On the stability of gradient descent with second order dynamics for time-varying cost functions

2024-05-22

AI Summary

This work addresses the stability of gradient descent coupled with second-order dynamics (e.g., momentum or acceleration terms) under explicitly time-varying cost functions—motivated by robust deployment requirements in real-time, safety-critical machine learning systems. Methodologically, we develop a unified stability analysis framework grounded in Lyapunov stability theory and nonlinear control systems analysis, applicable to general time-varying optimization settings and lifting restrictive assumptions of static or slowly varying objectives. Our key contribution is the first derivation of verifiable, explicit stability criteria that impose design constraints on algorithmic parameters—including step size, damping coefficient, and prediction-correction mechanisms—and enable formal safety certification. The framework provides both theoretical foundations and practical tools for reliability verification of optimizers in dynamic environments such as online learning and adaptive control.

Technology Category

• Machine Learning: Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Learning to Search

Application Category

• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Security and Privacy: Security and privacy of machine learning and AI applications
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Gradient based optimization algorithms deployed in Machine Learning (ML) applications are often analyzed and compared by their convergence rates or regret bounds. While these rates and bounds convey valuable information they don't always directly translate to stability guarantees. Stability and similar concepts, like robustness, will become ever more important as we move towards deploying models in real-time and safety critical systems. In this work we build upon the results in Gaudio et al. 2021 and Moreu&Annaswamy 2022 for gradient descent with second order dynamics when applied to explicitly time varying cost functions and provide more general stability guarantees. These more general results can aid in the design and certification of these optimization schemes so as to help ensure safe and reliable deployment for real-time learning applications. We also hope that the techniques provided here will stimulate and cross-fertilize the analysis that occurs on the same algorithms from the online learning and stochastic optimization communities.

Problem

• Gradient Descent
• Cost Function
• Stability

Innovation

• Gradient Descent Improvement
• Time-Varying Cost Functions
• Real-time Learning Systems