Convexity and strict convexity for compositional neural networks in high-dimensional optimal control

TL;DR

This paper extends theoretical results on neural network approximations for high-dimensional optimal control problems, focusing on convexity conditions and providing error bounds under weaker assumptions.

Contribution

It generalizes previous work by relaxing strict convexity to convexity and demonstrates how to reformulate problems to meet convexity requirements.

Findings

Reformulating control problems ensures strict convexity.

Weak error bounds are established for convex, not strictly convex, cost functionals.

Theoretical framework supports curse-of-dimensionality-free approximation.

Abstract

Neural networks (NNs) have emerged as powerful tools for solving high-dimensional optimal control problems. In particular, their compositional structure has been shown to enable efficient approximation of high-dimensional functions, helping to mitigate the curse of dimensionality in optimal control problems. In this work, we build upon the theoretical framework developed by Kang & Gong (SIAM J. Control Optim. 60(2):786-813, 2022), particularly their results on NN approximations for compositional functions in optimal control. Theorem 6.2 in Kang & Gong (SIAM J. Control Optim. 60(2):786-813, 2022) establishes that, under suitable assumptions on the compositional structure and its associated features, optimal control problems with strictly convex cost functionals admit a curse-of-dimensionality-free approximation of the optimal control by NNs. We extend this result in two directions. First, we analyze the strict convexity requirement on the cost functional and demonstrate that reformulating a discrete-time optimal control problem with linear transitions and stage costs as a terminal cost problem ensures the necessary strict convexity. Second, we establish a generalization of Theorem 6.2 in Kang & Gong (SIAM J. Control Optim. 60(2):786-813, 2022) which provides weak error bounds for optimal control approximations by NNs when the cost functional is only convex rather than strictly convex.