Interpolating between Optimal Transport and KL regularized Optimal Transport using Rényi Divergences

TL;DR

This work introduces a Rényi-divergence-based regularization of optimal transport to interpolate between the classical unregularized OT and KL-regularized OT. It develops two reformulations: a premetric defined by Rényi-ball constraints and a penalized objective, and derives a dual framework with explicit primal plans expressed via dual potentials. The authors prove fundamental properties, including existence/uniqueness of minimizers and interpolation theorems governing limits as $α→0,1$ and $ε→0,∞$, along with a nested mirror-descent algorithm for computation. Empirical results on synthetic and real data show that Rényi-regularized OT plans are tighter and closer to unregularized OT than KL or Tsallis counterparts, with improved ground-truth recovery in tasks such as voter migration forecasting.

Abstract

Regularized optimal transport (OT) has received much attention in recent years starting from Cuturi's introduction of Kullback-Leibler (KL) divergence regularized OT. In this paper, we propose regularizing the OT problem using the family of $α$-Rényi divergences for $α\in (0, 1)$. Rényi divergences are neither $f$-divergences nor Bregman distances, but they recover the KL divergence in the limit $α\nearrow 1$. The advantage of introducing the additional parameter $α$ is that for $α\searrow 0$ we obtain convergence to the unregularized OT problem. For the KL regularized OT problem, this was achieved by letting the regularization parameter $\varepsilon$ tend to zero, which causes numerical instabilities. We present two different ways to obtain premetrics on probability measures, namely by Rényi divergence constraints and by penalization. The latter premetric interpolates between the unregularized and the KL regularized OT problem with weak convergence of the unique minimizer, generalizing the interpolation property of KL regularized OT. We use a nested mirror descent algorithm to solve the primal formulation. Both on real and synthetic data sets Rényi regularized OT plans outperform their KL and Tsallis counterparts in terms of being closer to the unregularized transport plans and recovering the ground truth in inference tasks better.

Figures (2)

• Figure 1: Comparison of the unregularized plan (right) with the KL regularized plans for different regularization parameters, between two Gaussians, for a (scaled) squared Euclidean distance matrix FLA2021. The two plans on the right had to be computed using improved versions of the Sinkhorn algorithm CPSV17S19, because the vanilla version diverged.
• Figure 2: Plot of the discrete Rényi divergence (left), defined in \ref{['eq:discreteRenyi']}, and the discrete Tsallis divergence (right), both of $(p,1-p)$ to $(\frac{1}{4},\frac{3}{4})$ over $p \in [0, 1]$ for different values of $\alpha \in [0, 1]$ as indicated by the color bar. Note that in the left plot, the green band extends much higher and the red band is much thinner than in the right plot.

Theorems & Definitions (34)

• Definition 2.1: $\alpha$-Rényi divergence EH14
• Definition 2.2: Markov kernel
• Example 2.1: Deterministic Markov kernel
• Proposition 2.1: Properties of the Rényi divergence
• Definition 2.3: $\alpha$-Rényi ball of level $\gamma$ centered around $\mu\otimes\nu$
• Lemma 2.1
• proof
• Definition 2.4: Rényi-OT premetric
• Proposition 2.2
• proof