# On improving generalization in a class of learning problems with the method of small parameters for weakly-controlled optimal gradient systems

arXiv: https://arxiv.org/abs/2412.08772 · HTML: https://mlsanity.com/paper/2412.08772/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 1 of its 9 references that are in the map.
- First posted on arXiv 11 Dec 2024.

## Builds on (most cited first, 1 of 1)

- [A successive approximation method in functional spaces for hierarchical optimal control problems and its application to learning](https://mlsanity.com/paper/2410.20617/) (2024)

## Abstract (arXiv)

In this paper, we provide a mathematical framework for improving generalization in a class of learning problems which is related to point estimations for modeling of high-dimensional nonlinear functions. In particular, we consider a variational problem for a weakly-controlled gradient system, whose control input enters into the system dynamics as a coefficient to a nonlinear term which is scaled by a small parameter. Here, the optimization problem consists of a cost functional, which is associated with how to gauge the quality of the estimated model parameters at a certain fixed final time w.r.t. the model validating dataset, while the weakly-controlled gradient system, whose the time-evolution is guided by the model training dataset and its perturbed version with small random noise. Using the perturbation theory, we provide results that will allow us to solve a sequence of optimization problems, i.e., a set of decomposed optimization problems, so as to aggregate the corresponding approximate optimal solutions that are reasonably sufficient for improving generalization in such a class of learning problems. Moreover, we also provide an estimate for the rate of convergence for such approximate optimal solutions. Finally, we present some numerical results for a typical case of nonlinear regression problem.