On theoretical guarantees and a blessing of dimensionality for nonconvex sampling

TL;DR

This paper establishes that under certain conditions, sampling from nonlogconcave measures in high dimensions can be achieved with polynomial complexity, leveraging a phenomenon called a blessing of dimensionality, while other conditions lead to exponential complexity.

Contribution

It provides the first comprehensive theoretical analysis showing polynomial complexity for sampling under specific geometric conditions, highlighting the blessing of dimensionality.

Findings

Polynomial complexity when $R \

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,

Abstract

Existing guarantees for algorithms sampling from nonlogconcave measures on $\mathbb{R}^d$ are generally inexplicit or unscalable. Even for the class of measures with logdensities that have bounded Hessians and are strongly concave outside a Euclidean ball of radius $R$, no available theory is comprehensively satisfactory with respect to both $R$ and $d$. In this paper, it is shown that complete polynomial complexity can in fact be achieved if $R\leq c\sqrt{d}$, whilst an exponential number of point evaluations is generally necessary for any algorithm as soon as $R\geq C\sqrt{d}$ for constants $C>c>0$. Importance sampling with a tail-matching proposal achieves the former, owing to a blessing of dimensionality. On the other hand, if strong concavity outside a ball is replaced by a distant dissipativity condition, then sampling guarantees must generally scale exponentially with $d$ in all parameter regimes.