ZeroSARAH: Efficient Nonconvex Finite-Sum Optimization with Zero Full Gradient Computation

We propose ZeroSARAH -- a novel variant of the variance-reduced method SARAH (Nguyen et al., 2017) -- for minimizing the average of a large number of nonconvex functions $\frac{1}{n}\sum_{i=1}^{n}f_i(x)$. To the best of our knowledge, in this nonconvex finite-sum regime, all existing variance-reduced methods, including SARAH, SVRG, SAGA and their variants, need to compute the full gradient over all $n$ data samples at the initial point $x^0$, and then periodically compute the full gradient once every few iterations (for SVRG, SARAH and their variants). Moreover, SVRG, SAGA and their variants typically achieve weaker convergence results than variants of SARAH: $n^{2/3}/\epsilon^2$ vs. $n^{1/2}/\epsilon^2$. ZeroSARAH is the first variance-reduced method which does not require any full gradient computations, not even for the initial point. Moreover, ZeroSARAH obtains new state-of-the-art convergence results, which can improve the previous best-known result (given by e.g., SPIDER, SpiderBoost, SARAH, SSRGD and PAGE) in certain regimes. Avoiding any full gradient computations (which is a time-consuming step) is important in many applications as the number of data samples $n$ usually is very large. Especially in the distributed setting, periodic computation of full gradient over all data samples needs to periodically synchronize all machines/devices, which may be impossible or very hard to achieve. Thus, we expect that ZeroSARAH will have a practical impact in distributed and federated learning where full device participation is impractical.