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

The Polyak-Lojasiewicz (PL) constant of a function f colon ℝd → 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 limt→ 0^+ t-1 C_mathsfLS(μt) of the LS constant of μt 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 nabla2 f at the minimizer.