A Sufficient Condition for Convergences of Adam and RMSProp (1811.09358v3)

Abstract: Adam and RMSProp are two of the most influential adaptive stochastic algorithms for training deep neural networks, which have been pointed out to be divergent even in the convex setting via a few simple counterexamples. Many attempts, such as decreasing an adaptive learning rate, adopting a big batch size, incorporating a temporal decorrelation technique, seeking an analogous surrogate, etc., have been tried to promote Adam/RMSProp-type algorithms to converge. In contrast with existing approaches, we introduce an alternative easy-to-check sufficient condition, which merely depends on the parameters of the base learning rate and combinations of historical second-order moments, to guarantee the global convergence of generic Adam/RMSProp for solving large-scale non-convex stochastic optimization. Moreover, we show that the convergences of several variants of Adam, such as AdamNC, AdaEMA, etc., can be directly implied via the proposed sufficient condition in the non-convex setting. In addition, we illustrate that Adam is essentially a specifically weighted AdaGrad with exponential moving average momentum, which provides a novel perspective for understanding Adam and RMSProp. This observation coupled with this sufficient condition gives much deeper interpretations on their divergences. At last, we validate the sufficient condition by applying Adam and RMSProp to tackle a certain counterexample and train deep neural networks. Numerical results are exactly in accord with our theoretical analysis.

• The paper introduces a sufficient condition that guarantees global convergence of Adam and RMSProp based on specific settings of the base learning rate and second-moment parameters.
• It reformulates Adam as a variant of AdaGrad with momentum, clarifying how historical second-order moments affect divergence.
• Experimental results on synthetic examples and real-world tasks like MNIST and CIFAR-100 validate the theoretical findings and offer practical optimization insights.

Overview of "A Sufficient Condition for Convergences of Adam and RMSProp"

The paper "A Sufficient Condition for Convergences of Adam and RMSProp," authored by Fangyu Zou, Li Shen, Zequn Jie, Weizhong Zhang, and Wei Liu, addresses the convergence issues prevalent in popular adaptive stochastic algorithms like Adam and RMSProp, especially in non-convex optimization settings. These algorithms, while celebrated for their empirical success in training deep neural networks, have been demonstrated to suffer from divergence under certain circumstances, even in convex scenarios. The paper offers a new perspective by deriving an easy-to-check sufficient condition for the convergence of these algorithms.

Key Contributions

1. Sufficient Condition for Convergence: The core contribution of the paper is the introduction of a sufficient condition that ensures the global convergence of Adam and RMSProp algorithms. This condition is straightforward and depends solely on the settings of the base learning rate and the parameters used to compute the historical second-order moments. This differs from previous approaches that often require adjustment of learning rates or modifications to the algorithm itself.
2. Insights into Algorithm Behavior: By proposing this sufficient condition, the authors offer new insights into why Adam and RMSProp may diverge in certain settings. The divergence could be attributed to improper parameter settings related to the accumulation of historical second-order moments rather than to an unbalanced adaptive learning rate.
3. Reformulation and Comparison: The authors reformulate Adam as a specific variant of the AdaGrad algorithm, enhanced with exponential moving average momentum. This formulation offers a novel perspective for understanding both Adam and RMSProp, aligning with existing work on weighted AdaGrad methods with different momentum techniques.
4. Experimental Validation: To substantiate their theoretical claims, the authors conduct extensive experiments. They validate the proposed sufficient condition by applying Adam and RMSProp to both synthetic counterexamples and real-world neural network training tasks, such as on LeNet with MNIST and ResNet on CIFAR-100 datasets. Their numerical results are consistent with their theoretical predictions.

Implications and Future Directions

The sufficient condition presented offers a practical guideline for configuring Adam and RMSProp to ensure convergence, impacting how these algorithms are employed in training large-scale neural networks. The identification that improper parameter setting, rather than the adaptive learning rate, may cause divergence provides significant theoretical implications, prompting a reevaluation of how these methods are stabilized.

The acknowledgment of Adam as a variant of AdaGrad with momentum suggests further exploration of the synergies between different optimization paradigms, potentially leading to the development of new algorithms that inherit the benefits of both. Future research could focus on refining these conditions further or extending the insights gained here to other adaptive optimization algorithms and settings in machine learning.

By emphasizing a sufficient condition that can be easily verified, the paper stands to streamline the optimization process for non-experts employing these widely-used algorithms, facilitating more robust training procedures in diverse applications of machine learning and AI.