*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 $1-β_1$.

Alexandre Défossez, Léon Bottou, Francis Bach and Nicolas Usunier: **A simple convergence proof of Adam and Adagrad**, *arXiv preprint arXiv:2003.02395*, 2020.

@article{defossez-2020, title = {A simple convergence proof of {Adam} and {Adagrad}}, author = {D{\'e}fossez, Alexandre and Bottou, L{\'e}on and Bach, Francis and Usunier, Nicolas}, journal = {arXiv preprint arXiv:2003.02395}, year = {2020}, url = {http://leon.bottou.org/papers/defossez-2020}, }

papers/defossez-2020.txt · Last modified: 2022/04/19 09:52 by leonb