Adaptive Simulated Annealing: A Near-optimal Connection between Sampling and Counting

TL;DR

This paper introduces an improved cooling schedule for approximate counting and sampling in discrete systems, significantly reducing the number of steps needed and thus enhancing efficiency in estimating partition functions.

Contribution

It presents a near-optimal reduction from approximate counting to sampling and introduces a cooling schedule of length O*(√ln A), improving over previous methods.

Findings

Cooling schedule length is reduced from O*(ln A) to O*(√ln A).

For problems like Ising model and graph colorings, schedule length is O*(√n).

Overall running time savings of O*(n) in approximate counting algorithms.

Abstract

We present a near-optimal reduction from approximately counting the cardinality of a discrete set to approximately sampling elements of the set. An important application of our work is to approximating the partition function $Z$ of a discrete system, such as the Ising model, matchings or colorings of a graph. The typical approach to estimating the partition function $Z(\beta^*)$ at some desired inverse temperature $\beta^*$ is to define a sequence, which we call a {\em cooling schedule}, $\beta_0=0<\beta_1<...<\beta_\ell=\beta^*$ where Z(0) is trivial to compute and the ratios $Z(\beta_{i+1})/Z(\beta_i)$ are easy to estimate by sampling from the distribution corresponding to $Z(\beta_i)$. Previous approaches required a cooling schedule of length $O^*(\ln{A})$ where $A=Z(0)$, thereby ensuring that each ratio $Z(\beta_{i+1})/Z(\beta_i)$ is bounded. We present a cooling schedule of length $\ell=O^*(\sqrt{\ln{A}})$. For well-studied problems such as estimating the partition function of the Ising model, or approximating the number of colorings or matchings of a graph, our cooling schedule is of length $O^*(\sqrt{n})$, which implies an overall savings of $O^*(n)$ in the running time of the approximate counting algorithm (since roughly $\ell$ samples are needed to estimate each ratio).