L-SVRG and L-Katyusha with Arbitrary Sampling

We develop and analyze a new family of nonaccelerated and accelerated loopless variancereduced methods for finite-sum optimization problems. Our convergence analysis relies on a novel expected smoothness condition which upper bounds the variance of the stochastic gradient estimation by a constant times a distance-like function. This allows us to handle with ease arbitrary sampling schemes as well as the nonconvex case. We perform an indepth estimation of these expected smoothness parameters and propose new importance samplings which allow linear speedup when the expected minibatch size is in a certain range. Furthermore, a connection between these expected smoothness parameters and expected separable overapproximation (ESO) is established, which allows us to exploit data sparsity as well. Our general methods and results recover as special cases the loopless SVRG (Hofmann et al., 2015) and loopless Katyusha (Kovalev et al., 2019) methods. Keywords: L-SVRG, L-Katyusha, Arbitrary sampling, Expected smoothness, ESO