On Solution Functions of Optimization: Universal Approximation and Covering Number Bounds

• URL: http://arxiv.org/abs/2212.01314v1
• Date: Fri, 2 Dec 2022 17:16:04 GMT
• Title: On Solution Functions of Optimization: Universal Approximation and Covering Number Bounds
• Authors: Ming Jin, Vanshaj Khattar, Harshal Kaushik, Bilgehan Sel, and Ruoxi Jia
• Abstract summary: We study the expressability of convex optimization functions solution functions of linear targetibility(1) (LP) and approximable approximation power of QP. Our results provide the emph we model for rigorous analysis of the properties of some restricted programming and implications for algorithmic design and performance guarantees.
• Score: 6.3291148076593355
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We study the expressibility and learnability of convex optimization solution functions and their multi-layer architectural extension. The main results are: \emph{(1)} the class of solution functions of linear programming (LP) and quadratic programming (QP) is a universal approximant for the $C^k$ smooth model class or some restricted Sobolev space, and we characterize the rate-distortion, \emph{(2)} the approximation power is investigated through a viewpoint of regression error, where information about the target function is provided in terms of data observations, \emph{(3)} compositionality in the form of a deep architecture with optimization as a layer is shown to reconstruct some basic functions used in numerical analysis without error, which implies that \emph{(4)} a substantial reduction in rate-distortion can be achieved with a universal network architecture, and \emph{(5)} we discuss the statistical bounds of empirical covering numbers for LP/QP, as well as a generic optimization problem (possibly nonconvex) by exploiting tame geometry. Our results provide the \emph{first rigorous analysis of the approximation and learning-theoretic properties of solution functions} with implications for algorithmic design and performance guarantees.

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