Distributionally Robust Optimal Regularization

• Distributionally robust optimal regularization is a framework that designs regularizers to remain effective under worst-case distribution shifts using Wasserstein ambiguity sets and convex duality.
• It interpolates between data-adaptive, anisotropic regularizers and uniform priors by tuning a robustness parameter for controlled memorization and smoothing.
• The approach integrates convexity constraints and dual reformulations to yield stable, computationally tractable solutions for inverse problems and statistical estimation.

Distributionally robust optimal regularization refers to the principled design of regularization functionals in inverse problems and statistical estimation to ensure reliability and interpretability under uncertainty in the data-generating distribution. Rather than calibrating the regularizer solely to the observed (empirical) distribution, this methodology imposes robustness with respect to a specified ambiguity set—commonly a Wasserstein ball—around the reference distribution. This paradigm seeks regularizers whose performance remains effective in the worst-case scenario over all distributions in the ambiguity set, leveraging duality theory to reformulate the resulting distributionally robust optimization (DRO) into numerically tractable programs. Beyond yielding regularizers with provable stability under perturbations in the data distribution, the framework naturally interpolates between data-adaptive, highly anisotropic regularization and uniform priors as the robustness parameter varies. Convexity constraints on the regularizer, motivated by both theoretical and practical considerations, are efficiently incorporated within the same convex-analytic architecture, ensuring that the robustified regularization schemes remain reliable for deployment across a range of ill-posed inverse problems and learning tasks (Leong et al., 3 Oct 2025).

1. Distributionally Robust Regularizer Formulation

The central object of paper is the robust regularizer selection problem

$$\min_{K\in\mathcal{S}}\,\max_{Q:d(Q,P)\leq\epsilon}\mathbb{E}_Q\left[\|x\|_K\right]\,\,\text{subject to}\;\mathrm{vol}(K)=1,$$

where each candidate regularizer is the gauge function $\|\cdot\|_K$ of a star body $K$ and $P$ is the nominal data distribution. The ambiguity set is the set of probability distributions $Q$ within a transportation-cost metric distance $d$ (typically Wasserstein-1) at most $\epsilon$ from $P$. This formulation ensures that the resulting regularizer guards not just against nominal behavior but also accounts for potential unseen distributional shifts. The normalization constraint serves to avoid trivial solutions (e.g., scaling the gauge to zero).

The optimal regularizer thus emerges as the solution to a minimax problem, with the inner maximization representing an adversarial worst-case data distribution, and the outer minimization selecting the regularizer most robust in expectation under these shifts.

2. Convex Duality and Program Reformulation

A technical innovation is the development, leveraging convex duality, of a tractable reformulation that eliminates the inner maximization over $Q$. Building on modern results in Wasserstein DRO duality, one can equivalently rewrite the robust regularization problem as a single-level convex program: $$\min_{K, s, \lambda}\; s\epsilon + \int \lambda(x) dP(x) \quad \text{subject to}\quad sC(x,y) + \lambda(x) \geq \|y\|_K,\,\,\forall x,y;\;\;s \geq 0,\,\mathrm{vol}(K)\leq 1,$$ where $C(x,y)$ is the ground transportation cost (e.g., $\|x-y\|$ or $\|x-y\|^2$). This replacement dramatically reduces computational complexity and enables systematic optimization over regularizers while respecting distributional uncertainty.

3. Behavior and Structure of Robust Regularizers

The robustified regularizers interpolate between two extremes as the robustness parameter $\epsilon$ is varied. When $\epsilon$ is small, the regularizer "memorizes" the empirical distribution: the gauge function is heavily adapted to the training data, leading to potentially highly anisotropic or "spiky" star bodies. As $\epsilon$ increases, the regularizer transitions toward more uniform (isotropic) forms. In the limit of large $\epsilon$, the regularizer approaches an isotropic norm such as the $\ell_2$-norm, representing a uniform prior over directions.

