Cauchy-Schwarz Regularizers

2025-03-03

AI Summary

Modeling discrete, structured, and orthogonal prior constraints simultaneously within optimization frameworks remains challenging. Method: We propose a novel differentiable regularizer—the Cauchy–Schwarz (CS) regularizer—that systematically derives regularization principles from the Cauchy–Schwarz inequality, yielding an explicit, scale-adaptive, computationally lightweight penalty term free of spurious stationary points. Unlike existing approaches, it induces desired structural properties—such as orthogonality or feature vector structure—directly within gradient-based optimization, without auxiliary training or architectural modifications. Contribution/Results: Empirical evaluation demonstrates that the CS regularizer significantly improves structural fidelity in underdetermined linear inverse problems and achieves state-of-the-art accuracy in low-bit neural network weight quantization, outperforming mainstream quantization methods. Its analytical tractability, parameter-free adaptivity, and seamless integration into standard optimization pipelines make it broadly applicable across structured learning and inverse problem domains.

Technology Category

• Computer Vision: Learning & Optimization for CV
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

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.

Problem

• Introduces Cauchy-Schwarz regularizers for optimization problems.
• Promotes discrete-valued vectors, eigenvectors, and orthogonal matrices.
• Enhances neural network weight quantization and underdetermined systems.

Innovation

• Introduces Cauchy-Schwarz regularizers for optimization.
• Promotes discrete vectors, eigenvectors, and orthogonality.
• Adapts scale automatically, suitable for neural networks.