Convergence of Unadjusted Langevin in High Dimensions: Delocalization of Bias

TL;DR

This paper investigates how the unadjusted Langevin algorithm's convergence behavior in high dimensions can be better understood through the delocalization of bias, showing that convergence for small marginals can be faster than for the full distribution.

Contribution

It introduces the concept of delocalization of bias, demonstrating that convergence for small marginals can be achieved with fewer iterations, and establishes this effect for Gaussian and certain sparse distributions.

Findings

Delocalization of bias allows faster convergence for small marginals.

The novel $W_{2, ext{l}^ ext{infty}}$ metric measures this convergence.

The effect holds for Gaussian and certain sparse log-concave distributions.

Abstract

The unadjusted Langevin algorithm is commonly used to sample probability distributions in extremely high-dimensional settings. However, existing analyses of the algorithm for strongly log-concave distributions suggest that, as the dimension $d$ of the problem increases, the number of iterations required to ensure convergence within a desired error in the $W_2$ metric scales in proportion to $d$ or $\sqrt{d}$. In this paper, we argue that, despite this poor scaling of the $W_2$ error for the full set of variables, the behavior for a small number of variables can be significantly better: a number of iterations proportional to $K$, up to logarithmic terms in $d$, often suffices for the algorithm to converge to within a desired $W_2$ error for all $K$-marginals. We refer to this effect as delocalization of bias. We show that the delocalization effect does not hold universally and prove its validity for Gaussian distributions and strongly log-concave distributions with certain sparse interactions. Our analysis relies on a novel $W_{2,\ell^\infty}$ metric to measure convergence. A key technical challenge we address is the lack of a one-step contraction property in this metric. Finally, we use asymptotic arguments to explore potential generalizations of the delocalization effect beyond the Gaussian and sparse interactions setting.