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- W4310746603 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." @default.
- W4310746603 created "2022-12-17" @default.
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- W4310746603 date "2022-12-02" @default.
- W4310746603 modified "2023-10-06" @default.
- W4310746603 title "On Solution Functions of Optimization: Universal Approximation and Covering Number Bounds" @default.
- W4310746603 doi "https://doi.org/10.48550/arxiv.2212.01314" @default.
- W4310746603 hasPublicationYear "2022" @default.
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