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Stability and Sharper Risk Bounds with Convergence Rate $\tilde{O}(1/n^2)$

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📄 Abstract

Abstract: Prior work (Klochkov $\&$ Zhivotovskiy, 2021) establishes at most $O\left(\log (n)/n\right)$ excess risk bounds via algorithmic stability for strongly-convex learners with high probability. We show that under the similar common assumptions -- - Polyak-Lojasiewicz condition, smoothness, and Lipschitz continous for losses -- - rates of $O\left(\log^2(n)/n^2\right)$ are at most achievable. To our knowledge, our analysis also provides the tightest high-probability bounds for gradient-based generalization gaps in nonconvex settings.

Authors (4)

Bowei Zhu

Shaojie Li

Mingyang Yi

Yong Liu

Submitted

October 13, 2024

arXiv Category

cs.LG

arXiv PDF

Key Contributions

Establishes tighter excess risk bounds of $O(\log^2(n)/n^2)$ for strongly-convex learners using algorithmic stability, improving upon prior $O(\log(n)/n)$ bounds. It also provides the tightest high-probability bounds for generalization gaps in non-convex settings under common assumptions.

Business Value

Provides stronger theoretical guarantees for the performance and generalization of machine learning algorithms, which can lead to the development of more reliable and robust AI systems.

Paper Metadata

Innovation Type

Theoretical Analysis

Deployment Feasibility

N/A (Theoretical work)

Limitations Addressed

Improves upon existing excess risk bounds and generalization gap bounds, providing sharper theoretical guarantees for learning algorithms.

Performance Gains

Improved theoretical bounds (e.g., $O(\log^2(n)/n^2)$ vs $O(\log(n)/n)$).

Technical Tags

excess risk boundsalgorithmic stabilityconvergence rategeneralization gapsnon-convex optimizationhigh probability boundsoptimization theory

Research Topics

Machine Learning TheoryOptimizationGeneralization BoundsAlgorithmic Stability

Methods & Architectures

High-probability analysisAlgorithmic stability analysisAnalysis of convergence rates

Applications & Tasks

Theoretical Machine Learning Optimization Algorithms Deriving tighter generalization boundsAnalyzing convergence rates Establishing $O(\log^2(n)/n^2)$ excess risk boundsProviding tightest high-probability bounds for generalization gaps in non-convex settings

Related Fields

Machine Learning TheoryOptimizationStatisticsTheoretical Computer Science

Keywords

excess risk boundsalgorithmic stabilityconvergence rategeneralization gapnon-convex optimizationhigh probability boundsmachine learning theoryoptimizationPolyak-Lojasiewicz conditionsmoothnessLipschitz continuity

Academic Context

#Machine Learning Theory#Optimization#Generalization Bounds#Algorithmic Stability

Commercial Potential

Competitive Edge

Offers tighter theoretical bounds than previously established in the literature for specific learning settings.

Resource Requirements

Compute Needs

N/A (Theoretical)

Data Requirements

N/A (Theoretical)

Deployment Constraints

N/A (Theoretical)

Scalability

N/A (Theoretical)

Production Readiness

Maturity Level

Theoretical Foundation

Patent Potential

Low

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