On improving generalization in a class of learning problems with the method of small parameters for weakly-controlled optimal gradient systems
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866910742248161280 |
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| author | Befekadu, Getachew K. |
| author_facet | Befekadu, Getachew K. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_08772 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On improving generalization in a class of learning problems with the method of small parameters for weakly-controlled optimal gradient systems Befekadu, Getachew K. Optimization and Control Machine Learning 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. |
| title | On improving generalization in a class of learning problems with the method of small parameters for weakly-controlled optimal gradient systems |
| topic | Optimization and Control Machine Learning |
| url | https://arxiv.org/abs/2412.08772 |