On characterizing optimal learning trajectories in a class of learning problems

TL;DR

This paper develops a mathematical framework combining maximum principle and dynamic programming to characterize optimal learning trajectories in high-dimensional nonlinear function modeling, providing insights into model training and validation.

Contribution

It introduces a novel optimal control approach for learning trajectories, linking control theory with model estimation in complex learning problems.

Findings

Framework successfully characterizes optimal trajectories.

Algorithmic method for constructing optimal learning paths.

Enhanced understanding of model performance optimization.

Abstract

In this brief paper, we provide a mathematical framework that exploits the relationship between the maximum principle and dynamic programming for characterizing optimal learning trajectories in a class of learning problem, which is related to point estimations for modeling of high-dimensional nonlinear functions. Here, such characterization for the optimal learning trajectories is associated with the solution of an optimal control problem for a weakly-controlled gradient system with small parameters, whose time-evolution is guided by a model training dataset and its perturbed version, while the optimization problem consists of a cost functional that summarizes how to gauge the quality/performance of the estimated model parameters at a certain fixed final time w.r.t. a model validating dataset. Moreover, using a successive Galerkin approximation method, we provide an algorithmic recipe how to construct the corresponding optimal learning trajectories leading to the optimal estimated model parameters for such a class of learning problem.