Sampling from the Continuous Random Energy Model in Total Variation Distance

TL;DR

This paper presents two polynomial-time algorithms for approximately sampling from the Gibbs distribution of the continuous random energy model (CREM) in the high-temperature regime, achieving total variation distance guarantees and analyzing their efficiency.

Contribution

It introduces new algorithms for sampling from CREM in regimes previously not accessible, with bounds on running time and accuracy, and explores phase transition thresholds.

Findings

Algorithms work up to the critical temperature for concave covariance functions.

Sampling algorithms achieve polynomial time in TV distance and failure probability.

Spectral gap analysis indicates exponential smallness, implying inherent Markov chain limitations.

Abstract

The continuous random energy model (CREM) is a toy model of spin glasses on $\{0,1\}^N$ that, in the limit, exhibits an infinitely hierarchical correlation structure. We give two polynomial-time algorithms to approximately sample from the Gibbs distribution of the CREM in the high-temperature regime $\beta<\beta_{\min}:=\min\{\beta_c,\beta_G\}$, based on a Markov chain and a sequential sampler. The running time depends algebraically on the desired TV distance and failure probability and exponentially in $(1/g)^{O(1)}$, where $g$ is the gap to a certain inverse temperature threshold $\beta_{\min}$; this contrasts with previous results which only attain $o(N)$ accuracy in KL divergence. If the covariance function $A$ of the CREM is concave, the algorithms work up to the critical threshold $\beta_c$, which is the static phase transition point; while for $A$ non-concave, if $\beta_G<\beta_c$, the algorithms work up to the known algorithmic threshold $\beta_G$ proposed in Addario-Berry and Maillard (2020) for non-trivial sampling guarantees. Our result depends on quantitative bounds for the fluctuation of the partition function and a new contiguity result of the ``tilted" CREM obtained from sampling, which is of independent interest. We also show that the spectral gap is exponentially small with high probability, suggesting that the algebraic dependence is unavoidable with a Markov chain approach.