Shifted Composition III: Local Error Framework for KL Divergence

• URL: http://arxiv.org/abs/2412.17997v1
• Date: Mon, 23 Dec 2024 21:40:01 GMT
• Title: Shifted Composition III: Local Error Framework for KL Divergence
• Authors: Jason M. Altschuler, Sinho Chewi,
• Abstract summary: Coupling arguments are a central tool for bounding the deviation between two processes.<n>We adapt coupling arguments to the Kullback-Leibler (KL) divergence.
• Score: 12.93725028754563
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Coupling arguments are a central tool for bounding the deviation between two stochastic processes, but traditionally have been limited to Wasserstein metrics. In this paper, we apply the shifted composition rule--an information-theoretic principle introduced in our earlier work--in order to adapt coupling arguments to the Kullback-Leibler (KL) divergence. Our framework combine the strengths of two previously disparate approaches: local error analysis and Girsanov's theorem. Akin to the former, it yields tight bounds by incorporating the so-called weak error, and is user-friendly in that it only requires easily verified local assumptions; and akin to the latter, it yields KL divergence guarantees and applies beyond Wasserstein contractivity. We apply this framework to the problem of sampling from a target distribution $\pi$. Here, the two stochastic processes are the Langevin diffusion and an algorithmic discretization thereof. Our framework provides a unified analysis when $\pi$ is assumed to be strongly log-concave (SLC), weakly log-concave (WLC), or to satisfy a log-Sobolev inequality (LSI). Among other results, this yields KL guarantees for the randomized midpoint discretization of the Langevin diffusion. Notably, our result: (1) yields the optimal $\tilde O(\sqrt d/\epsilon)$ rate in the SLC and LSI settings; (2) is the first result to hold beyond the 2-Wasserstein metric in the SLC setting; and (3) is the first result to hold in \emph{any} metric in the WLC and LSI settings.

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