Potential Hessian Ascent III: Sampling the Sherrington--Kirkpatrick Model at Beta < 1/2

TL;DR

This paper introduces a polynomial-time sampling algorithm for the Sherrington-Kirkpatrick model at inverse temperatures below 1/2, improving error guarantees and extending the regime of efficient sampling.

Contribution

It adapts potential Hessian ascent for sampling, achieving negligible TVD error in the entire replica-symmetric regime, previously only accessible in Wasserstein distance or at lower temperatures.

Findings

Achieves $O(1)$ Wasserstein error for finite-time localization.

Refines error to $o(1)$ in TVD using Jarzynski's equality and rejection sampling.

Extends efficient sampling guarantees up to inverse temperature $eta < 1/2$.

Abstract

We give a polynomial-time algorithm to sample from the Gibbs measure of the Sherrington-Kirkpatrick model with negligible total-variation distance (TVD) error up to inverse temperature $\beta < 1/2$. Prior work obtained TVD error guarantees only up to $\beta\approx 0.295$, while results covering the entire replica-symmetric regime $\beta < 1$ gave guarantees only in Wasserstein distance. Our approach demonstrates that the same potential Hessian ascent previously developed for optimization also functions as a sampling algorithm by implementing algorithmic stochastic localization at high temperature. By estimating the covariance of the tilted Gibbs distribution via Gaussian integration by parts, overlap concentration, and precise cavity estimates, we show that a Hessian-ascent process achieves an $O(1)$ Wasserstein error guarantee for finite-time localization, improving on the previous $o(n)$. A careful comparison of stochastic localization with the Hessian ascent process and a free probability argument controlling the diagonal sub-algebra of the Hessian improves this to $O(1)$ in KL divergence. We then use Jarzynski's equality with rejection sampling, along with entropy contraction on the time-$T$ localized distribution, to refine the error to $o(1)$ in TVD up to a constant time $T$ and to complete the sampling with the polarized walk.