Stochastic Gradient Richardson-Romberg Markov Chain Monte Carlo

Alain Durmus (1) , Umut Şimşekli (1) , Éric Moulines (2) , Roland Badeau (1) , Gael Richard (1)

1 LTCI - Laboratoire Traitement et Communication de l'Information
2 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique

Stochastic Gradient Markov Chain Monte Carlo (SG-MCMC) algorithms have become increasingly popular for Bayesian inference in large-scale applications. Even though these methods have proved useful in several scenarios, their performance is often limited by their bias. In this study, we propose a novel sampling algorithm that aims to reduce the bias of SG-MCMC while keeping the variance at a reasonable level. Our approach is based on a numerical sequence acceleration method, namely the Richardson-Romberg extrapolation, which simply boils down to running almost the same SG-MCMC algorithm twice in parallel with different step sizes. We illustrate our framework on the popular Stochastic Gradient Langevin Dynamics (SGLD) algorithm and propose a novel SG-MCMC algorithm referred to as Stochastic Gradient Richardson-Romberg Langevin Dynamics (SGRRLD). We provide formal theoretical analysis and show that SGRRLD is asymptotically consistent, satisfies a central limit theorem, and its non-asymptotic bias and the mean squared-error can be bounded. Our results show that SGRRLD attains higher rates of convergence than SGLD in both finite-time and asymptotically, and it achieves the theoretical accuracy of the methods that are based on higher-order integrators. We support our findings using both synthetic and real data experiments.

Domaines: Traitement du signal et de l'image [eess.SP]

Alain Durmus, Umut Şimşekli, Éric Moulines, Roland Badeau, Gael Richard. Stochastic Gradient Richardson-Romberg Markov Chain Monte Carlo. Thirtieth Annual Conference on Neural Information Processing Systems (NIPS), Dec 2016, Barcelone, Spain. ⟨hal-01354064⟩