STORM+: Fully Adaptive SGD with Recursive Momentum for Nonconvex Optimization
Authors: Kfir Levy, Ali Kavis, Volkan Cevher

In this work we investigate stochastic non-convex optimization problems where the objective is an expectation over smooth loss functions, and the goal is to find an approximate stationary point. The most popular approach to handling such problems is variance reduction techniques, which are also known to obtain tight convergence rates, matching the lower bounds in this case. Nevertheless, these techniques require a careful maintenance of anchor points in conjunction with appropriately selected ``mega-batchsizes". This leads to a challenging hyperparameter tuning problem, that weakens their practicality. Recently, [Cutkosky and Orabona, 2019] have shown that one can employ recursive momentum in order to avoid the use of anchor points and large batchsizes, and still obtain the optimal rate for this setting. Yet, their method called $\rm{STORM}$ crucially relies on the knowledge of the smoothness, as well a bound on the gradient norms. In this work we propose $\rm{STORM}^{+}$, a new method that is completely parameter-free, does not require large batch-sizes, and obtains the optimal $O(1/T^{1/3})$ rate for finding an approximate stationary point. Our work builds on the $\rm{STORM}$ algorithm, in conjunction with a novel approach to adaptively set the learning rate and momentum parameters.