Improving Sampling Accuracy of Stochastic Gradient MCMC Methods via Non-uniform Subsampling of Gradients

Ruilin Li, Xin Wang, Hongyuan Zha, Molei Tao

Keywords: SG-MCMC, Non-uniform weight

Abstract: Common Stochastic Gradient MCMC methods approximate gradients by stochastic ones via uniformly subsampled data points. A non-uniform subsampling scheme, however, can reduce the variance introduced by the stochastic approximation and make the sampling of a target distribution more accurate. For this purpose, an exponentially weighted stochastic gradient approach (EWSG) is developed to match the transition kernel of a non-uniform-SG-MCMC method with that of a batch-gradient-MCMC method. If needed to be put in the importance sampling (IS) category, EWSG can be viewed as a way to extend the IS+SG approach successful for optimization to the sampling setup. EWSG works for a range of MCMC methods, and a demonstration on Stochastic-Gradient 2nd-order Langevin is provided. In our practical implementation of EWSG, the non-uniform subsampling is performed efficiently via a Metropolis-Hasting chain on the data index, which is coupled to the sampling algorithm. The fact that our method has reduced local variance with high probability is theoretically analyzed. A non-asymptotic global error analysis is also presented. As a practical implementation contains hyperparameters, numerical experiments based on both synthetic and real world data sets are provided, to both demonstrate the empirical performances and recommend hyperparameter choices. Notably, while statistical accuracy has improved, the speed of convergence, with appropriately chosen hyper-parameters, was empirically observed to be at least comparable to the uniform version, which renders EWSG a practically useful alternative to common variance reduction treatments.

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One-sentence Summary: A new way for improving the sampling accuracy of SG-MCMC methods is proposed, together with theoretical analysis and empirical evaluations.

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