Faster Convergence of Local SGD for Over-Parameterized Models
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arXiv
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866910478160101376 |
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| author | Qin, Tiancheng Etesami, S. Rasoul Uribe, César A. |
| author_facet | Qin, Tiancheng Etesami, S. Rasoul Uribe, César A. |
| contents | Modern machine learning architectures are often highly expressive. They are usually over-parameterized and can interpolate the data by driving the empirical loss close to zero. We analyze the convergence of Local SGD (or FedAvg) for such over-parameterized models in the heterogeneous data setting and improve upon the existing literature by establishing the following convergence rates. For general convex loss functions, we establish an error bound of $Ø(1/T)$ under a mild data similarity assumption and an error bound of $Ø(K/T)$ otherwise, where $K$ is the number of local steps and $T$ is the total number of iterations. For non-convex loss functions we prove an error bound of $Ø(K/T)$. These bounds improve upon the best previous bound of $Ø(1/\sqrt{nT})$ in both cases, where $n$ is the number of nodes, when no assumption on the model being over-parameterized is made. We complete our results by providing problem instances in which our established convergence rates are tight to a constant factor with a reasonably small stepsize. Finally, we validate our theoretical results by performing large-scale numerical experiments that reveal the convergence behavior of Local SGD for practical over-parameterized deep learning models, in which the $Ø(1/T)$ convergence rate of Local SGD is clearly shown. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2201_12719 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Faster Convergence of Local SGD for Over-Parameterized Models Qin, Tiancheng Etesami, S. Rasoul Uribe, César A. Machine Learning Optimization and Control Modern machine learning architectures are often highly expressive. They are usually over-parameterized and can interpolate the data by driving the empirical loss close to zero. We analyze the convergence of Local SGD (or FedAvg) for such over-parameterized models in the heterogeneous data setting and improve upon the existing literature by establishing the following convergence rates. For general convex loss functions, we establish an error bound of $Ø(1/T)$ under a mild data similarity assumption and an error bound of $Ø(K/T)$ otherwise, where $K$ is the number of local steps and $T$ is the total number of iterations. For non-convex loss functions we prove an error bound of $Ø(K/T)$. These bounds improve upon the best previous bound of $Ø(1/\sqrt{nT})$ in both cases, where $n$ is the number of nodes, when no assumption on the model being over-parameterized is made. We complete our results by providing problem instances in which our established convergence rates are tight to a constant factor with a reasonably small stepsize. Finally, we validate our theoretical results by performing large-scale numerical experiments that reveal the convergence behavior of Local SGD for practical over-parameterized deep learning models, in which the $Ø(1/T)$ convergence rate of Local SGD is clearly shown. |
| title | Faster Convergence of Local SGD for Over-Parameterized Models |
| topic | Machine Learning Optimization and Control |
| url | https://arxiv.org/abs/2201.12719 |