Progressive Entropic Optimal Transport Solvers

TL;DR

This paper introduces ProgOT, a new class of entropic optimal transport solvers that improve speed, robustness, and scalability by using a progressive, discretized approach inspired by dynamic OT, and proves their statistical consistency.

Contribution

ProgOT offers a novel progressive framework for entropic OT that enhances computational efficiency and robustness, and includes theoretical guarantees for transport map estimation.

Findings

ProgOT outperforms standard solvers in speed and robustness at large scales.

ProgOT surpasses neural network-based approaches in practical applications.

The approach is statistically consistent for estimating optimal transport maps.

Abstract

Optimal transport (OT) has profoundly impacted machine learning by providing theoretical and computational tools to realign datasets. In this context, given two large point clouds of sizes $n$ and $m$ in $\mathbb{R}^d$, entropic OT (EOT) solvers have emerged as the most reliable tool to either solve the Kantorovich problem and output a $n\times m$ coupling matrix, or to solve the Monge problem and learn a vector-valued push-forward map. While the robustness of EOT couplings/maps makes them a go-to choice in practical applications, EOT solvers remain difficult to tune because of a small but influential set of hyperparameters, notably the omnipresent entropic regularization strength $\varepsilon$. Setting $\varepsilon$ can be difficult, as it simultaneously impacts various performance metrics, such as compute speed, statistical performance, generalization, and bias. In this work, we propose a new class of EOT solvers (ProgOT), that can estimate both plans and transport maps. We take advantage of several opportunities to optimize the computation of EOT solutions by dividing mass displacement using a time discretization, borrowing inspiration from dynamic OT formulations, and conquering each of these steps using EOT with properly scheduled parameters. We provide experimental evidence demonstrating that ProgOT is a faster and more robust alternative to standard solvers when computing couplings at large scales, even outperforming neural network-based approaches. We also prove statistical consistency of our approach for estimating optimal transport maps.