Sampling from Spherical Spin Glasses in Total Variation via Algorithmic Stochastic Localization

We consider the problem of algorithmically sampling from the Gibbs measure of a mixed p-spin spherical spin glass. We give a polynomial-time algorithm that samples from the Gibbs measure up to vanishing total variation error, for any model whose mixture satisfies ξ”(s) < 1/(1-s)^2, ∀ s∈ [0,1). This includes the pure p-spin glasses above a critical temperature that is within an absolute (p-independent) constant of the so-called shattering phase transition. Our algorithm follows the algorithmic stochastic localization approach introduced in (Alaoui, Montanari, Sellke, 20022). A key step of this approach is to estimate the mean of a sequence of tilted measures. We produce an improved estimator for this task by identifying a suitable correction to the TAP fixed point selected by approximate message passing (AMP). As a consequence, we improve the algorithm's guarantee over previous work, from normalized Wasserstein to total variation error. In particular, the new algorithm and analysis opens the way to perform inference about one-dimensional projections of the measure.