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

• URL: http://arxiv.org/abs/2412.08772v1
• Date: Wed, 11 Dec 2024 20:50:29 GMT
• Title: On improving generalization in a class of learning problems with the method of small parameters for weakly-controlled optimal gradient systems
• Authors: Getachew K. Befekadu,
• Abstract summary: 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. Using the perturbation theory, we provide results that will allow us to solve a sequence of optimization problems. We also provide an estimate for the rate of convergence for such approximate optimal solutions.
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• Abstract: 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.

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