Variants of RMSProp and Adagrad with Logarithmic Regret Bounds (1706.05507v2)

Abstract: Adaptive gradient methods have become recently very popular, in particular as they have been shown to be useful in the training of deep neural networks. In this paper we have analyzed RMSProp, originally proposed for the training of deep neural networks, in the context of online convex optimization and show $\sqrt{T}$-type regret bounds. Moreover, we propose two variants SC-Adagrad and SC-RMSProp for which we show logarithmic regret bounds for strongly convex functions. Finally, we demonstrate in the experiments that these new variants outperform other adaptive gradient techniques or stochastic gradient descent in the optimization of strongly convex functions as well as in training of deep neural networks.

• The paper introduces SC-Adagrad and SC-RMSProp that achieve logarithmic regret bounds, marking a significant theoretical advancement for adaptive gradient methods.
• It rigorously examines RMSProp's parameter decay and its encapsulation of Adagrad’s logic to bridge key gaps in online convex optimization theory.
• Empirical evaluations reveal that the novel variants outperform traditional methods such as SGD and standard RMSProp across diverse deep learning tasks.

Variants of RMSProp and Adagrad with Logarithmic Regret Bounds: An Analysis

This paper presents an in-depth analysis of adaptive gradient methods, specifically focusing on RMSProp and Adagrad, within the context of online convex optimization. It proposes novel variants, SC-Adagrad and SC-RMSProp, which demonstrate logarithmic regret bounds for strongly convex functions. Such results mark a significant contribution to the theoretical underpinnings of adaptive optimization techniques widely applied in machine learning, particularly in training deep neural networks.

Overview of Analysis

1. Adaptive Gradient Methods Background: The paper begins by contextualizing the relevance of adaptive learning rates, as initially suggested by Adagrad. RMSProp, another method under examination, was originally developed to enhance deep neural network training. Despite its empirical success, a rigorous theoretical justification of RMSProp's efficacy had yet to be established. This paper addresses this gap by analyzing RMSProp using the perspective of online convex optimization.
2. Regret Bounds for Convex and Strongly Convex Functions:
   • RMSProp: The analysis reveals that RMSProp can achieve a regret bound of the order $O(\sqrt{T})$ under specific parameter conditions, particularly when the parameter $\beta_t$ adheres to a specified decay condition, diverging from the constant factor often used in practice.
   • SC-Adagrad and SC-RMSProp: The paper proposes SC-Adagrad as an adaptation of Adagrad specifically for strongly convex functions, achieving logarithmic regret bounds. Similarly, SC-RMSProp extends RMSProp to strongly convex scenarios with a focus on data-dependent regret bounds that are considerably tighter than data-independent ones.
3. Key Theoretical Contributions:
   • RMSProp’s parameter scheme was shown to encapsulate Adagrad's approach as a particular case, providing a new theoretical justification for this widely-used optimization method in deep learning.
   • The development of SC-Adagrad and SC-RMSProp highlights the importance of a dampening factor and adaptive decay schemes for achieving optimal regret bounds in strongly convex settings.
4. Empirical Evaluation: The experimental results further substantiate the theoretical analyses. For strongly convex settings, the novel variants SC-Adagrad and SC-RMSProp outperformed traditional methods including stochastic gradient descent (SGD), Adagrad, and conventional RMSProp. The paper demonstrated their competitiveness across multiple datasets for both strongly convex and non-convex optimization problems. Practical experiments included convolutional neural networks and multilayer perceptrons, suggesting these methods as viable alternatives for optimizing deep learning systems.

Implications and Future Directions

The paper's findings bear crucial implications:

• Theoretical and Practical Insights for RMSProp: Providing a theoretical basis for RMSProp in convex optimization augments its credibility as a tool for deep learning. The suggested adjustment in parameter decay rate introduces an opportunity for improved performance in deep learning applications, especially when typical settings do not guarantee convergence.
• Advancement in Strongly Convex Optimization: By achieving logarithmic regret bounds, SC-Adagrad and SC-RMSProp enhance the applicability of adaptive gradient methods to strongly convex problems, potentially benefitting a range of machine learning tasks beyond traditional neural network training.

Looking forward, this research lays the groundwork for further exploration into adaptive algorithms tailored for specific problem settings in machine learning, particularly those incorporating deeper structural assumptions about the loss functions, such as smoothness or even additional strong convexity properties. Future studies could also delve into fine-tuning the decay schemes of dampening factors, with a focus on enhancing empirical performance while preserving computational efficiency.

In summary, this paper makes a notable contribution by bridging the gap between theoretical optimality and practical utility in adaptive optimization methods, offering a path forward for both improved theoretical understanding and applied machine learning practices.