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Motivated by applications in machine learning and operations research, westudy regret minimization with stochastic first-order oracle feedback in onlineconstrained, and possibly non-smooth, non-convex problems. In this setting, theminimization of external regret is beyond reach for first-order methods, so wefocus on a local regret measure defined via a proximal-gradient mapping. Toachieve no (local) regret in this setting, we develop a prox-grad method basedon stochastic first-order feedback, and a simpler method for when access to aperfect 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 methodsrequire. As an important application of our results, we also obtain a linkbetween online and offline non-convex stochastic optimization manifested as anew prox-grad scheme with complexity guarantees matching those obtained viavariance reduction techniques.

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