Faster Convergence of Stochastic Gradient Langevin Dynamics for Non-Log-Concave Sampling

• URL: http://arxiv.org/abs/2010.09597v2
• Date: Tue, 23 Feb 2021 07:15:34 GMT
• Title: Faster Convergence of Stochastic Gradient Langevin Dynamics for Non-Log-Concave Sampling
• Authors: Difan Zou and Pan Xu and Quanquan Gu
• Abstract summary: We provide a new convergence analysis of gradient Langevin dynamics (SGLD) for sampling from a class of distributions that can be non-log-concave. At the core of our approach is a novel conductance analysis of SGLD using an auxiliary time-reversible Markov Chain.
• Score: 110.88857917726276
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We provide a new convergence analysis of stochastic gradient Langevin dynamics (SGLD) for sampling from a class of distributions that can be non-log-concave. At the core of our approach is a novel conductance analysis of SGLD using an auxiliary time-reversible Markov Chain. Under certain conditions on the target distribution, we prove that $\tilde O(d^4\epsilon^{-2})$ stochastic gradient evaluations suffice to guarantee $\epsilon$-sampling error in terms of the total variation distance, where $d$ is the problem dimension. This improves existing results on the convergence rate of SGLD (Raginsky et al., 2017; Xu et al., 2018). We further show that provided an additional Hessian Lipschitz condition on the log-density function, SGLD is guaranteed to achieve $\epsilon$-sampling error within $\tilde O(d^{15/4}\epsilon^{-3/2})$ stochastic gradient evaluations. Our proof technique provides a new way to study the convergence of Langevin-based algorithms and sheds some light on the design of fast stochastic gradient-based sampling algorithms.

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