Regret minimization in stochastic non-convex learning via a proximal-gradient approach

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