Stochastic Conjugate Frameworks for Nonconvex and Nonsmooth Optimization

We introduce two new stochastic conjugate frameworks for a class of nonconvex and possibly also nonsmooth optimization problems. These frameworks are built upon Stochastic Recursive Gradient Algorithm (SARAH) and we thus refer to them as Acc-Prox-CG-SARAH and Acc-Prox-CG-SARAH-RS, respectively. They are efficiently accelerated, easy to implement, tune free and can be smoothly extended and modified. We devise a deterministic restart scheme for stochastic optimization and apply it in our second stochastic conjugate framework, which serves the key difference between the two approaches. In addition, we apply the ProbAbilistic Gradient Estimator (PAGE) and further develop a practical variant, denoted as Acc-Prox-CG-SARAH-ST, in order to reduce potential computational overhead. We provide comprehensive and rigorous convergence analysis for all three approaches and establish linear convergence rates for unconstrained minimization problem with nonconvex and nonsmooth objective functions. Experiments have demonstrated that Acc-Prox-CG-SARAH and Acc-Prox-CG-SARAH-RS both outperform state-of-art methods consistently and Acc-Prox-CG-SARAH-ST can as well achieve comparable convergence speed. In terms of theory and experiments, we verify the strong computational efficiency of the deterministic restart scheme in stochastic optimization methods.