On the Interaction of Noise, Compression Role, and Adaptivity under $(L_0, L_1)$-Smoothness: An SDE-based Approach

• URL: http://arxiv.org/abs/2506.00181v1
• Date: Fri, 30 May 2025 19:35:15 GMT
• Title: On the Interaction of Noise, Compression Role, and Adaptivity under $(L_0, L_1)$-Smoothness: An SDE-based Approach
• Authors: Enea Monzio Compagnoni, Rustem Islamov, Antonio Orvieto, Eduard Gorbunov,
• Abstract summary: We study the dynamics of Distributed SGD, Distributed Compressed SGD, and Distributed SignSGD.<n>Our analysis provides insights into the intricate interactions between batch noise, gradient compression, and adaptivity.
• Score: 20.77655203511758
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Using stochastic differential equation (SDE) approximations, we study the dynamics of Distributed SGD, Distributed Compressed SGD, and Distributed SignSGD under $(L_0,L_1)$-smoothness and flexible noise assumptions. Our analysis provides insights -- which we validate through simulation -- into the intricate interactions between batch noise, stochastic gradient compression, and adaptivity in this modern theoretical setup. For instance, we show that \textit{adaptive} methods such as Distributed SignSGD can successfully converge under standard assumptions on the learning rate scheduler, even under heavy-tailed noise. On the contrary, Distributed (Compressed) SGD with pre-scheduled decaying learning rate fails to achieve convergence, unless such a schedule also accounts for an inverse dependency on the gradient norm -- de facto falling back into an adaptive method.

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