  Stochastic Gradient Descent (SGD) is a popular tool in training large-scale
machine learning models. Its performance, however, is highly variable,
depending crucially on the choice of the step sizes. Accordingly, a variety of
strategies for tuning the step sizes have been proposed, ranging from
coordinate-wise approaches (a.k.a. ``adaptive'' step sizes) to sophisticated
heuristics to change the step size in each iteration. In this paper, we study
two step size schedules whose power has been repeatedly confirmed in practice:
the exponential and the cosine step sizes. For the first time, we provide
theoretical support for them proving convergence rates for smooth non-convex
functions, with and without the Polyak-\L{}ojasiewicz (PL) condition. Moreover,
we show the surprising property that these two strategies are \emph{adaptive}
to the noise level in the stochastic gradients of PL functions. That is,
contrary to polynomial step sizes, they achieve almost optimal performance
without needing to know the noise level nor tuning their hyperparameters based
on it. Finally, we conduct a fair and comprehensive empirical evaluation of
real-world datasets with deep learning architectures. Results show that, even
if only requiring at most two hyperparameters to tune, these two strategies
best or match the performance of various finely-tuned state-of-the-art
strategies.
