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

2025-05-30

AI Summary

This work investigates the convergence dynamics of distributed SGD, compressed SGD, and SignSGD under $(L_0, L_1)$-smooth objectives and generalized (including heavy-tailed) gradient noise. Adopting a stochastic differential equation (SDE) framework for continuous-time modeling, we systematically characterize the coupled effects of batch noise, gradient compression, and learning-rate adaptivity. We establish, for the first time, that adaptive methods—including Distributed SignSGD—remain convergent under heavy-tailed noise, whereas standard non-adaptive step-size decay schemes provably fail unless they implicitly depend on gradient norms; this unifies and rigorously justifies the theoretical necessity of adaptivity. Furthermore, we quantify how compression interacts with noise to affect convergence rates, and validate the high fidelity of our SDE approximation via dynamical simulations. The analysis provides a novel theoretical framework and design principles for robust distributed training.

Technology Category

• Machine Learning: Distributed Machine Learning & Federated Learning
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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 extit{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.

Problem

• Analyze noise, compression, adaptivity in Distributed SGD dynamics
• Study convergence of adaptive methods under heavy-tailed noise
• Compare pre-scheduled vs adaptive learning rate strategies

Innovation

• SDE-based analysis of distributed SGD dynamics
• Study of noise and compression interaction
• Adaptive methods convergence under heavy-tailed noise