On the Convergence of Stochastic Gradient MCMC Algorithms with High-Order Integrators

Recent advances in Bayesian learning with large-scale data have witnessed emergence of stochastic gradient MCMC algorithms (SG-MCMC), such as stochastic gradient Langevin dynamics (SGLD), stochastic gradient Hamiltonian MCMC (SGHMC), and the stochastic gradient thermostat. While finite-time convergence properties of the SGLD with a 1st-order Euler integrator have recently been studied, corresponding theory for general SG-MCMCs has not been explored. In this paper we consider general SG-MCMCs with high-order integrators, and develop theory to analyze finite-time convergence properties and their asymptotic invariant measures. Our theoretical results show faster convergence rates and more accurate invariant measures for SG-MCMCs with higher-order integrators. For example, with the proposed efficient 2nd-order symmetric splitting integrator, the mean square error (MSE) of the posterior average for the SGHMC achieves an optimal convergence rate of L-4/5 at L iterations, compared to L-2/3 for the SGHMC and SGLD with 1st-order Euler integrators. Furthermore, convergence results of decreasing-step-size SG-MCMCs are also developed, with the same convergence rates as their fixed-step-size counterparts for a specific decreasing sequence. Experiments on both synthetic and real datasets verify our theory, and show advantages of the proposed method in two large-scale real applications.

[1]  N. Kryloff,et al.  La Theorie Generale De La Mesure Dans Son Application A L'Etude Des Systemes Dynamiques De la Mecanique Non Lineaire , 1937 .

[2]  R. Khasminskii.  Stochastic Stability of Differential Equations , 1980 .

[3]  Michael I. Jordan,et al.  Latent Dirichlet Allocation , 2001, J. Mach. Learn. Res..

[4]  P. Giesl.  Construction of Global Lyapunov Functions Using Radial Basis Functions , 2007 .

[5]  Ruslan Salakhutdinov,et al.  On the quantitative analysis of deep belief networks , 2008, ICML '08.

[6]  Francis R. Bach,et al.  Online Learning for Latent Dirichlet Allocation , 2010, NIPS.

[7]  Andrew M. Stuart,et al.  Convergence of Numerical Time-Averaging and Stationary Measures via Poisson Equations , 2009, SIAM J. Numer. Anal..

[8]  S. Sharma,et al.  The Fokker-Planck Equation , 2010 .

[9]  Yee Whye Teh,et al.  Bayesian Learning via Stochastic Gradient Langevin Dynamics , 2011, ICML.

[10]  Erwan Faou,et al.  Weak Backward Error Analysis for SDEs , 2011, SIAM J. Numer. Anal..

[11]  B. Leimkuhler,et al.  Rational Construction of Stochastic Numerical Methods for Molecular Sampling , 2012, 1203.5428.

[12]  Yee Whye Teh,et al.  Stochastic Gradient Riemannian Langevin Dynamics on the Probability Simplex , 2013, NIPS.

[13]  M. Kopec.  Weak backward error analysis for overdamped Langevin processes , 2013, 1310.2404.

[14]  Karol Gregor,et al.  Neural Variational Inference and Learning in Belief Networks , 2014, ICML.

[15]  Hiroshi Nakagawa,et al.  Approximation Analysis of Stochastic Gradient Langevin Dynamics by using Fokker-Planck Equation and Ito Process , 2014, ICML.

[16]  Ryan Babbush,et al.  Bayesian Sampling Using Stochastic Gradient Thermostats , 2014, NIPS.

[17]  Tianqi Chen,et al.  Stochastic Gradient Hamiltonian Monte Carlo , 2014, ICML.

[18]  Zhe Gan,et al.  Learning Deep Sigmoid Belief Networks with Data Augmentation , 2015, AISTATS.

[19]  M. Betancourt.  The Fundamental Incompatibility of Hamiltonian Monte Carlo and Data Subsampling , 2015, 1502.01510.

[20]  Assyr Abdulle,et al.  Long Time Accuracy of Lie-Trotter Splitting Methods for Langevin Dynamics , 2015, SIAM J. Numer. Anal..

[21]  Zhe Gan,et al.  Scalable Deep Poisson Factor Analysis for Topic Modeling , 2015, ICML.

[22]  Michael Betancourt,et al.  The Fundamental Incompatibility of Scalable Hamiltonian Monte Carlo and Naive Data Subsampling , 2015, ICML.

[23]  K. Zygalakis,et al.  (Non-) asymptotic properties of Stochastic Gradient Langevin Dynamics , 2015, 1501.00438.

[24]  Yee Whye Teh,et al.  Consistency and Fluctuations For Stochastic Gradient Langevin Dynamics , 2014, J. Mach. Learn. Res..

[25]  BENEDICT LEIMKUHLER,et al.  Adaptive Thermostats for Noisy Gradient Systems , 2015, SIAM J. Sci. Comput..