Computational Separations between Sampling and Optimization

TL;DR

This paper investigates the computational complexity differences between sampling and optimization in Bayesian learning, revealing that they are often incomparable and can exhibit phase transitions in difficulty, especially in non-convex settings.

Contribution

It demonstrates that sampling and optimization are provably incomparable in certain continuous function families and identifies phase transitions in sampling complexity based on temperature.

Findings

Sampling can be NP-hard where optimization is easy.

Optimization can be computationally easy while sampling is NP-hard.

Sampling complexity exhibits a sharp phase transition with temperature.

Abstract

Two commonly arising computational tasks in Bayesian learning are Optimization (Maximum A Posteriori estimation) and Sampling (from the posterior distribution). In the convex case these two problems are efficiently reducible to each other. Recent work (Ma et al. 2019) shows that in the non-convex case, sampling can sometimes be provably faster. We present a simpler and stronger separation. We then compare sampling and optimization in more detail and show that they are provably incomparable: there are families of continuous functions for which optimization is easy but sampling is NP-hard, and vice versa. Further, we show function families that exhibit a sharp phase transition in the computational complexity of sampling, as one varies the natural temperature parameter. Our results draw on a connection to analogous separations in the discrete setting which are well-studied.