  Rapid advances in data collection and processing capabilities have allowed
for the use of increasingly complex models that give rise to nonconvex
optimization problems. These formulations, however, can be arbitrarily
difficult to solve in general, in the sense that even simply verifying that a
given point is a local minimum can be NP-hard [1]. Still, some relatively
simple algorithms have been shown to lead to surprisingly good empirical
results in many contexts of interest. Perhaps the most prominent example is the
success of the backpropagation algorithm for training neural networks. Several
recent works have pursued rigorous analytical justification for this phenomenon
by studying the structure of the nonconvex optimization problems and
establishing that simple algorithms, such as gradient descent and its
variations, perform well in converging towards local minima and avoiding
saddle-points. A key insight in these analyses is that gradient perturbations
play a critical role in allowing local descent algorithms to efficiently
distinguish desirable from undesirable stationary points and escape from the
latter. In this article, we cover recent results on second-order guarantees for
stochastic first-order optimization algorithms in centralized, federated, and
decentralized architectures.
