Large Deviation Upper Bounds and Improved MSE Rates of Nonlinear SGD: Heavy-tailed Noise and Power of Symmetry
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| Format: | Preprint |
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2024
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| _version_ | 1866917964811337728 |
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| author | Armacki, Aleksandar Yu, Shuhua Bajovic, Dragana Jakovetic, Dusan Kar, Soummya |
| author_facet | Armacki, Aleksandar Yu, Shuhua Bajovic, Dragana Jakovetic, Dusan Kar, Soummya |
| contents | We study large deviation upper bounds and mean-squared error (MSE) guarantees of a general framework of nonlinear stochastic gradient methods in the online setting, in the presence of heavy-tailed noise. Unlike existing works that rely on the closed form of a nonlinearity (typically clipping), our framework treats the nonlinearity in a black-box manner, allowing us to provide unified guarantees for a broad class of bounded nonlinearities, including many popular ones, like sign, quantization, normalization, as well as component-wise and joint clipping. We provide several strong results for a broad range of step-sizes in the presence of heavy-tailed noise with symmetric probability density function, positive in a neighbourhood of zero and potentially unbounded moments. In particular, for non-convex costs we provide a large deviation upper bound for the minimum norm-squared of gradients, showing an asymptotic tail decay on an exponential scale, at a rate $\sqrt{t} / \log(t)$. We establish the accompanying rate function, showing an explicit dependence on the choice of step-size, nonlinearity, noise and problem parameters. Next, for non-convex costs and the minimum norm-squared of gradients, we derive the optimal MSE rate $\widetilde{\mathcal{O}}(t^{-1/2})$. Moreover, for strongly convex costs and the last iterate, we provide an MSE rate that can be made arbitrarily close to the optimal rate $\mathcal{O}(t^{-1})$, improving on the state-of-the-art results in the presence of heavy-tailed noise. Finally, we establish almost sure convergence of the minimum norm-squared of gradients, providing an explicit rate, which can be made arbitrarily close to $o(t^{-1/4})$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_15637 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Large Deviation Upper Bounds and Improved MSE Rates of Nonlinear SGD: Heavy-tailed Noise and Power of Symmetry Armacki, Aleksandar Yu, Shuhua Bajovic, Dragana Jakovetic, Dusan Kar, Soummya Machine Learning Optimization and Control Probability We study large deviation upper bounds and mean-squared error (MSE) guarantees of a general framework of nonlinear stochastic gradient methods in the online setting, in the presence of heavy-tailed noise. Unlike existing works that rely on the closed form of a nonlinearity (typically clipping), our framework treats the nonlinearity in a black-box manner, allowing us to provide unified guarantees for a broad class of bounded nonlinearities, including many popular ones, like sign, quantization, normalization, as well as component-wise and joint clipping. We provide several strong results for a broad range of step-sizes in the presence of heavy-tailed noise with symmetric probability density function, positive in a neighbourhood of zero and potentially unbounded moments. In particular, for non-convex costs we provide a large deviation upper bound for the minimum norm-squared of gradients, showing an asymptotic tail decay on an exponential scale, at a rate $\sqrt{t} / \log(t)$. We establish the accompanying rate function, showing an explicit dependence on the choice of step-size, nonlinearity, noise and problem parameters. Next, for non-convex costs and the minimum norm-squared of gradients, we derive the optimal MSE rate $\widetilde{\mathcal{O}}(t^{-1/2})$. Moreover, for strongly convex costs and the last iterate, we provide an MSE rate that can be made arbitrarily close to the optimal rate $\mathcal{O}(t^{-1})$, improving on the state-of-the-art results in the presence of heavy-tailed noise. Finally, we establish almost sure convergence of the minimum norm-squared of gradients, providing an explicit rate, which can be made arbitrarily close to $o(t^{-1/4})$. |
| title | Large Deviation Upper Bounds and Improved MSE Rates of Nonlinear SGD: Heavy-tailed Noise and Power of Symmetry |
| topic | Machine Learning Optimization and Control Probability |
| url | https://arxiv.org/abs/2410.15637 |