AdaShift: Decorrelation and Convergence of Adaptive Learning Rate Methods

Zhiming Zhou*, Qingru Zhang*, Guansong Lu, Hongwei Wang, Weinan Zhang, Yong Yu

Abstract: Adam is shown not being able to converge to the optimal solution in certain cases. Researchers recently propose several algorithms to avoid the issue of non-convergence of Adam, but their efficiency turns out to be unsatisfactory in practice. In this paper, we provide a new insight into the non-convergence issue of Adam as well as other adaptive learning rate methods. We argue that there exists an inappropriate correlation between gradient $g_t$ and the second moment term $v_t$ in Adam ($t$ is the timestep), which results in that a large gradient is likely to have small step size while a small gradient may have a large step size. We demonstrate that such unbalanced step sizes are the fundamental cause of non-convergence of Adam, and we further prove that decorrelating $v_t$ and $g_t$ will lead to unbiased step size for each gradient, thus solving the non-convergence problem of Adam. Finally, we propose AdaShift, a novel adaptive learning rate method that decorrelates $v_t$ and $g_t$ by temporal shifting, i.e., using temporally shifted gradient $g_{t-n}$ to calculate $v_t$. The experiment results demonstrate that AdaShift is able to address the non-convergence issue of Adam, while still maintaining a competitive performance with Adam in terms of both training speed and generalization.

Keywords: optimizer, Adam, convergence, decorrelation

TL;DR: We analysis and solve the non-convergence issue of Adam.

Code: [ 3 community implementations](https://paperswithcode.com/paper/?openreview=HkgTkhRcKQ)

Community Implementations: [ 3 code implementations](https://www.catalyzex.com/paper/arxiv:1810.00143/code)