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Error estimate for regularized optimal transport problems via Bregman divergence

Keiichi Morikuni, Koya Sakakibara, Asuka Takatsu

TL;DR

The paper develops a non-asymptotic error bound for regularized optimal transport on finite sets when regularization is performed via a general Bregman divergence D_U. It introduces key quantities D_U, _U(x,y), _C(x,y), R_U(x,y), and  u_U(x,y), and proves that the regularized objective converges to the true OT value with a bound governed by the inverse derivative e_U of U, decaying faster than exponential under suitable assumptions. The results recover known KL-based entropic regularization (U = U_o) and extend to a broad class of divergences, offering potentially sharper error decay in numerical experiments. The work also analyzes normalization and scaling properties, establishing when KL-invariance under data scaling holds and highlighting the distinctive behavior of non-KL Bregman regularizers. Overall, the framework provides both theoretical guarantees and practical guidance for choosing regularizers to improve convergence in finite-set OT problems.

Abstract

Regularization by the Shannon entropy enables us to efficiently and approximately solve optimal transport problems on a finite set. This paper is concerned with regularized optimal transport problems via Bregman divergence. We introduce the required properties for Bregman divergences, provide a non-asymptotic error estimate for the regularized problem, and show that the error estimate becomes faster than exponentially.

Error estimate for regularized optimal transport problems via Bregman divergence

TL;DR

The paper develops a non-asymptotic error bound for regularized optimal transport on finite sets when regularization is performed via a general Bregman divergence D_U. It introduces key quantities D_U, _U(x,y), _C(x,y), R_U(x,y), and  u_U(x,y), and proves that the regularized objective converges to the true OT value with a bound governed by the inverse derivative e_U of U, decaying faster than exponential under suitable assumptions. The results recover known KL-based entropic regularization (U = U_o) and extend to a broad class of divergences, offering potentially sharper error decay in numerical experiments. The work also analyzes normalization and scaling properties, establishing when KL-invariance under data scaling holds and highlighting the distinctive behavior of non-KL Bregman regularizers. Overall, the framework provides both theoretical guarantees and practical guidance for choosing regularizers to improve convergence in finite-set OT problems.

Abstract

Regularization by the Shannon entropy enables us to efficiently and approximately solve optimal transport problems on a finite set. This paper is concerned with regularized optimal transport problems via Bregman divergence. We introduce the required properties for Bregman divergences, provide a non-asymptotic error estimate for the regularized problem, and show that the error estimate becomes faster than exponentially.
Paper Structure (23 sections, 7 theorems, 160 equations, 2 figures)

This paper contains 23 sections, 7 theorems, 160 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. On Assumption \ref{['xy']} with the positivity and finiteness of $\mathfrak{D}_U(x,y), \Delta_C(x,y)$
  4. On Definition \ref{['def:rU']}
  5. On Assumption \ref{['assumption']}
  6. Asymptotic behavior of the error
  7. Proof of Theorem \ref{['thm:convergence']}
  8. Normalization and scaling
  9. Normalization
  10. Scaling
  11. Invariance of the Kullback--Leibler divergence under scaling of data
  12. Examples and comparison
  13. Model case
  14. q-logarithmic function
  15. Upper incomplete gamma function
  16. ...and 8 more sections

Key Result

Theorem 1.7

Under Assumptions xy and assumption, the interval is well-defined and nonempty. In addition, holds for $\varepsilon$ in the above interval.

Figures (2)

  • Figure 1: Error, estimate$/\Delta_C(x,y)$ vs. regularization parameter $\varepsilon$ for the incomplete gamma function $U_\alpha$. Solid line: error, dashed line: estimate, red: $\alpha=1.0$, blue: $\alpha=1/2$, green: $\alpha=1/3$, purple: $\alpha=1/4$.
  • Figure 3: Error, estimate$/\Delta_C(x,y)$ vs. regularization parameter $\varepsilon$ for a scaled version of $U_{\mathrm{s}}$. Solid line: error, dashed line: estimate, red: $a=2$, blue: $a=3$, green: $a=4$, purple: $a=5$, black: $U_o$.

Theorems & Definitions (13)

  • Definition 1.1
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Theorem 1.7
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • ...and 3 more