High-dimensional limit theorems for SGD: Momentum and Adaptive Step-sizes
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arXiv
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| Format: | Preprint |
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2025
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| author | Jagannath, Aukosh Jones-McCormick, Taj Sarangian, Varnan |
| author_facet | Jagannath, Aukosh Jones-McCormick, Taj Sarangian, Varnan |
| contents | We develop a high-dimensional scaling limit for Stochastic Gradient Descent with Polyak Momentum (SGD-M) and adaptive step-sizes. This provides a framework to rigourously compare online SGD with some of its popular variants. We show that the scaling limits of SGD-M coincide with those of online SGD after an appropriate time rescaling and a specific choice of step-size. However, if the step-size is kept the same between the two algorithms, SGD-M will amplify high-dimensional effects, potentially degrading performance relative to online SGD. We demonstrate our framework on two popular learning problems: Spiked Tensor PCA and Single Index Models. In both cases, we also examine online SGD with an adaptive step-size based on normalized gradients. In the high-dimensional regime, this algorithm yields multiple benefits: its dynamics admit fixed points closer to the population minimum and widens the range of admissible step-sizes for which the iterates converge to such solutions. These examples provide a rigorous account, aligning with empirical motivation, of how early preconditioners can stabilize and improve dynamics in settings where online SGD fails. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_03952 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | High-dimensional limit theorems for SGD: Momentum and Adaptive Step-sizes Jagannath, Aukosh Jones-McCormick, Taj Sarangian, Varnan Machine Learning We develop a high-dimensional scaling limit for Stochastic Gradient Descent with Polyak Momentum (SGD-M) and adaptive step-sizes. This provides a framework to rigourously compare online SGD with some of its popular variants. We show that the scaling limits of SGD-M coincide with those of online SGD after an appropriate time rescaling and a specific choice of step-size. However, if the step-size is kept the same between the two algorithms, SGD-M will amplify high-dimensional effects, potentially degrading performance relative to online SGD. We demonstrate our framework on two popular learning problems: Spiked Tensor PCA and Single Index Models. In both cases, we also examine online SGD with an adaptive step-size based on normalized gradients. In the high-dimensional regime, this algorithm yields multiple benefits: its dynamics admit fixed points closer to the population minimum and widens the range of admissible step-sizes for which the iterates converge to such solutions. These examples provide a rigorous account, aligning with empirical motivation, of how early preconditioners can stabilize and improve dynamics in settings where online SGD fails. |
| title | High-dimensional limit theorems for SGD: Momentum and Adaptive Step-sizes |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2511.03952 |