A uniform rate of convergence for the entropic potentials in the quadratic Euclidean setting

1. Problem Statement

The paper addresses the Quadratic Euclidean Optimal Transport (OT) problem between two absolutely continuous probability measures, $\mu$ and $\nu$, on $\mathbb{R}^d$.

• The Unregularized Problem: The goal is to find the Brenier potential $\phi_0$ (a convex function) such that its gradient $\nabla \phi_0$ pushes $\mu$ forward to $\nu$. This is equivalent to solving a second-order nonlinear PDE (Monge-Ampère equation), which is computationally difficult.
• The Entropic Regularization: To make the problem tractable, the authors consider the Entropic Optimal Transport problem with a regularization parameter $\varepsilon > 0$. This introduces an entropy term to the objective function, leading to a unique solution characterized by entropic potentials $(\phi_\varepsilon, \psi_\varepsilon)$.
• The Core Question: As $\varepsilon \to 0$, the entropic potentials $\phi_\varepsilon$ and their gradients $\nabla \phi_\varepsilon$ converge to the Brenier potential $\phi_0$ and its gradient $\nabla \phi_0$. While convergence is known (often via $\Gamma$-convergence or compactness arguments), the paper seeks to quantify the rate of this convergence uniformly on compact sets. Specifically, the authors aim to bound:
  $$\|\nabla \phi_\varepsilon - \nabla \phi_0\|_{K, \infty} = \sup_{x \in K} |\nabla \phi_\varepsilon(x) - \nabla \phi_0(x)|$$
  and similarly for the potentials themselves, where $K \subset \mathbb{R}^d$ is a compact set.

2. Methodology

The author employs a multi-step analytical approach combining convex analysis, functional inequalities, and interpolation theory.

A. Convexity and Regularity

A foundational observation is that the entropic potentials $\phi_\varepsilon$ are convex functions (Proposition 2.1). This is crucial because, for convex functions, uniform convergence on compact sets implies uniform convergence of their gradients on compact sets.
The paper relies on specific convexity assumptions on the measures $\mu$ and $\nu$:

1. Log-Concavity/Convexity of Potentials (A1): The measures have densities $e^{-V}$ and $e^{-W}$ where $V$ and $W$ are smooth. Specifically, $\nabla^2 V \preceq \alpha I$ and $\nabla^2 W \succeq \beta I$. This ensures that both the Brenier map $\nabla \phi_0$ and the entropic map $\nabla \phi_\varepsilon$ are globally Lipschitz with explicit constants (Theorems 2.2 and 2.3).
2. Poincaré Inequality (A2): The measure $\mu$ satisfies a Poincaré inequality, which allows controlling the $L^2$ norm of a function by the $L^2$ norm of its gradient (given zero mean).
3. Finite Differential Entropy (A3): Ensures the existence of necessary moments and entropy bounds.

B. $L^2$ Convergence Baseline

The paper first establishes that under assumptions (A1)-(A3), the difference between the gradients converges in the $L^2(\mu)$ norm at a rate of $O(\sqrt{\varepsilon})$ (specifically $\|\nabla \phi_\varepsilon - \nabla \phi_0\|_{L^2(\mu)}^2 \leq C\varepsilon$). This is derived from known results on the convergence of the transport costs and the Lipschitz property of the optimal map.

C. Gagliardo-Nirenberg Interpolation

To upgrade the convergence from $L^2$ to uniform convergence ($L^\infty$) on compact sets, the author utilizes the Gagliardo-Nirenberg inequality.

• The inequality interpolates between the $L^\infty$ norm of the gradient, the $L^\infty$ norm of the Hessian, and the $L^2$ norm of the function.
• By choosing specific parameters ($j=1, k=2, p=\infty, q=2, r=\infty$), the author derives an exponent $\theta = \frac{d+2}{d+4}$.
• The strategy involves:
  1. Multiplying the difference $(\phi_\varepsilon - \phi_0)$ by a smooth cutoff function $w$ supported on a ball containing $K$.
  2. Bounding the $L^\infty$ norm of the Hessian of the product using the global Lipschitz constants derived from (A1).
  3. Bounding the $L^2$ norm of the product using the Poincaré inequality and the known $L^2$ convergence rate ($O(\varepsilon)$).

D. Gaussian Case (Appetizer)

Before the general proof, the author explicitly solves the problem for Gaussian measures ($\mu = \mathcal{N}(0, A), \nu = \mathcal{N}(0, B)$). In this case, the potentials are quadratic, and the gradients are linear maps. Using matrix Taylor expansions, the author proves an explicit $O(\varepsilon)$ convergence rate, which serves as a benchmark and motivates the general case.

3. Key Contributions and Results

Main Theorem 1.2 (Gradient Convergence)

For measures satisfying (A1), (A2), and (A3), and for any compact set $K \subseteq \mathbb{R}^d$, there exists a constant $C_{grad}$ such that:
$$\|\nabla \phi_\varepsilon - \nabla \phi_0\|_{K, \infty} \leq C_{grad} \, \varepsilon^{\frac{1}{d+4}}$$
This provides the first quantitative bound for the uniform convergence of entropic gradients in the continuous setting.

Main Theorem 1.3 (Potential Convergence)

Under the same assumptions and a normalization condition ($\int \phi_\varepsilon d\mu = 0$), the potentials themselves converge uniformly on compact, connected sets $K$ with rate:
$$\|\phi_\varepsilon - \phi_0\|_{K, \infty} \leq C_{pot} \left( \varepsilon^{\frac{1}{d+4}} + \varepsilon \right)$$

Proposition 1.1 (Gaussian Case)

For Gaussian measures, the convergence rate is strictly $O(\varepsilon)$, which is faster than the general bound. The constant depends on the operator norms of the covariance matrices and the radius of the compact set.

4. Significance and Context

• Bridging Theory and Practice: Entropic OT is widely used in machine learning (e.g., Sinkhorn algorithm) because it is numerically efficient. However, understanding how well it approximates the true OT map (Brenier map) is critical for applications requiring high precision. This paper quantifies exactly how small $\varepsilon$ needs to be to guarantee a specific error bound on the transport map.
• First Continuous Uniform Rate: Previous literature had established rates for:
  • Discrete and semi-discrete settings.
  • $L^2$ norms of gradients (e.g., $O(\varepsilon)$ or $O(-\varepsilon \log \varepsilon)$).
  • Convergence of the value function (cost).
  • This work is the first to provide a rate for uniform convergence on compact sets in the continuous setting.
• Methodological Innovation: The use of the Gagliardo-Nirenberg inequality to bridge $L^2$ convergence (derived from cost differences) to $L^\infty$ convergence (uniform error) is a novel technique in this specific context of entropic regularization.
• Limitations: The rate $\varepsilon^{\frac{1}{d+4}}$ is likely not optimal (as seen in the Gaussian case where it is $\varepsilon^1$), but it is the first explicit bound for general measures. The dependence on dimension $d$ in the exponent suggests the "curse of dimensionality" affects the uniform convergence rate.

In summary, this paper rigorously establishes that entropic regularization provides a consistent approximation of the optimal transport map, providing explicit error bounds that decay as $\varepsilon \to 0$, thereby justifying the use of entropic OT for approximating the Brenier map in continuous spaces.