In this paper, we propose a new stochastic 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., 2017a) and consist of two steps: a proximal gradient step and an averaging step that are different from existing nonconvex proximal-type algorithms. The algorithms only require a smoothness assumption of the nonconvex objective term. In the finite-sum case, we show that our algorithm achieves optimal convergence rate by matching the lower-bound worst-case complexity, while in the expectation case, it attains the best-known convergence rate under only standard smoothness and bounded variance assumptions. One key step of our algorithms is a new constant step-size that helps to achieve desired convergence rate. Our step-size is much larger than existing methods including proximal SVRG schemes in the single sample case. We generalize our algorithm to mini-batches for both inner and outer loops, and adaptive step-sizes. We also specify the algorithm to the non-composite case that covers and dominates existing state-of-the-arts in terms of convergence rate. We test the proposed algorithms on two composite nonconvex optimization problems and feedforward neural networks using several well-known datasets.

提出了一种新的随机一阶算法框架来解决随机复合非凸优化问题，该算法覆盖了有限和期望设置，其中算法仅需要非凸目标项的平均光滑性假设和附加的有界方差假设，并证明了算法可以实现最佳复杂度界限。

ProxSARAH：随机复合非凸优化的高效算法框架