Simple Convergence Proof of Adam From a Sign-like Descent Perspective

Abstract

Adam is widely recognized as one of the most effective optimizers for training deep neural networks (DNNs). Despite its remarkable empirical success, its theoretical convergence analysis remains unsatisfactory. Existing works predominantly interpret Adam as a preconditioned stochastic gradient descent with momentum (SGDM), formulated as $\bm{x}_{t+1} = \bm{x}_t - \frac{\gamma_t}{{\sqrt{\bm{v}_t}+\epsilon}} \circ \bm{m}_t$. This perspective necessitates strong assumptions and intricate techniques, resulting in lengthy and opaque convergence proofs that are difficult to verify and extend. In contrast, we propose a novel interpretation by treating Adam as a sign-like optimizer, expressed as $\bm{x}_{t+1} = \bm{x}_t - \gamma_t \frac{|\bm{m}_t|}{{\sqrt{\bm{v}_t}+\epsilon}} \circ {\rm Sign}(\bm{m}_t)$. This reformulation significantly simplifies the convergence analysis. For the first time, with some mild conditions, we prove that Adam achieves the optimal rate of ${\cal O}(\frac{1}{T^{\sfrac{1}{4}}})$ rather than the previous ${\cal O} \left(\frac{\ln T}{T^{\sfrac{1}{4}}}\right)$ under weak assumptions of the generalized $p$-affine variance and $(L_0, L_1, q)$-smoothness, without dependence on the model dimensionality or the numerical stability parameter $\epsilon$. Additionally, our theoretical analysis provides new insights into the role of momentum as a key factor ensuring convergence and offers practical guidelines for tuning learning rates in Adam, further bridging the gap between theory and practice.