On characterizing optimal learning trajectories in a class of learning problems
Getachew K Befekadu
TL;DR
The paper addresses how to characterize optimal learning trajectories in a class of high-dimensional nonlinear estimation problems by linking Pontryagin's maximum principle with dynamic programming. It formulates learning as an optimal-control problem over a weakly-controlled gradient flow with small $\epsilon$, driven by training data $\mathcal{Z}^{(1)}$ and its perturbation $\tilde{\mathcal{Z}}^{(1)}$, and evaluated on a validation set $\mathcal{Z}^{(2)}$ via the final cost $\Phi$. The main contributions include deriving Euler–Lagrange/maximum-principle conditions, relating the adjoint to the dynamic-programming value function (via the Hamilton–Jacobi–Bellman framework), and providing a practical successive Galerkin algorithm to approximate the optimal learning trajectories by expressing the control as $u(t)=C\Psi(t)$ and updating coefficients until convergence, which yields the optimal final parameters $\theta^{*}$. This work offers a principled, constructively computable bridge between optimal-control theory and learning generalization, enabling principled design of learning dynamics under perturbations.
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.