Stochastic gradient descent (SGD) is widely used in machine learning. Although being commonly viewed as a fast but not accurate version of gradient descent (GD), it always finds better solutions than GD for modern neural networks. In order to understand this phenomenon, we take an alternative view that SGD is working on the convolved (thus smoothed) version of the loss function. We show that, even if the function $f$ has many bad local minima or saddle points, as long as for every point $x$, the weighted average of the gradients of its neighborhoods is one point convex with respect to the desired solution $x^*$, SGD will get close to, and then stay around $x^*$ with constant probability. Our result identifies a set of functions that SGD provably works, which is much larger than the set of convex functions. Empirically, we observe that the loss surface of neural networks enjoys nice one point convexity properties locally, therefore our theorem helps explain why SGD works so well for neural networks.

We analyze the variance of stochastic gradients along negative curvature directions in certain non-convex machine learning models and show that stochastic gradients indeed exhibit a strong component along these directions. Furthermore, we show that - contrary to the case of isotropic noise - this variance is proportional to the magnitude of the corresponding eigenvalues and not decreasing in the dimensionality. Based upon this bservation we propose a new assumption under which we show that the injection of explicit, isotropic noise usually applied to make gradient descent escape saddle points can successfully be replaced by a simple SGD step. Additionally - and under the same condition - we derive the first convergence rate for plain SGD to a second-order stationary point in a number of iterations that is independent of the problem dimension.

We propose a stochastic variance-reduced cubic regularized Newton method (SVRC) for non-convex optimization. At the core of our algorithm is a novel semi-stochastic gradient along with a semi-stochastic Hessian, which are specifically designed for cubic regularization method. We show that our algorithm is guaranteed to converge to an $(\epsilon,\sqrt{\epsilon})$-approximate local minimum within $\tilde{O}(n^{4/5}/\epsilon^{3/2})$ second-order oracle calls, which outperforms the state-of-the-art cubic regularization algorithms including subsampled cubic regularization. Our work also sheds light on the application of variance reduction technique to high-order non-convex optimization methods. Thorough experiments on various non-convex optimization problems support our theory.

We investigate a projection free optimization method, namely non-convex conditional gradient sliding (NCGS) for non-convex optimization problems on the batch, stochastic and finite-sum settings. Conditional gradient sliding (CGS) method, by integrating Nesterov's accelerated gradient method with Frank-Wolfe (FW) method in a smart way, outperforms FW for convex optimization, by reducing the amount of gradient computations. However, the study of CGS in the non-convex setting is limited. In this paper, we propose the non-convex conditional gradient sliding (NCGS) methods and analyze their convergence properties. We also leverage the idea of variance reduction from the recent progress in convex optimization to obtain a new algorithm termed {\em variance reduced NCGS} (NCGS-VR), and obtain faster convergence rate than the batch NCGS in the finite-sum setting. We show that NCGS algorithms outperform their Frank-Wolfe counterparts both in theory and in practice, for all three settings, namely the batch, stochastic and finite-sum setting. This significantly improves our understanding of optimizing non-convex functions with complicated feasible sets (where projection is prohibitively expensive).