Stochastic Nested Variance Reduction for Nonconvex Optimization

We study finite-sum nonconvex optimization problems, where the objective function is an average of n nonconvex functions. We propose a new stochastic gradient descent algorithm based on nested variance reduction. Compared with conventional stochastic variance reduced gradient (SVRG) algorithm that uses two reference points to construct a semi-stochastic gradient with diminishing variance in each iteration, our algorithm uses K+1 nested reference points to build a semi-stochastic gradient to further reduce its variance in each iteration. For smooth nonconvex functions, the proposed algorithm converges to an ε-approximate first-order stationary point (i.e., |nabla F(x)|2≤ ε) within tilde O(nland ε-2+ε-3land n1/2ε-2) number of stochastic gradient evaluations. This improves the best known gradient complexity of SVRG O(n+n2/3ε-2) and that of SCSG O(nland ε-2+ε-10/3land n2/3ε-2). For gradient dominated functions, our algorithm also achieves better gradient complexity than the state-of-the-art algorithms. Thorough experimental results on different nonconvex optimization problems back up our theory.