arxiv_ml 70% Match Research Paper Machine learning theorists,Researchers in optimization,Statistical physicists,Deep learning researchers 19 hours ago

Limit Theorems for Stochastic Gradient Descent in High-Dimensional Single-Layer Networks

reinforcement-learning › robotics-rl

📄 Abstract

Abstract: This paper studies the high-dimensional scaling limits of online stochastic gradient descent (SGD) for single-layer networks. Building on the seminal work of Saad and Solla, which analyzed the deterministic (ballistic) scaling limits of SGD corresponding to the gradient flow of the population loss, we focus on the critical scaling regime of the step size. Below this critical scale, the effective dynamics are governed by ballistic (ODE) limits, but at the critical scale, new correction term appears that changes the phase diagram. In this regime, near the fixed points, the corresponding diffusive (SDE) limits of the effective dynamics reduces to an Ornstein-Uhlenbeck process under certain conditions. These results highlight how the information exponent controls sample complexity and illustrates the limitations of deterministic scaling limit in capturing the stochastic fluctuations of high-dimensional learning dynamics.

Key Contributions

This paper analyzes the high-dimensional scaling limits of online SGD for single-layer networks, focusing on the critical step size regime. It shows that below this scale, dynamics follow ballistic (ODE) limits, but at the critical scale, new correction terms appear. Near fixed points, the diffusive (SDE) limits reduce to an Ornstein-Uhlenbeck process, revealing how the information exponent controls sample complexity and highlighting limitations of deterministic scaling limits.

Business Value

A deeper theoretical understanding of SGD dynamics can lead to the development of more efficient and stable training algorithms for deep learning models, potentially reducing training time and improving performance.

Paper Metadata

Innovation Type

Theoretical

Deployment Feasibility

This is a theoretical analysis, not a direct implementation. Its value lies in guiding the design of future algorithms and understanding existing ones.

Limitations Addressed

Limitations of deterministic scaling limits (gradient flow) in capturing stochastic fluctuations,Understanding the critical role of step size in high-dimensional SGD,Characterizing the behavior of SGD in the diffusive regime

Performance Gains

Provides theoretical insights into sample complexity and the behavior of SGD dynamics.

Technical Tags

stochastic gradient descent (SGD)high-dimensionalsingle-layer networksscaling limitsgradient flowdiffusive limitsOrnstein-Uhlenbeck processsample complexity

Research Topics

Machine Learning TheoryOptimization AlgorithmsDeep Learning AnalysisStatistical MechanicsStochastic Processes

Methods & Architectures

Analysis of SGD scaling limitsDerivation of diffusive (SDE) limitsIdentification of critical scaling regimes Single-layer Neural Networks

Applications & Tasks

Machine Learning Theory Statistical Physics Understanding high-dimensional SGD dynamicsAnalyzing scaling limitsCharacterizing stochastic fluctuationsDetermining sample complexity Deriving theoretical scaling limits of SGDAnalyzing the impact of step size on dynamicsCharacterizing diffusive limits near fixed pointsUnderstanding information exponent's role

Related Fields

Machine Learning TheoryOptimizationStatisticsStatistical PhysicsDeep Learning

Keywords

Stochastic Gradient DescentSGDHigh-Dimensional LearningNeural NetworksScaling LimitsGradient FlowDiffusive LimitsSDEOrnstein-Uhlenbeck ProcessSample ComplexityMachine Learning TheoryOptimization Dynamics

Academic Context

#Machine Learning Theory#Optimization Algorithms#Deep Learning Analysis#Statistical Mechanics#Stochastic Processes

Commercial Potential

Potential Products

More robust and efficient optimization algorithmsTheoretical frameworks for analyzing deep learning training

Target Industries

TechnologyAI ResearchSoftware Development

Use Case Examples

Designing new optimizers that better handle high-dimensional, noisy loss landscapesUnderstanding the theoretical limits of training large neural networks

Competitive Edge

Extends foundational work on SGD scaling limits by analyzing the critical regime and diffusive dynamics, providing a more nuanced understanding than purely deterministic ODE limits.

Market Opportunity

Fundamental research underpinning the entire ML industry.

Revenue Models

Academic publicationsguiding future algorithm development

Resource Requirements

Compute Needs

Low (theoretical analysis)

Data Requirements

Theoretical models of loss landscapes

Deployment Constraints

Theoretical findings need to be translated into practical algorithmic improvements.

Scalability

Focuses on high-dimensional scaling limits.

Production Readiness

Maturity Level

Theoretical Foundation

Time to Market

Ongoing research and development

Patent Potential

Low (theoretical explanation)

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