  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.
