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Adam with model exponential moving average is effective for nonconvex optimization (2405.18199v2)

Published 28 May 2024 in cs.LG and math.OC

Abstract: In this work, we offer a theoretical analysis of two modern optimization techniques for training large and complex models: (i) adaptive optimization algorithms, such as Adam, and (ii) the model exponential moving average (EMA). Specifically, we demonstrate that a clipped version of Adam with model EMA achieves the optimal convergence rates in various nonconvex optimization settings, both smooth and nonsmooth. Moreover, when the scale varies significantly across different coordinates, we demonstrate that the coordinate-wise adaptivity of Adam is provably advantageous. Notably, unlike previous analyses of Adam, our analysis crucially relies on its core elements -- momentum and discounting factors -- as well as model EMA, motivating their wide applications in practice.

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Authors (2)
  1. Kwangjun Ahn (33 papers)
  2. Ashok Cutkosky (50 papers)
Citations (2)

Summary

  • The paper demonstrates that a clipped variant of Adam paired with model EMA attains optimal convergence rates in nonconvex optimization.
  • It employs an online-to-nonconvex framework to effectively address gradient variance challenges with dynamic, coordinate-wise adaptivity.
  • The findings offer practical insights for large-scale machine learning tasks, enhancing adaptive optimization for neural network training.

Analysis of Adam with Model Exponential Moving Average in Nonconvex Optimization

The paper "Adam with Model Exponential Moving Average is Effective for Nonconvex Optimization" provides a theoretical exploration of contemporary optimization methodologies—specifically, employing the Adam algorithm in combination with the Model Exponential Moving Average (EMA) technique. The authors, Kwangjun Ahn and Ashok Cutkosky, undertake a comprehensive analysis to showcase the convergence benefits of these methods across a range of nonconvex optimization contexts, including both smooth and nonsmooth landscapes.

Core Contributions

The analysis reveals that a clipped variant of Adam, when integrated with model EMA, achieves optimal convergence rates for nonconvex optimization problems. The paper delineates the advantages of Adam's coordinate-wise adaptivity, particularly in scenarios where the problem's scale differs across coordinates. This finding underlines an advantage over traditional analyses of the Adam algorithm, which often overlook the momentum and adaptive learning rates as pivotal components, as well as the essential role of model EMA.

Two main strategies underpin the research. Firstly, the authors develop their results using an online-to-nonconvex conversion framework, emphasizing the utility of online learning algorithms in nonconvex settings. Secondly, the paper combines insights from recent works to refine the analysis of Adam and clarify its efficacy through mechanism design that naturally leads to the EMA in practice.

Key Results and Implications

The principal result of this paper highlights the capacity of Adam with model EMA to reach (λ,)(\lambda, )-stationary points with iteration complexity that aligns with theoretical lower bounds for nonconvex optimization. Particularly, by tuning parameters such as the learning rate discount factor β and leveraging a scale-free follow-the-regularized-leader (FTRL) method for online learning, the approach effectively addresses gradient variance challenges prominent in nonconvex environments.

The research advances the understanding of Adam's adaptivity to varied scales by harnessing the EMA. This adaptivity implies significant practical utility in large-scale machine learning tasks like neural network training, where parameter scales can vary substantially.

Theoretical Insights

The theoretical underpinning emphasizes the importance of discounting and adapting parameters dynamically, leveraging discounted regret bounds from online learning theory. This approach contrasts with conventional analyses, which are typically confined to fixed-rate methods and smooth convex settings. Through the lens of regret minimization, the paper effectively lays the groundwork for improved adaptive optimization algorithms that may outperform well-established methods like stochastic gradient descent (SGD) in practice.

Future Directions and Challenges

The research identifies some gaps in existing analyses of Adam and extends theoretical rigor to practical optimization scenarios. However, it also raises questions regarding the precise role of parameter settings, such as the relationship between momentum factors and practical defaults used in real-world applications. Future work could focus on refining these components to further enhance the practical applicability of such adaptive optimization techniques.

In conclusion, the paper contributes substantively to the field of optimization in machine learning by offering a robust theoretical foundation for the use of adaptive methods in nonconvex settings. These insights could guide the future development of algorithms that are both theoretically sound and empirically effective in diverse, large-scale optimization problems.

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