A Unified Analysis of AdaGrad with Weighted Aggregation and Momentum Acceleration
This work provides a theoretical foundation for understanding and improving adaptive optimization algorithms in deep learning, though it is incremental in nature.
The authors tackled the theoretical gap in convergence analysis for adaptive stochastic algorithms in non-convex settings by proposing AdaUSM, a unified framework that incorporates momentum and weighted adaptive learning rates, achieving an O(log(T)/√T) convergence rate with polynomially growing weights.
Integrating adaptive learning rate and momentum techniques into SGD leads to a large class of efficiently accelerated adaptive stochastic algorithms, such as AdaGrad, RMSProp, Adam, AccAdaGrad, \textit{etc}. In spite of their effectiveness in practice, there is still a large gap in their theories of convergences, especially in the difficult non-convex stochastic setting. To fill this gap, we propose \emph{weighted AdaGrad with unified momentum}, dubbed AdaUSM, which has the main characteristics that (1) it incorporates a unified momentum scheme which covers both the heavy ball momentum and the Nesterov accelerated gradient momentum; (2) it adopts a novel weighted adaptive learning rate that can unify the learning rates of AdaGrad, AccAdaGrad, Adam, and RMSProp. Moreover, when we take polynomially growing weights in AdaUSM, we obtain its $\mathcal{O}(\log(T)/\sqrt{T})$ convergence rate in the non-convex stochastic setting. We also show that the adaptive learning rates of Adam and RMSProp correspond to taking exponentially growing weights in AdaUSM, thereby providing a new perspective for understanding Adam and RMSProp. Lastly, comparative experiments of AdaUSM against SGD with momentum, AdaGrad, AdaEMA, Adam, and AMSGrad on various deep learning models and datasets are also carried out.