  Motivated by applications in machine learning and operations research, we
study regret minimization with stochastic first-order oracle feedback in online
constrained, and possibly non-smooth, non-convex problems. In this setting, the
minimization of external regret is beyond reach for first-order methods, so we
focus on a local regret measure defined via a proximal-gradient mapping. To
achieve no (local) regret in this setting, we develop a prox-grad method based
on stochastic first-order feedback, and a simpler method for when access to a
perfect first-order oracle is possible. Both methods are min-max order-optimal,
and we also establish a bound on the number of prox-grad queries these methods
require. As an important application of our results, we also obtain a link
between online and offline non-convex stochastic optimization manifested as a
new prox-grad scheme with complexity guarantees matching those obtained via
variance reduction techniques.
