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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). Note that SVRG, SAGA and their variants typically achieve weaker convergence results than variants of SARAH: $n^{2/3}/\epsilon^2$ vs. $n^{1/2}/\epsilon^2$. Thus we focus on the variant of SARAH. The proposed ZeroSARAH and its distributed variant D-ZeroSARAH are the \emph{first} variance-reduced algorithms which \emph{do not require any full gradient computations}, not even for the initial point. Moreover, for both standard and distributed settings, we show that ZeroSARAH and D-ZeroSARAH obtain new state-of-the-art convergence results, which can improve the previous best-known result (given by e.g., SPIDER, SARAH, and PAGE) in certain regimes. Avoiding any full gradient computations (which are time-consuming steps) 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 clients/devices/machines, which may be impossible or unaffordable. Thus, we expect that ZeroSARAH/D-ZeroSARAH will have a practical impact in distributed and federated learning where full device participation is impractical.

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