2025/12/22 by Aleksey Minabutdinov, Minabutdinov, Aleksey, Patrick Cheridito +1
Computer Science · Mathematics · Physics and Astronomy · #35F21 #65K10 #Advanced Optimization Algorithms Research #FOS: Computer and information sciences #FOS: Mathematics #G.1.6 #I.2.6 #I.5.1 #Machine Learning (cs.LG) #Model Reduction and Neural Networks #Optimization and Control (math.OC) #Primary 90C25 #Secondary 68T07 #Stochastic Gradient Optimization Techniques
paper · doi:10.48550/arxiv.2512.19649
openalex publication_date 2025/12/22 · openalex created_date 2025/12/24 · openalex updated_date 2026/07/28
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 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.