Convergence Guarantees for RMSProp and Adam in Generalized-smooth Non-convex Optimization with Affine Noise Variance

2024-04-01

arXiv.org

AI Summary

This work addresses the convergence of RMSProp and Adam for generalized smooth nonconvex optimization. Under the weakest known assumptions—coordinate-wise generalized smoothness and affine noise variance—we establish the first tight theoretical guarantees. We introduce a novel descent lemma that overcomes critical challenges: adaptive step-size dependence, unbounded gradient estimates, and mismatched Lipschitz constants. Rigorously, we prove that both algorithms converge to an ε-stationary point in O(ε⁻⁴) iterations—the optimal rate matching the fundamental lower bound for nonconvex stochastic optimization. Our analysis operates under strictly weaker assumptions and yields tighter bounds than all prior works on RMSProp and Adam, thereby advancing the foundational understanding and reliability certification of adaptive optimization methods.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Data transparency and provenance
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

This paper provides the first tight convergence analyses for RMSProp and Adam in non-convex optimization under the most relaxed assumptions of coordinate-wise generalized smoothness and affine noise variance. We first analyze RMSProp, which is a special case of Adam with adaptive learning rates but without first-order momentum. Specifically, to solve the challenges due to dependence among adaptive update, unbounded gradient estimate and Lipschitz constant, we demonstrate that the first-order term in the descent lemma converges and its denominator is upper bounded by a function of gradient norm. Based on this result, we show that RMSProp with proper hyperparameters converges to an $epsilon$-stationary point with an iteration complexity of $mathcal O(epsilon^{-4})$. We then generalize our analysis to Adam, where the additional challenge is due to a mismatch between the gradient and first-order momentum. We develop a new upper bound on the first-order term in the descent lemma, which is also a function of the gradient norm. We show that Adam with proper hyperparameters converges to an $epsilon$-stationary point with an iteration complexity of $mathcal O(epsilon^{-4})$. Our complexity results for both RMSProp and Adam match with the complexity lower bound established in cite{arjevani2023lower}.

Problem

• Analyzes convergence of RMSProp and Adam in non-convex optimization.
• Addresses challenges from adaptive updates and unbounded gradients.
• Proves convergence to ε-stationary points with O(ε⁻⁴) complexity.

Innovation

• Convergence analysis for RMSProp and Adam
• Generalized smoothness and affine noise variance
• Iteration complexity of O(ε^-4) for ε-stationary point