Title: A Proximal-Sinkhorn-Newton Method for Entropic Optimal Transport
Primary area: optimization
Keywords: Entropic Optimal Transport, Sinkhorn Algorithm, Proximal Point Method, Newton Method, Matrix Scaling

Abstract: Entropic optimal transport (OT) enables efficient distribution alignment through the Sinkhorn method. However, it suffers from numerical instability and slow convergence under weak entropic regularization. We propose a two-stage framework that establishes an inexact-to-exact paradigm to address these challenges. The first stage employs an inexact proximal point method to decompose the entropic OT into simpler subproblems, yielding approximate solutions with superior numerical stability. The second stage employs a sparse Newton method with global convergence and a locally quadratic rate to refine the approximate solutions. Compared to previous Newton-based algorithms, it accelerates updates and prevents the objective score from plateauing during optimization. With numerical instability handled in the first stage,  Sinkhorn iterations can provide an alternative to the Newton method under relatively heavy entropic regularization. The yielding Proximal-Sinkhorn-Newton method enjoys the strengths of three approaches and outperforms the baselines across various regularizations and error tolerances, achieving over 20× speedups on some problems compared to other Newton-based methods.

TL;DR: We propose a hybrid solver for entropic optimal transport that ensures both stability and efficiency across a wide range of entropic regularization parameters and error tolerances.