Finding Local Minima via Stochastic Nested Variance Reduction

We propose two algorithms that can find local minima faster than the state-of-the-art algorithms in both finite-sum and general stochastic nonconvex optimization. At the core of the proposed algorithms is One-epoch-SNVRG^+ using stochastic nested variance reduction (Zhou et al., 2018a), which outperforms the state-of-the-art variance reduction algorithms such as SCSG (Lei et al., 2017). In particular, for finite-sum optimization problems, the proposed SNVRG++Neon2finite algorithm achieves tildeO(n1/2ε-2+nεH-3+n3/4εH-7/2) gradient complexity to converge to an (ε, εH)-second-order stationary point, which outperforms SVRG+Neon2finite (Allen-Zhu and Li, 2017) , the best existing algorithm, in a wide regime. For general stochastic optimization problems, the proposed SNVRG++Neon2online achieves tildeO(ε-3+εH-5+ε-2εH-3) gradient complexity, which is better than both SVRG+Neon2online (Allen-Zhu and Li, 2017) and Natasha2 (Allen-Zhu, 2017) in certain regimes. Furthermore, we explore the acceleration brought by third-order smoothness of the objective function.