  We propose a new stochastic first-order algorithmic framework to solve
stochastic composite nonconvex optimization problems that covers both
finite-sum and expectation settings. Our algorithms rely on the SARAH estimator
introduced in (Nguyen et al, 2017) and consist of two steps: a proximal
gradient and an averaging step making them different from existing nonconvex
proximal-type algorithms. The algorithms only require an average smoothness
assumption of the nonconvex objective term and additional bounded variance
assumption if applied to expectation problems. They work with both constant and
adaptive step-sizes, while allowing single sample and mini-batches. In all
these cases, we prove that our algorithms can achieve the best-known complexity
bounds. One key step of our methods is new constant and adaptive step-sizes
that help to achieve desired complexity bounds while improving practical
performance. Our constant step-size is much larger than existing methods
including proximal SVRG schemes in the single sample case. We also specify the
algorithm to the non-composite case that covers existing state-of-the-arts in
terms of complexity bounds. Our update also allows one to trade-off between
step-sizes and mini-batch sizes to improve performance. We test the proposed
algorithms on two composite nonconvex problems and neural networks using
several well-known datasets.
