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.

I like the fact that the method is very simple to understand and implement (see my summary), and does not require any major changes to the base SG-MCMC algorithm. Also, this seems very general and applies to a large class of SG-MCMC algorithms, and is therefore potentially very impactful to the Stochastic Gradient MCMC community. Novelty: Although Richardson-Romberg extrapolation is well known in numerical analysis, it is not widely known in the machine learning / stochastic gradient MCMC community. Clarity: The paper is well written and the presentation is clear. Comments/questions: - Can this technique be directly applied to all SG-MCMC methods?

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).