The ballistic limit of the log-Sobolev constant equals the Polyak-Łojasiewicz constant

Abstract: The Polyak-Lojasiewicz (PL) constant of a function $f \colon \mathbb{R}^d \to \mathbb{R}$ characterizes the best exponential rate of convergence of gradient flow for $f$, uniformly over initializations. Meanwhile, in the theory of Markov diffusions, the log-Sobolev (LS) constant plays an analogous role, governing the exponential rate of convergence for the Langevin dynamics from arbitrary initialization in the Kullback-Leibler divergence. We establish a new connection between optimization and sampling by showing that the low temperature limit $\lim_{t\to 0^+} t^{-1} C_{\mathsf{LS}}(\mu_t)$ of the LS constant of $\mu_t \propto \exp(-f/t)$ is exactly the PL constant of $f$, under mild assumptions. In contrast, we show that the corresponding limit for the Poincar\'e constant is the inverse of the smallest eigenvalue of $\nabla^2 f$ at the minimizer.