相关论文: A Deterministic Gradient-Based Approach to Avoid S…

Gradient descent (GD) and stochastic gradient descent (SGD) are the workhorses of large-scale machine learning. While classical theory focused on analyzing the performance of these methods in convex optimization problems, the most notable…

Stochastically controlled stochastic gradient (SCSG) methods have been proved to converge efficiently to first-order stationary points which, however, can be saddle points in nonconvex optimization. It has been observed that a stochastic…

This paper shows that a perturbed form of gradient descent converges to a second-order stationary point in a number iterations which depends only poly-logarithmically on dimension (i.e., it is almost "dimension-free"). The convergence rate…

In centralized settings, it is well known that stochastic gradient descent (SGD) avoids saddle points and converges to local minima in nonconvex problems. However, similar guarantees are lacking for distributed first-order algorithms. The…

Stochastic gradient descent (SGD) is a standard optimization method to minimize a training error with respect to network parameters in modern neural network learning. However, it typically suffers from proliferation of saddle points in the…

In this paper, we give a sharp analysis for Stochastic Gradient Descent (SGD) and prove that SGD is able to efficiently escape from saddle points and find an $(\epsilon, O(\epsilon^{0.5}))$-approximate second-order stationary point in…

Nonconvex optimization underlies many modern machine learning and control tasks, where saddle points pose the dominant obstacle to reliable convergence in high-dimensional settings. Escaping these saddle points deterministically using…

A variant of consensus based distributed gradient descent (\textbf{DGD}) is studied for finite sums of smooth but possibly non-convex functions. In particular, the local gradient term in the fixed step-size iteration of each agent is…

Gradient descent is a popular algorithm in optimization, and its performance in convex settings is mostly well understood. In non-convex settings, it has been shown that gradient descent is able to escape saddle points asymptotically and…

Adaptive methods such as Adam and RMSProp are widely used in deep learning but are not well understood. In this paper, we seek a crisp, clean and precise characterization of their behavior in nonconvex settings. To this end, we first…

We analyze the variance of stochastic gradients along negative curvature directions in certain non-convex machine learning models and show that stochastic gradients exhibit a strong component along these directions. Furthermore, we show…

We consider minimizing a nonconvex, smooth function $f$ on a Riemannian manifold $\mathcal{M}$. We show that a perturbed version of Riemannian gradient descent algorithm converges to a second-order stationary point (and hence is able to…

A major challenge in training large-scale machine learning models is configuring the training process to maximize model performance, i.e., finding the best training setup from a vast design space. In this work, we unlock a gradient-based…

This paper considers a class of distributed resource allocation problems where each agent privately holds a smooth, potentially non-convex local objective, subject to a globally coupled equality constraint. Built upon the existing method,…

We analyze stochastic gradient descent for optimizing non-convex functions. In many cases for non-convex functions the goal is to find a reasonable local minimum, and the main concern is that gradient updates are trapped in saddle points.…

Momentum Stochastic Gradient Descent (MSGD) algorithm has been widely applied to many nonconvex optimization problems in machine learning, e.g., training deep neural networks, variational Bayesian inference, and etc. Despite its empirical…

We provide larger step-size restrictions for which gradient descent based algorithms (almost surely) avoid strict saddle points. In particular, consider a twice differentiable (non-convex) objective function whose gradient has Lipschitz…

Stochastic gradient descent (SGD) with stochastic momentum is popular in nonconvex stochastic optimization and particularly for the training of deep neural networks. In standard SGD, parameters are updated by improving along the path of the…

The note considers normalized gradient descent (NGD), a natural modification of classical gradient descent (GD) in optimization problems. A serious shortcoming of GD in non-convex problems is that GD may take arbitrarily long to escape from…

Gradient descent and its variants are widely used in machine learning. However, oracle access of gradient may not be available in many applications, limiting the direct use of gradient descent. This paper proposes a method of estimating…