Fast Stochastic Composite Minimization and an Accelerated Frank-Wolfe Algorithm under Parallelization
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866913583982444544 |
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| author | Dubois-Taine, Benjamin Bach, Francis Berthet, Quentin Taylor, Adrien |
| author_facet | Dubois-Taine, Benjamin Bach, Francis Berthet, Quentin Taylor, Adrien |
| contents | We consider the problem of minimizing the sum of two convex functions. One of those functions has Lipschitz-continuous gradients, and can be accessed via stochastic oracles, whereas the other is "simple". We provide a Bregman-type algorithm with accelerated convergence in function values to a ball containing the minimum. The radius of this ball depends on problem-dependent constants, including the variance of the stochastic oracle. We further show that this algorithmic setup naturally leads to a variant of Frank-Wolfe achieving acceleration under parallelization. More precisely, when minimizing a smooth convex function on a bounded domain, we show that one can achieve an $ε$ primal-dual gap (in expectation) in $\tilde{O}(1/ \sqrtε)$ iterations, by only accessing gradients of the original function and a linear maximization oracle with $O(1/\sqrtε)$ computing units in parallel. We illustrate this fast convergence on synthetic numerical experiments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_12751 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Fast Stochastic Composite Minimization and an Accelerated Frank-Wolfe Algorithm under Parallelization Dubois-Taine, Benjamin Bach, Francis Berthet, Quentin Taylor, Adrien Optimization and Control Machine Learning We consider the problem of minimizing the sum of two convex functions. One of those functions has Lipschitz-continuous gradients, and can be accessed via stochastic oracles, whereas the other is "simple". We provide a Bregman-type algorithm with accelerated convergence in function values to a ball containing the minimum. The radius of this ball depends on problem-dependent constants, including the variance of the stochastic oracle. We further show that this algorithmic setup naturally leads to a variant of Frank-Wolfe achieving acceleration under parallelization. More precisely, when minimizing a smooth convex function on a bounded domain, we show that one can achieve an $ε$ primal-dual gap (in expectation) in $\tilde{O}(1/ \sqrtε)$ iterations, by only accessing gradients of the original function and a linear maximization oracle with $O(1/\sqrtε)$ computing units in parallel. We illustrate this fast convergence on synthetic numerical experiments. |
| title | Fast Stochastic Composite Minimization and an Accelerated Frank-Wolfe Algorithm under Parallelization |
| topic | Optimization and Control Machine Learning |
| url | https://arxiv.org/abs/2205.12751 |