Beyond Smoothness and Convexity: Optimization via sampling

This work explores a novel perspective on solving nonconvex and nonsmooth optimization problems by leveraging sampling based methods. Instead of treating the objective function purely through traditional (often deterministic) optimization approaches, we view it as inducing a target distribution.We then draw samples from this distribution using Markov Chain Monte Carlo (MCMC) techniques, particularly Langevin Dynamics (LD), to locate regions of low function values. By analyzing the convergence properties of LD in both KL divergence and total variation distance, we establish explicit bounds on how many iterations are required for the induced distribution to approximate the target. We also provide probabilistic guarantees that an appropriately drawn sample will lie within a desired neighborhood of the global minimizer, even when the objective is nonconvex or nonsmooth.