Deep Legendre Transform
This addresses a fundamental operation in convex analysis with applications in optimization, control theory, physics, and economics, offering a scalable solution for high-dimensional problems.
The paper tackles the problem of computing convex conjugates of differentiable convex functions in high dimensions, where traditional methods are intractable, and demonstrates that their deep learning algorithm delivers accurate results across high-dimensional examples and can obtain exact conjugates for specific functions using symbolic regression.
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 error estimates for the approximation quality. Numerical experiments demonstrate our method's ability to deliver accurate results across different high-dimensional examples. Moreover, by employing symbolic regression with Kolmogorov--Arnold networks, it is able to obtain the exact convex conjugates of specific convex functions.