Deep Legendre Transform
Aleksey Minabutdinov, Patrick Cheridito

TL;DR
This paper presents a new deep learning algorithm for efficiently computing convex conjugates of differentiable convex functions, overcoming high-dimensional computational challenges and providing accurate approximations with error estimates.
Contribution
The paper introduces an implicit Fenchel formulation-based neural network method for convex conjugation, enabling efficient high-dimensional computation and exact solutions via symbolic regression.
Findings
Accurate convex conjugate computations in high dimensions
Provides a posteriori error estimates for approximations
Achieves exact solutions for specific functions using symbolic regression
Abstract
We introduce a novel deep learning algorithm for computing convex conjugates of differentiable convex functions, a fundamental operation in convex analysis with various applications in different fields such as optimization, control theory, physics and economics. While traditional numerical methods suffer from the curse of dimensionality and become computationally intractable in high dimensions, more recent neural network--based approaches scale better, but have mostly been studied with the aim of solving optimal transport problems and require the solution of complicated optimization or max--min problems. Using an implicit Fenchel formulation of convex conjugation, our approach facilitates an efficient gradient--based framework for the minimization of approximation errors and, as a byproduct, also provides a posteriori estimates of the approximation accuracy. Numerical experiments…
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Taxonomy
TopicsStochastic Gradient Optimization Techniques · Advanced Optimization Algorithms Research · Model Reduction and Neural Networks
