Provable Convergence and Limitations of Geometric Tempering for Langevin Dynamics

Sampling from a target distribution π whose density is known up to a normalizing constant is a challenging problem in statistics and machine learning, and is currently the subject of intense interest due to applications in Bayesian statistics [19] and energy-based models in deep learning [68], among other areas. In these settings, the normalizing constant of the target distribution π is typically intractable, and Markov Chain Monte Carlo (MCMC) algorithms [58, 57] are commonly used to generate Markov chains in the ambient space, whose law eventually approximates the target distribution. Among MCMC algorithms, the Unadjusted Langevin Algorithm (ULA), which corresponds to a time discretization of a Langevin diffusion process, has attracted considerable attention due to its simplicity, theoretical grounding, and utility in practice [59, 76, 23, 66]. For example, ULA can be proven to converge quickly when the target distribution π is smooth and strongly log-concave [23]. However, many cases in practice require to sample from distributions which are not log-concave, and indeed potentially even multi-modal [54, 82]. In such settings, the convergence of ULA is governed by functional inequalities which effectively quantify the convexity, or lack thereof, of the target distribution [75]. Nonetheless, truly multi-modal target distributions generally have poor functional inequalities, thus leading to weak convergence guarantees for ULA. This phenomenon is not merely a theoretical artifact, and it is wellknown amongst practitioners that when sampling from multi-modal distributions, algorithms based on ULA can get stuck in local modes and suffer from slow convergence [20]. Tempering or annealing is a popular technique [51, 27, 69] to overcome the deficiencies of ULA and other MCMC methods in the multi-modal setting.