On the Convergence of Adam and Beyond (1904.09237v1)

Abstract: Several recently proposed stochastic optimization methods that have been successfully used in training deep networks such as RMSProp, Adam, Adadelta, Nadam are based on using gradient updates scaled by square roots of exponential moving averages of squared past gradients. In many applications, e.g. learning with large output spaces, it has been empirically observed that these algorithms fail to converge to an optimal solution (or a critical point in nonconvex settings). We show that one cause for such failures is the exponential moving average used in the algorithms. We provide an explicit example of a simple convex optimization setting where Adam does not converge to the optimal solution, and describe the precise problems with the previous analysis of Adam algorithm. Our analysis suggests that the convergence issues can be fixed by endowing such algorithms with `long-term memory' of past gradients, and propose new variants of the Adam algorithm which not only fix the convergence issues but often also lead to improved empirical performance.

Convergence Issues in the Adam Optimization Algorithm

The paper "On the Convergence of Adam and Beyond" by Sashank J. Reddi, Satyen Kale, and Sanjiv Kumar rigorously analyzes the convergence properties of popular stochastic optimization methods used to train deep neural networks, particularly focusing on the Adam algorithm and its variants. The paper identifies fundamental flaws in these methods and proposes modifications aimed at ensuring consistent convergence to optimal solutions.

Key Contributions

1. Analysis of Exponential Moving Averages: The paper scrutinizes the exponential moving averages mechanism in algorithms such as RMSprop, Adam, Adadelta, and Nadam. It reveals that these algorithms can fail to converge in specific practical settings due to their reliance on a limited window of past gradients, which can cause a rapid decay of informative large gradient signals. The authors demonstrate this issue by providing a simple convex optimization example where Adam does not achieve convergence.
2. Long-Term Memory for Convergence: To address the observed failures, the paper suggests incorporating long-term memory of past gradients into these algorithms. The authors introduce new variants of Adam, specifically designed to retain historical gradient information over a more extended period, thereby mitigating the rapid decay issue and ensuring convergence.
3. AMSGrad Algorithm: The authors propose the AMSGrad algorithm as a principled variant of Adam. AMSGrad maintains the maximum of all exponential moving averages of squared gradients up to the current time step, which prevents the learning rate from increasing and ensures asymptotic convergence. The convergence analysis for AMSGrad in convex settings shows a regret bound similar to Adagrad, demonstrating its theoretical soundness.
4. Empirical Validation: The paper also includes a preliminary empirical evaluation of the proposed AMSGrad algorithm on standard machine learning problems, showing that AMSGrad performs better or similarly to Adam in practice, providing both stability and reliability in convergence.

Implications and Speculation on Future Developments

Practical Implications:

The identified convergence issues in algorithms like Adam and RMSprop call for a reassessment of their use in deep learning training, especially under non-standard conditions or with high-dimensional data. The introduction of AMSGrad presents practitioners with a robust alternative that addresses the pitfalls without sacrificing the computational efficiency and practical benefits of Adam.

Theoretical Implications:

The paper significantly impacts optimization theory in machine learning by highlighting the necessity of long-term memory in gradient-based optimization methods. It further refines the understanding of the interplay between step size adaptation and convergence guarantees.

Speculations on Future AI Developments:

Future research might extend the principles from this paper to more complex, non-convex optimization landscapes commonly encountered in deep learning. Additionally, integrating AMSGrad-like mechanisms with other advanced optimization techniques could yield even more powerful and reliable training algorithms.

Conclusion

The paper offers a crucial exploration of the convergence properties of Adam and similar stochastic optimization methods. By diagnosing the inherent issues in their design and proposing effective solutions, such as the AMSGrad algorithm, it advances both theoretical insights and practical tools for training deep neural networks. These contributions will likely spur further investigations and innovations in the development of robust optimization algorithms in the field of machine learning.