A Generalized Version of Chung's Lemma and its Applications
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917261301776384 |
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| author | Jiang, Li Li, Xiao Milzarek, Andre Qiu, Junwen |
| author_facet | Jiang, Li Li, Xiao Milzarek, Andre Qiu, Junwen |
| contents | Chung's Lemma is a classical tool for establishing asymptotic convergence rates of (stochastic) optimization methods under strong convexity-type assumptions and appropriate polynomial diminishing step sizes. In this work, we develop a generalized version of Chung's Lemma, which provides a simple non-asymptotic convergence framework for a more general family of step size rules. We demonstrate broad applicability of the proposed generalized lemma by deriving tight non-asymptotic convergence rates for a large variety of stochastic methods. In particular, we obtain partially new non-asymptotic complexity results for stochastic optimization methods, such as Stochastic Gradient Descent (SGD) and Random Reshuffling (RR), under a general $(θ,μ)$-Polyak-Lojasiewicz (PL) condition and for various step sizes strategies, including polynomial, constant, exponential, and cosine step sizes rules. Notably, as a by-product of our analysis, we observe that exponential step sizes exhibit superior adaptivity to both landscape geometry and gradient noise; specifically, they achieve optimal convergence rates without requiring exact knowledge of the underlying landscape or separate parameter selection strategies for noisy and noise-free regimes. Our results demonstrate that the developed variant of Chung's Lemma offers a versatile, systematic, and streamlined approach to establish non-asymptotic convergence rates under general step size rules. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_05637 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Generalized Version of Chung's Lemma and its Applications Jiang, Li Li, Xiao Milzarek, Andre Qiu, Junwen Optimization and Control Machine Learning Probability 90C15, 90C30, 90C26 Chung's Lemma is a classical tool for establishing asymptotic convergence rates of (stochastic) optimization methods under strong convexity-type assumptions and appropriate polynomial diminishing step sizes. In this work, we develop a generalized version of Chung's Lemma, which provides a simple non-asymptotic convergence framework for a more general family of step size rules. We demonstrate broad applicability of the proposed generalized lemma by deriving tight non-asymptotic convergence rates for a large variety of stochastic methods. In particular, we obtain partially new non-asymptotic complexity results for stochastic optimization methods, such as Stochastic Gradient Descent (SGD) and Random Reshuffling (RR), under a general $(θ,μ)$-Polyak-Lojasiewicz (PL) condition and for various step sizes strategies, including polynomial, constant, exponential, and cosine step sizes rules. Notably, as a by-product of our analysis, we observe that exponential step sizes exhibit superior adaptivity to both landscape geometry and gradient noise; specifically, they achieve optimal convergence rates without requiring exact knowledge of the underlying landscape or separate parameter selection strategies for noisy and noise-free regimes. Our results demonstrate that the developed variant of Chung's Lemma offers a versatile, systematic, and streamlined approach to establish non-asymptotic convergence rates under general step size rules. |
| title | A Generalized Version of Chung's Lemma and its Applications |
| topic | Optimization and Control Machine Learning Probability 90C15, 90C30, 90C26 |
| url | https://arxiv.org/abs/2406.05637 |