On the Convergence of A Class of Adam-Type Algorithms for Non-Convex Optimization (1808.02941v2)

Abstract: This paper studies a class of adaptive gradient based momentum algorithms that update the search directions and learning rates simultaneously using past gradients. This class, which we refer to as the "Adam-type", includes the popular algorithms such as the Adam, AMSGrad and AdaGrad. Despite their popularity in training deep neural networks, the convergence of these algorithms for solving nonconvex problems remains an open question. This paper provides a set of mild sufficient conditions that guarantee the convergence for the Adam-type methods. We prove that under our derived conditions, these methods can achieve the convergence rate of order $O(\log{T}/\sqrt{T})$ for nonconvex stochastic optimization. We show the conditions are essential in the sense that violating them may make the algorithm diverge. Moreover, we propose and analyze a class of (deterministic) incremental adaptive gradient algorithms, which has the same $O(\log{T}/\sqrt{T})$ convergence rate. Our study could also be extended to a broader class of adaptive gradient methods in machine learning and optimization.

On the Convergence of a Class of Adam-Type Algorithms for Non-Convex Optimization

This paper presents a theoretical analysis of a broad class of adaptive gradient-based momentum algorithms, labeled as "Adam-type" methods, for non-convex optimization problems. This class includes notable algorithms such as Adam, AMSGrad, and AdaGrad, which are widely adopted in deep learning for their practical efficiency. Nevertheless, a comprehensive theoretical understanding of their convergence properties, particularly in non-convex settings, was lacking prior to this paper.

Analytical Framework and Results

The authors put forth an analytical framework that establishes mild sufficient conditions under which Adam-type methods can be ensured to converge to first-order stationary points for non-convex stochastic optimization problems. A key result of their analysis is a convergence rate of order $O(\log{T}/\sqrt{T})$, where $T$ denotes the number of iterations.

A significant contribution of this paper is the convergence analysis of a new algorithm called AdaFom (AdaGrad with First Order Momentum), which showcases convergence properties that align well with practical observations. The analysis identifies essential conditions which, if violated, result in algorithmic divergence. These conditions provide valuable insights for practitioners aiming to monitor algorithm progress and validate convergence behavior.

Numerical Implications and Empirical Validation

Throughout the analysis, assumptions include the boundedness of the noisy gradient and the Lipschitz continuity of the differentiable function's gradient. These assumptions are standard in stochastic optimization settings, making the theoretical results broadly applicable. Furthermore, the analysis reveals the importance of the "effective stepsize" and its oscillation, highlighting conditions where certain Adam-type algorithms can outperform stochastic gradient descent (SGD).

The paper also explores the oscillatory behavior of adaptive learning rates, asserting that ensuring bounded oscillation is pivotal for ensuring convergence. This line of argument is exemplified through concrete theoretical examples and numerical experiments.

Implications for AI and Future Work

The implications of this research are multifaceted. Practically, it equips practitioners with robust tools for deploying Adam-type algorithms in non-convex settings, enhancing scalability and performance for deep learning applications. Theoretically, it bridges a gap in understanding the behavior of adaptive gradient methods beyond convex optimization, offering a foundation for future studies that might extend these results to more generalized classes of problems, such as constrained optimization.

The authors suggest that future work could explore sharpening these convergence results to better quantify the practical benefits of Adam-type methods, especially in non-convex landscapes where these algorithms have demonstrated substantial empirical success. Addressing sharpness in worst-case convergence rates would significantly contribute to the theoretical landscape of optimization in high-dimensional spaces.

This paper sets a precedent for structured analysis of adaptive optimization methods, presenting convincing arguments and results that enrich both the theoretical and practical understanding of non-convex optimization in machine learning.