On the Convergence of Adam and Beyond
This addresses convergence problems in widely used optimization algorithms for training deep networks, which is crucial for researchers and practitioners in machine learning.
The paper identifies that Adam and similar stochastic optimization methods fail to converge to optimal solutions in some convex settings due to their use of exponential moving averages, and proposes new variants with long-term memory that fix these issues and improve empirical performance.
Several recently proposed stochastic optimization methods that have been successfully used in training deep networks such as RMSProp, Adam, Adadelta, Nadam are based on using gradient updates scaled by square roots of exponential moving averages of squared past gradients. In many applications, e.g. learning with large output spaces, it has been empirically observed that these algorithms fail to converge to an optimal solution (or a critical point in nonconvex settings). We show that one cause for such failures is the exponential moving average used in the algorithms. We provide an explicit example of a simple convex optimization setting where Adam does not converge to the optimal solution, and describe the precise problems with the previous analysis of Adam algorithm. Our analysis suggests that the convergence issues can be fixed by endowing such algorithms with `long-term memory' of past gradients, and propose new variants of the Adam algorithm which not only fix the convergence issues but often also lead to improved empirical performance.