Concave Certificates: Geometric Framework for Distributionally Robust Risk and Complexity Analysis

Distributionally Robust (DR) optimization aims to certify worst-case risk within a Wasserstein uncertainty set. Current certifications typically rely either on global Lipschitz bounds, which are often conservative, or on local gradient information, which provides only a first-order approximation. This paper introduces a novel geometric framework based on the least concave majorants of the growth rate function. Our proposed concave certificate establishes a tight bound of DR risk that remains applicable to non-Lipschitz and non-differentiable losses. We extend this framework to complexity analysis, introducing a deterministic bound that complements standard statistical generalization bound. Furthermore, we utilize this certificate to bound the gap between adversarial and empirical Rademacher complexity, demonstrating that dependencies on input diameter, network width, and depth can be eliminated. For practical application in deep learning, we introduce the adversarial score as a tractable relaxation of the concave certificate that enables efficient and layer-wise analysis of neural networks. We validate our theoretical results in various numerical experiments on classification and regression tasks on real-world data.

Implicit Regularization via Spectral Neural Networks and Non-linear Matrix Sensing

The phenomenon of implicit regularization has attracted interest in recent years as a fundamental aspect of the remarkable generalizing ability of neural networks. In a nutshell, it entails that gradient descent dynamics in many neural nets, even without any explicit regularizer in the loss function, converges to the solution of a regularized learning problem. However, known results attempting to theoretically explain this phenomenon focus overwhelmingly on the setting of linear neural nets, and the simplicity of the linear structure is particularly crucial to existing arguments. In this paper, we explore this problem in the context of more realistic neural networks with a general class of non-linear activation functions, and rigorously demonstrate the implicit regularization phenomenon for such networks in the setting of matrix sensing problems, together with rigorous rate guarantees that ensure exponentially fast convergence of gradient descent.In this vein, we contribute a network architecture called Spectral Neural Networks (abbrv. SNN) that is particularly suitable for matrix learning problems. Conceptually, this entails coordinatizing the space of matrices by their singular values and singular vectors, as opposed to by their entries, a potentially fruitful perspective for matrix learning. We demonstrate that the SNN architecture is inherently much more amenable to theoretical analysis than vanilla neural nets and confirm its effectiveness in the context of matrix sensing, via both mathematical guarantees and empirical investigations. We believe that the SNN architecture has the potential to be of wide applicability in a broad class of matrix learning scenarios.