ and its adaptive version on classical sampling benchmarks as well as in sampling high-dimensional distributions defined by Generative Adversarial Networks seen as Energy Based Models. We provide the code to reproduce the experiments at the link: https://github.com/svsamsonov/ex2mcmc_new.