Adaptive Biased Stochastic Optimization

This work develops and analyzes a class of adaptive biased stochastic optimization (ABSO) algorithms from the perspective of the GEneralized Adaptive gRadient (GEAR) method that contains Adam, AdaGrad, RMSProp, etc. Particularly, two preferred biased stochastic optimization (BSO) algorithms, the biased stochastic variance reduction gradient (BSVRG) algorithm and the stochastic recursive gradient algorithm (SARAH), equipped with GEAR, are first considered in this work, leading to two ABSO algorithms: BSVRG-GEAR and SARAH-GEAR. We present a uniform analysis of ABSO algorithms for minimizing strongly convex (SC) and Polyak-Łojasiewicz (PŁ) composite objective functions. Second, we also use our framework to develop another novel BSO algorithm, adaptive biased stochastic conjugate gradient (coined BSCG-GEAR), which achieves the well-known oracle complexity. Specifically, under mild conditions, we prove that the resulting ABSO algorithms attain a linear convergence rate on both PŁ and SC cases. Moreover, we show that the complexity of the resulting ABSO algorithms is comparable to that of advanced stochastic gradient-based algorithms. Finally, we demonstrate the empirical superiority and the numerical stability of the resulting ABSO algorithms by conducting numerical experiments on different applications of machine learning.