Kurdyka-Łojasiewicz exponent via Hadamard parametrization
Fuente:
arXiv
Salvato in:
| Autori principali: | , , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866913393127981056 |
|---|---|
| author | Ouyang, Wenqing Liu, Yuncheng Pong, Ting Kei Wang, Hao |
| author_facet | Ouyang, Wenqing Liu, Yuncheng Pong, Ting Kei Wang, Hao |
| contents | We consider a class of $\ell_1$-regularized optimization problems and the associated smooth "over-parameterized" optimization problems built upon the Hadamard parametrization, or equivalently, the Hadamard difference parametrization (HDP). We characterize the set of second-order stationary points of the HDP-based model and show that they correspond to some stationary points of the corresponding $\ell_1$-regularized model. More importantly, we show that the Kurdyka-Lojasiewicz (KL) exponent of the HDP-based model at a second-order stationary point can be inferred from that of the corresponding $\ell_1$-regularized model under suitable assumptions. Our assumptions are general enough to cover a wide variety of loss functions commonly used in $\ell_1$-regularized models, such as the least squares loss function and the logistic loss function. Since the KL exponents of many $\ell_1$-regularized models are explicitly known in the literature, our results allow us to leverage these known exponents to deduce the KL exponents at second-order stationary points of the corresponding HDP-based models, which were previously unknown. Finally, we demonstrate how these explicit KL exponents at second-order stationary points can be applied to deducing the explicit local convergence rate of a standard gradient descent method for minimizing the HDP-based model. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_00377 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Kurdyka-Łojasiewicz exponent via Hadamard parametrization Ouyang, Wenqing Liu, Yuncheng Pong, Ting Kei Wang, Hao Optimization and Control 90C25, 90C26, 68Q25 We consider a class of $\ell_1$-regularized optimization problems and the associated smooth "over-parameterized" optimization problems built upon the Hadamard parametrization, or equivalently, the Hadamard difference parametrization (HDP). We characterize the set of second-order stationary points of the HDP-based model and show that they correspond to some stationary points of the corresponding $\ell_1$-regularized model. More importantly, we show that the Kurdyka-Lojasiewicz (KL) exponent of the HDP-based model at a second-order stationary point can be inferred from that of the corresponding $\ell_1$-regularized model under suitable assumptions. Our assumptions are general enough to cover a wide variety of loss functions commonly used in $\ell_1$-regularized models, such as the least squares loss function and the logistic loss function. Since the KL exponents of many $\ell_1$-regularized models are explicitly known in the literature, our results allow us to leverage these known exponents to deduce the KL exponents at second-order stationary points of the corresponding HDP-based models, which were previously unknown. Finally, we demonstrate how these explicit KL exponents at second-order stationary points can be applied to deducing the explicit local convergence rate of a standard gradient descent method for minimizing the HDP-based model. |
| title | Kurdyka-Łojasiewicz exponent via Hadamard parametrization |
| topic | Optimization and Control 90C25, 90C26, 68Q25 |
| url | https://arxiv.org/abs/2402.00377 |