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On the SDEs and Scaling Rules for Adaptive Gradient Algorithms

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arxiv 2205.10287 v3 pith:QBJWHBWP submitted 2022-05-20 cs.LG

classification cs.LG
keywords adamgradientrmspropadaptiveapproximationsmethodsoptimizationscaling
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abstract

Approximating Stochastic Gradient Descent (SGD) as a Stochastic Differential Equation (SDE) has allowed researchers to enjoy the benefits of studying a continuous optimization trajectory while carefully preserving the stochasticity of SGD. Analogous study of adaptive gradient methods, such as RMSprop and Adam, has been challenging because there were no rigorously proven SDE approximations for these methods. This paper derives the SDE approximations for RMSprop and Adam, giving theoretical guarantees of their correctness as well as experimental validation of their applicability to common large-scaling vision and language settings. A key practical result is the derivation of a $\textit{square root scaling rule}$ to adjust the optimization hyperparameters of RMSprop and Adam when changing batch size, and its empirical validation in deep learning settings.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Improving Neural Network Training by Decoupling the Magnitude and Direction of Weight Vectors

    cs.LG 2026-06 unverdicted novelty 6.0 of 10

    Splitting weight matrices into a fixed-norm direction and learnable per-row/column magnitudes improves LLM training over AdamW/Muon, removes weight decay and warmup, and transfers the optimal LR across width.

  2. Is your batch size the problem? Revisiting the Adam-SGD gap in language modeling

    cs.LG 2025-06 conditional novelty 6.0 of 10

    SGD with momentum can match Adam's performance in language modeling when trained with small batches and careful tuning, a result that contradicts several popular explanations for the optimizer gap.

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