Proximal Sampler with Adaptive Step Size

We consider the problem of sampling from a target unnormalized distribution $\exp(-f(x))$ defined on $\mathbb{R}^d$ where $f(x)$ is smooth, but the smoothness parameter is unknown. As a key design parameter of Markov chain Monte Carlo (MCMC) algorithms, the step size is crucial for the convergence guarantee. Existing non-asymptotic analysis on MCMC with fixed step sizes indicates that the step size heavily relies on global smoothness. However, this choice does not utilize the local information and fails when the smoothness coefficient is hard to estimate. A tuning-free algorithm that can adaptively update stepsize is highly desirable. In this work, we propose an \textbf{adaptive} proximal sampler that can utilize the local geometry to adjust step sizes and is guaranteed to converge to the target distribution. Experiments demonstrate the comparable or superior performance of our algorithm over various baselines.