Variational Mirror Descent for Robust Learning in Schrödinger Bridge
Keywords: optimal transport, mirror descent, variational methods
TL;DR: We propose a variational mirror descent framework for the Schrödinger bridge problem.
Abstract: Schrödinger bridge (SB) has evolved into a universal class of probabilistic generative models. Recent studies regarding the Sinkhorn algorithm through mirror descent (MD) have gained attention, revealing geometric insights into solution acquisition of the SB problems. In this paper, we propose a variational online MD framework for the SB problems, which provides further stability to SB solvers. We formally prove convergence and a regret bound $\mathcal{O}(\textrm{\small$\sqrt{T}$})$ of online mirror descent under mild assumptions. As a result of analysis, we propose a simulation-free SB algorithm called Variational Mirrored Schrödinger Bridge (VMSB) by utilizing the Wasserstein-Fisher-Rao geometry of the Gaussian mixture parameterization for Schrödinger potentials. Based on the Wasserstein gradient flow theory, our variational MD framework offers tractable gradient-based learning dynamics that precisely approximate a subsequent update. We demonstrate the performance of the proposed VMSB algorithm in an extensive suite of benchmarks.
Supplementary Material: zip
Primary Area: learning theory
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Submission Number: 5991
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