Cauchy-Schwarz Regularizers
Sueda Taner, Ziyi Wang, Christoph Studer
TL;DR
This work introduces Cauchy–Schwarz (CS) regularizers that leverage the CS inequality to construct nonnegative losses which vanish exactly on targeted solution-sets, enabling explicit control over the structure of optimal solutions. The CS framework yields concrete instances such as symmetric binary, one-sided binary, and symmetric ternary regularizers for discrete-valued vectors, as well as eigenvector and orthogonal-column matrix regularizers, all of which are differentiable and automatically adapt their scale to the input without extra parameters. The authors analyze invexity considerations and the potential absence of spurious stationary points, and demonstrate applicability to underdetermined linear systems and neural network weight quantization, with scalable gradient-based optimization in mind. They further discuss generalizations (e.g., Hölder-type regularizers) and acknowledge limitations, including nonconvexity and possible spurious stationary points in some cases, while highlighting the practical impact of a flexible, scalable regularization framework for large-scale optimization tasks. Overall, CS regularizers offer a versatile, scale-adaptive toolkit for enforcing diverse solution structures in modern optimization problems and neural network training workflows.
Abstract
We introduce a novel class of regularization functions, called Cauchy-Schwarz (CS) regularizers, which can be designed to induce a wide range of properties in solution vectors of optimization problems. To demonstrate the versatility of CS regularizers, we derive regularization functions that promote discrete-valued vectors, eigenvectors of a given matrix, and orthogonal matrices. The resulting CS regularizers are simple, differentiable, and can be free of spurious stationary points, making them suitable for gradient-based solvers and large-scale optimization problems. In addition, CS regularizers automatically adapt to the appropriate scale, which is, for example, beneficial when discretizing the weights of neural networks. To demonstrate the efficacy of CS regularizers, we provide results for solving underdetermined systems of linear equations and weight quantization in neural networks. Furthermore, we discuss specializations, variations, and generalizations, which lead to an even broader class of new and possibly more powerful regularizers.