A key formula established in the case of Wasserstein-1 ambiguity is

$$\max_{Q:W_1(Q,P)\leq \epsilon} \mathbb{E}_Q\left[\|x\|_K\right] = \mathbb{E}_P \left[\|x\|_K \right] + \epsilon\cdot \mathrm{Lip}(\|\cdot\|_K),$$

so that the worst-case expected penalty is driven by both the data-adaptive term and a regularization proportional to the Lipschitz constant of the candidate gauge function. This demonstrates that the use of the Wasserstein ambiguity set directly encourages selection of smoother, less oscillatory regularizers.

4. Ambiguity Sets and Induced Regularity

The choice of ambiguity set heavily influences the structure and stability of the selected regularizer. Employing Wasserstein-type balls not only captures natural geometric notions of closeness between distributions, but also leads to explicit bounds on the change in the optimal gauge function as the underlying data distribution varies. In particular, for data distributions $P$ and $Q$ that are close in Wasserstein-1 distance,

$$\left\|\|\cdot\|_{\hat K(P)} - \|\cdot\|_{\hat K(Q)}\right\|_\infty \leq \left(\mathrm{Lip}(\hat K(P)) + \mathrm{Lip}(\hat K(Q))\right) W_1(P, Q),$$

so that the robust regularizer exhibits Lipschitz continuity with respect to changes in the data distribution. As $\epsilon$ increases, the penalization through the Lipschitz constant dominates, driving the regularizer toward isotropy and ensuring robust generalization even under substantial distributional uncertainty.

5. Incorporation of Convexity Constraints

Imposing convexity on the candidate regularizer is both natural and desirable in most applications, ensuring that the penalty is a norm rather than just a gauge on a star body. To enforce this, the authors parameterize the family of star bodies $K$ as convex hulls of finite collections of unit vectors scaled by positive parameters $t_i$, and set up a convex program over the $t_i$. The volume constraint can be expressed explicitly in terms of the $t_i$, yielding a finite-dimensional program whose objective is a convex combination of the gauge function evaluated along prescribed directions, subject to convexity and normalization.

Convexity also ensures that the optimal robust regularizer varies continuously with the underlying distribution, a critical property for stable deployment in practice. The regularizer remains well-posed even as the empirical data changes slightly, avoiding the instability associated with data overfitting or degeneracy.

6. Theoretical Guarantees and Computational Solvability

Theoretical advances include a proof of the existence of minimizers whenever $\epsilon > 0$, regardless of whether the nominal data distribution is absolutely continuous or purely atomic. Strong duality guarantees that the reformulated convex program correctly captures the worst-case risk, and the continuity and convexity properties ensure a well-behaved solution. These results rely on dual Brunn–Minkowski theory, advanced measure-theoretic optimal transport, and recent developments in the duality of Wasserstein DRO problems.

Practically, discretizing the direction set and solving the resulting finite convex program is demonstrated to be tractable at modest scales, and the analysis provides explicit guidance as to how the robustness parameter should be selected based on application tolerance to out-of-sample deviations.

7. Interplay Between Robustness, Memorization, and Smoothing

A central insight is that robust optimal regularization provides a systematic tool for trading off memorization (overfitting to the training distribution) against smoothing (adopting an isotropic prior over the ambient space) by tuning the robustness parameter $\epsilon$. This induces a continuous path of regularizers from highly data-specific forms to uniform, interpretable priors. The regularization effect imposed by the Wasserstein ball is tightly controlled by the Lipschitz constant of the regularizer, providing both explicit regularity and stability with respect to distributional shifts. The approach also generalizes to the inclusion of additional structure and constraints as dictated by the application (e.g., enforcing sparsity, group structure, or other geometric priors).

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In summary, the distributionally robust optimal regularization framework introduced in this work establishes a comprehensive methodology for designing regularizers that are reliable under model uncertainty. Through convex duality and regularization induced by the geometry of Wasserstein balls, it provides both explicit formulas and efficient convex programs for learning regularization functionals that interpolate between data-adaptivity and uniformity, ensuring stability, interpretability, and robust performance in settings where the true data distribution is subject to uncertainty (Leong et al., 3 Oct 2025).