arXiv Releases v2 Version of Paper on Solving Convex Optimization Problems with Neural Operators
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Anastasis Kratsios, Ariel Neufeld, and Philipp Schmocker have released the v2 version of a research paper on Proximal Equilibrium Operators (PEO) on arXiv. This operator is constructed through finite-dimensional deep equilibrium layers and can solve a family of convex optimization problems on separable Hilbert spaces. The paper proves that when the input loss function belongs to a suitable infinite-dimensional set, the trained PO can uniformly approximate the corresponding solution with arbitrary precision, with rank and width growing only logarithmically with the reciprocal of the approximation error, and depth growing linearly. The theoretical conclusions of this research and the trainability of PEO have been validated in three types of applications: nonlinear PDEs, stochastic optimal control, and financial portfolio optimization and hedging.
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- arXiv q-finCertifiable Neural Operators for Simultaneously Solving Infinitely Many Convex Programs: Logarithmic Rank and Linear Complexity
This arXiv paper proposes proximal equilibrium operators (PEOs), constructed via finite-dimensional deep equilibrium layers, that can solve a family of convex optimization problems on separable Hilbert spaces. The paper proves that when input loss functions lie in a suitable infinite-dimensional set, PEOs can uniformly approximate corresponding solutions to arbitrary precision, with rank and width growing only logarithmically with the inverse of approximation error, and depth growing linearly. The related theory and trainability have been validated across three application domains: nonlinear PDEs, stochastic optimal control, and financial portfolio optimization and hedging.
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