Quantization through Piecewise-Affine Regularization: Optimization and Statistical Guarantees

2025-08-14

AI Summary

This work addresses the combinatorial optimization challenge inherent in discrete/quantized parameter learning under supervised settings. We propose a continuous optimization framework based on piecewise affine regularization (PAR). Theoretically, we establish that, under overparameterization, critical points of the PAR loss exhibit intrinsic strong quantization properties and achieve statistical performance comparable to classical regularized estimators. Methodologically, we unify convex, quasiconvex, and nonconvex PAR regularizers, derive closed-form proximal mappings for each, and integrate them with proximal gradient descent, its accelerated variants, and the alternating direction method of multipliers (ADMM). Experiments demonstrate that the framework maintains computational efficiency while providing rigorous statistical guarantees: on linear regression tasks, it matches the accuracy of conventional quantization methods. Overall, this work introduces a new paradigm for discrete optimization—grounded in theoretical rigor yet computationally tractable.

Technology Category

• Machine Learning: Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Computer Vision: Learning & Optimization for CV

Application Category

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

Abstract

Optimization problems over discrete or quantized variables are very challenging in general due to the combinatorial nature of their search space. Piecewise-affine regularization (PAR) provides a flexible modeling and computational framework for quantization based on continuous optimization. In this work, we focus on the setting of supervised learning and investigate the theoretical foundations of PAR from optimization and statistical perspectives. First, we show that in the overparameterized regime, where the number of parameters exceeds the number of samples, every critical point of the PAR-regularized loss function exhibits a high degree of quantization. Second, we derive closed-form proximal mappings for various (convex, quasi-convex, and non-convex) PARs and show how to solve PAR-regularized problems using the proximal gradient method, its accelerated variant, and the Alternating Direction Method of Multipliers. Third, we study statistical guarantees of PAR-regularized linear regression problems; specifically, we can approximate classical formulations of $ell_1$-, squared $ell_2$-, and nonconvex regularizations using PAR and obtain similar statistical guarantees with quantized solutions.

Problem

• Optimizing discrete variables via continuous piecewise-affine regularization
• Proving quantization at critical points in overparameterized learning regimes
• Deriving statistical guarantees for quantized linear regression solutions

Innovation

• Piecewise-affine regularization enables continuous optimization quantization
• Proximal gradient methods solve PAR-regularized problems efficiently
• PAR provides statistical guarantees for quantized linear regression