Almost Tune-Free Variance Reduction

The variance reduction class of algorithms including the representative ones, abbreviated as SVRG and SARAH, have well documented merits for empirical risk minimization tasks. However, they require grid search to optimally tune parameters (step size and the number of iterations per inner loop) for best performance. This work introduces `almost tune-free' SVRG and SARAH schemes by equipping them with Barzilai-Borwein (BB) step sizes. To achieve the best performance, both i) averaging schemes; and, ii) the inner loop length are adjusted according to the BB step size. SVRG and SARAH are first reexamined through an `estimate sequence' lens. Such analysis provides new averaging methods that tighten the convergence rates of both SVRG and SARAH theoretically, and improve their performance empirically when the step size is chosen large. Then a simple yet effective means of adjusting the number of iterations per inner loop is developed, which completes the tune-free variance reduction together with BB step sizes. Numerical tests corroborate the proposed methods.