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

arXiv: https://arxiv.org/abs/2411.11415 · HTML: https://mlsanity.com/paper/2411.11415/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Nov 2024.

## Built on by (most cited first, 2 of 2)

- [Predictability Enables Parallelization of Nonlinear State Space Models](https://mlsanity.com/paper/2508.16817/) (2025)
- [The Tamed Subgradient Unadjusted Langevin Algorithm beyond Convexity](https://mlsanity.com/paper/2608.06283/) (2026)

## Abstract (arXiv)

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.