Rapid advances in data collection and processing capabilities have allowed\nfor the use of increasingly complex models that give rise to nonconvex\noptimization problems. These formulations, however, can be arbitrarily\ndifficult to solve in general, in the sense that even simply verifying that a\ngiven point is a local minimum can be NP-hard [1]. Still, some relatively\nsimple algorithms have been shown to lead to surprisingly good empirical\nresults in many contexts of interest. Perhaps the most prominent example is the\nsuccess of the backpropagation algorithm for training neural networks. Several\nrecent works have pursued rigorous analytical justification for this phenomenon\nby studying the structure of the nonconvex optimization problems and\nestablishing that simple algorithms, such as gradient descent and its\nvariations, perform well in converging towards local minima and avoiding\nsaddle-points. A key insight in these analyses is that gradient perturbations\nplay a critical role in allowing local descent algorithms to efficiently\ndistinguish desirable from undesirable stationary points and escape from the\nlatter. In this article, we cover recent results on second-order guarantees for\nstochastic first-order optimization algorithms in centralized, federated, and\ndecentralized architectures.\n