Abstract: We provide a simple proof of convergence covering both the Adam and Adagrad adaptive optimization algorithms when applied to smooth (possibly non-convex) objective functions with bounded gradients. We show that in expectation, the squared norm of the objective gradient averaged over the trajectory has an upper-bound which is explicit in the constants of the problem, parameters of the optimizer and the total number of iterations N. This bound can be made arbitrarily small: Adam with a learning rate $\alpha=1/\sqrt{N}$ and a momentum parameter on squared gradients $\beta_2 = 1 − 1/N$ achieves the same rate of convergence $O(\ln(N)/\sqrt{N})$ as Adagrad. Finally, we obtain the tightest dependency on the heavy ball momentum among all previous convergence bounds for non-convex Adam and Adagrad, improving from $O((1 − \beta_1)^{−3})$ to $O((1 − \beta_1)^{-1})$. Our technique also improves the best known dependency for standard SGD by a factor $1−β_1$.

@article{defossez-2022,
  title = {A Simple Convergence Proof of Adam and Adagrad},
  author = {D{\'e}fossez, Alexandre and Bottou, Leon and Bach, Francis and Usunier, Nicolas},
  journal = {Transactions on Machine Learning Research},
  issn = {2835-8856},
  year = {2022},
  url = {http://leon.bottou.org/papers/defossez-2022},
}