Adam Can Converge Without Any Modification On Update Rules
Ever since Reddi et al. 2018 pointed out the divergence issue of Adam, many new variants have been designed to obtain convergence. However, vanilla Adam remains exceptionally popular and it works well in practice. Why is there a gap between theory and practice? We point out there is a mismatch between the settings of theory and practice: Reddi et al. 2018 pick the problem after picking the hyperparameters of Adam, i.e., $(β_1, β_2)$; while practical applications often fix the problem first and then tune $(β_1, β_2)$. Due to this observation, we conjecture that the empirical convergence can be theoretically justified, only if we change the order of picking the problem and hyperparameter. In this work, we confirm this conjecture. We prove that, when $β_2$ is large and $β_1 < \sqrt{β_2}<1$, Adam converges to the neighborhood of critical points. The size of the neighborhood is propositional to the variance of stochastic gradients. Under an extra condition (strong growth condition), Adam converges to critical points. It is worth mentioning that our results cover a wide range of hyperparameters: as $β_2$ increases, our convergence result can cover any $β_1 \in [0,1)$ including $β_1=0.9$, which is the default setting in deep learning libraries. To our knowledge, this is the first result showing that Adam can converge without any modification on its update rules. Further, our analysis does not require assumptions of bounded gradients or bounded 2nd-order momentum. When $β_2$ is small, we further point out a large region of $(β_1,β_2)$ where Adam can diverge to infinity. Our divergence result considers the same setting as our convergence result, indicating a phase transition from divergence to convergence when increasing $β_2$. These positive and negative results can provide suggestions on how to tune Adam hyperparameters.