Analysis of Momentum-based Accelerated Variants of Stochastic Gradient Descent
In this article, the authors discuss the theoretical understanding and generalization error of momentum-based accelerated variants of stochastic gradient descent (SGD) when training machine learning models. They present several key findings and propose improvements to enhance the generalization error of these methods.
Stability Gap in SGD with Standard Heavy-Ball Momentum (SGDM)
The authors first demonstrate that there exists a convex loss function for which the stability gap for multiple epochs of SGD with standard heavy-ball momentum (SGDM) becomes unbounded. This highlights a limitation of using SGDM for training machine learning models, especially in cases where stability is crucial.
Improved Generalization with SGD and Early Momentum (SGDEM)
To address the generalization issue, the authors introduce a modified momentum-based update rule called SGD with early momentum (SGDEM). They evaluate SGDEM under a broad range of step-sizes for smooth Lipschitz loss functions and show that it can train machine learning models for multiple epochs while guaranteeing generalization. This improvement is significant as it provides a practical solution that can be applied to various scenarios.
Generalization of Standard SGDM for Strongly Convex Loss Functions
In the case of strongly convex loss functions, the authors find that there exists a range of momentum values for which multiple epochs of standard SGDM can also generalize. This discovery presents a specific condition where SGDM is effective, reinforcing its applicability in certain contexts.
Upper Bound on Expected True Risk
In addition to the findings on generalization, the authors develop an upper bound on the expected true risk. This bound takes into account the number of training steps, sample size, and momentum. By providing an analytical estimation of the true risk, this bound offers insights into the reliability and performance of momentum-based variants of SGD.
Experimental Evaluations and Consistency with Theoretical Bounds
The authors conclude their work by showcasing experimental evaluations that verify the consistency between their theoretical bounds and numerical results. They specifically focus on the application of SGDEM to improve the generalization error of SGDM when training ResNet-18 on the ImageNet dataset in practical distributed settings. This practical validation further strengthens the significance and relevance of their findings.
In summary, this article sheds light on the theoretical understanding and generalization error of momentum-based accelerated variants of stochastic gradient descent. It introduces an improved momentum-based update rule (SGDEM) to enhance generalization, identifies a condition where standard SGDM can also generalize, and provides an upper bound on the expected true risk. The experimental evaluations validate the theoretical findings and highlight the potential practical applications. This work contributes to advancing the understanding and optimization of machine learning model training algorithms.