From Halpern's Fixed-Point Iterations to Nesterov's Accelerated Interpretations for Root-Finding Problems
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arXiv
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866915938186559488 |
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| author | Tran-Dinh, Quoc |
| author_facet | Tran-Dinh, Quoc |
| contents | We derive an equivalent form of Halpern's fixed-point iteration scheme for solving a co-coercive equation (also called a root-finding problem), which can be viewed as a Nesterov's accelerated interpretation. We show that one method is equivalent to another via a simple transformation, leading to a straightforward convergence proof for Nesterov's accelerated scheme. Alternatively, we directly establish convergence rates of Nesterov's accelerated variant, and as a consequence, we obtain a new convergence rate of Halpern's fixed-point iteration. Next, we apply our results to different methods to solve monotone inclusions, where our convergence guarantees are applied. Since the gradient/forward scheme requires the co-coerciveness of the underlying operator, we derive new Nesterov's accelerated variants for both recent extra-anchored gradient and past-extra anchored gradient methods in the literature. These variants alleviate the co-coerciveness condition by only assuming the monotonicity and Lipschitz continuity of the underlying operator. Interestingly, our new Nesterov's accelerated interpretation of the past-extra anchored gradient method involves two past-iterate correction terms. This formulation is expected to guide us developing new Nesterov's accelerated methods for minimax problems and their continuous views without co-coericiveness. We test our theoretical results on two numerical examples, where the actual convergence rates match well the theoretical ones up to a constant factor. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_04869 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | From Halpern's Fixed-Point Iterations to Nesterov's Accelerated Interpretations for Root-Finding Problems Tran-Dinh, Quoc Optimization and Control We derive an equivalent form of Halpern's fixed-point iteration scheme for solving a co-coercive equation (also called a root-finding problem), which can be viewed as a Nesterov's accelerated interpretation. We show that one method is equivalent to another via a simple transformation, leading to a straightforward convergence proof for Nesterov's accelerated scheme. Alternatively, we directly establish convergence rates of Nesterov's accelerated variant, and as a consequence, we obtain a new convergence rate of Halpern's fixed-point iteration. Next, we apply our results to different methods to solve monotone inclusions, where our convergence guarantees are applied. Since the gradient/forward scheme requires the co-coerciveness of the underlying operator, we derive new Nesterov's accelerated variants for both recent extra-anchored gradient and past-extra anchored gradient methods in the literature. These variants alleviate the co-coerciveness condition by only assuming the monotonicity and Lipschitz continuity of the underlying operator. Interestingly, our new Nesterov's accelerated interpretation of the past-extra anchored gradient method involves two past-iterate correction terms. This formulation is expected to guide us developing new Nesterov's accelerated methods for minimax problems and their continuous views without co-coericiveness. We test our theoretical results on two numerical examples, where the actual convergence rates match well the theoretical ones up to a constant factor. |
| title | From Halpern's Fixed-Point Iterations to Nesterov's Accelerated Interpretations for Root-Finding Problems |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2203.04869 |