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Accelerating Convergence of Replica Exchange Stochastic Gradient MCMC via Variance Reduction

Wei Deng · Qi Feng · Georgios Karagiannis · Guang Lin · Faming Liang

Keywords: [ Markov jump process ] [ Dirichlet form ] [ generalized Girsanov theorem ] [ change of measure ] [ stochastic gradient Langevin dynamics ] [ parallel tempering ] [ replica exchange ] [ uncertainty quantification ] [ variance reduction ]

Abstract: Replica exchange stochastic gradient Langevin dynamics (reSGLD) has shown promise in accelerating the convergence in non-convex learning; however, an excessively large correction for avoiding biases from noisy energy estimators has limited the potential of the acceleration. To address this issue, we study the variance reduction for noisy energy estimators, which promotes much more effective swaps. Theoretically, we provide a non-asymptotic analysis on the exponential convergence for the underlying continuous-time Markov jump process; moreover, we consider a generalized Girsanov theorem which includes the change of Poisson measure to overcome the crude discretization based on the Gr\"{o}wall's inequality and yields a much tighter error in the 2-Wasserstein ($\mathcal{W}_2$) distance. Numerically, we conduct extensive experiments and obtain state-of-the-art results in optimization and uncertainty estimates for synthetic experiments and image data.

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