  Despite their overwhelming capacity to overfit, deep neural networks trained
by specific optimization algorithms tend to generalize well to unseen data.
Recently, researchers explained it by investigating the implicit regularization
effect of optimization algorithms. A remarkable progress is the work (Lyu&Li,
2019), which proves gradient descent (GD) maximizes the margin of homogeneous
deep neural networks. Except GD, adaptive algorithms such as AdaGrad, RMSProp
and Adam are popular owing to their rapid training process. However,
theoretical guarantee for the generalization of adaptive optimization
algorithms is still lacking. In this paper, we study the implicit
regularization of adaptive optimization algorithms when they are optimizing the
logistic loss on homogeneous deep neural networks. We prove that adaptive
algorithms that adopt exponential moving average strategy in conditioner (such
as Adam and RMSProp) can maximize the margin of the neural network, while
AdaGrad that directly sums historical squared gradients in conditioner can not.
It indicates superiority on generalization of exponential moving average
strategy in the design of the conditioner. Technically, we provide a unified
framework to analyze convergent direction of adaptive optimization algorithms
by constructing novel adaptive gradient flow and surrogate margin. Our
experiments can well support the theoretical findings on convergent direction
of adaptive optimization algorithms.
