Faster Convergence of Stochastic Gradient Langevin Dynamics for Non-Log-Concave Sampling
This work addresses the challenge of efficient sampling in high-dimensional, non-log-concave settings for machine learning and statistics, representing an incremental improvement over prior methods.
The paper tackles the problem of sampling from non-log-concave distributions using stochastic gradient Langevin dynamics (SGLD), achieving improved convergence rates of $ ilde O(d^4ε^{-2})$ and $ ilde O(d^{15/4}ε^{-3/2})$ stochastic gradient evaluations under specific conditions.
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 $\tilde O(d^4ε^{-2})$ stochastic gradient evaluations suffice to guarantee $ε$-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 $ε$-sampling error within $\tilde O(d^{15/4}ε^{-3/2})$ 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.