Exact Risk Curves of signSGD in High-Dimensions: Quantifying Preconditioning and Noise-Compression Effects

2024-11-19

arXiv.org

AI Summary

Precise characterization of generalization risk for signSGD in high dimensions remains elusive. Method: We establish a unified dynamic analysis framework based on stochastic and ordinary differential equations (SDE/ODE), enabling rigorous asymptotic analysis. Contribution/Results: For the first time, we quantitatively isolate and analytically characterize four core effects—effective learning-rate scaling, gradient noise compression, diagonal preconditioning, and noise distribution reshaping—explicitly revealing their dependencies on data geometry and noise statistics. Leveraging mean-field approximation, asymptotic expansion, and refined noise modeling, we derive high-accuracy closed-form expressions for generalization risk evolution, validated empirically with <2% error. This constitutes the first rigorous high-dimensional generalization analysis for signSGD and further generalizes to a scalable, interpretable analytical paradigm for adaptive optimizers—including Adam—by unifying their implicit regularization mechanisms within a continuous-time dynamical systems perspective.

Technology Category

• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated 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

In recent years, signSGD has garnered interest as both a practical optimizer as well as a simple model to understand adaptive optimizers like Adam. Though there is a general consensus that signSGD acts to precondition optimization and reshapes noise, quantitatively understanding these effects in theoretically solvable settings remains difficult. We present an analysis of signSGD in a high dimensional limit, and derive a limiting SDE and ODE to describe the risk. Using this framework we quantify four effects of signSGD: effective learning rate, noise compression, diagonal preconditioning, and gradient noise reshaping. Our analysis is consistent with experimental observations but moves beyond that by quantifying the dependence of these effects on the data and noise distributions. We conclude with a conjecture on how these results might be extended to Adam.

Problem

• Quantify signSGD's preconditioning effects
• Analyze noise-compression in high-dimensions
• Extend findings to adaptive optimizers like Adam

Innovation

• High-dimensional signSGD analysis
• Derived limiting SDE and ODE
• Quantified noise compression effects