Convergence rates of stochastic gradient method with independent sequences of step-size and momentum weight
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| Format: | Preprint |
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2024
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| _version_ | 1866909279969083392 |
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| author | Hwang, Wen-Liang |
| author_facet | Hwang, Wen-Liang |
| contents | In large-scale learning algorithms, the momentum term is usually included in the stochastic sub-gradient method to improve the learning speed because it can navigate ravines efficiently to reach a local minimum. However, step-size and momentum weight hyper-parameters must be appropriately tuned to optimize convergence. We thus analyze the convergence rate using stochastic programming with Polyak's acceleration of two commonly used step-size learning rates: ``diminishing-to-zero" and ``constant-and-drop" (where the sequence is divided into stages and a constant step-size is applied at each stage) under strongly convex functions over a compact convex set with bounded sub-gradients. For the former, we show that the convergence rate can be written as a product of exponential in step-size and polynomial in momentum weight. Our analysis justifies the convergence of using the default momentum weight setting and the diminishing-to-zero step-size sequence in large-scale machine learning software. For the latter, we present the condition for the momentum weight sequence to converge at each stage. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_02678 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Convergence rates of stochastic gradient method with independent sequences of step-size and momentum weight Hwang, Wen-Liang Machine Learning Optimization and Control In large-scale learning algorithms, the momentum term is usually included in the stochastic sub-gradient method to improve the learning speed because it can navigate ravines efficiently to reach a local minimum. However, step-size and momentum weight hyper-parameters must be appropriately tuned to optimize convergence. We thus analyze the convergence rate using stochastic programming with Polyak's acceleration of two commonly used step-size learning rates: ``diminishing-to-zero" and ``constant-and-drop" (where the sequence is divided into stages and a constant step-size is applied at each stage) under strongly convex functions over a compact convex set with bounded sub-gradients. For the former, we show that the convergence rate can be written as a product of exponential in step-size and polynomial in momentum weight. Our analysis justifies the convergence of using the default momentum weight setting and the diminishing-to-zero step-size sequence in large-scale machine learning software. For the latter, we present the condition for the momentum weight sequence to converge at each stage. |
| title | Convergence rates of stochastic gradient method with independent sequences of step-size and momentum weight |
| topic | Machine Learning Optimization and Control |
| url | https://arxiv.org/abs/2408.02678 |