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Stochastic gradient descent (SGD) is a prevalent optimization technique for large-scale distributed machine learning. While SGD computation can be efficiently divided between multiple machines, communication typically becomes a bottleneck…

The graduated optimization approach, also known as the continuation method, is a popular heuristic to solving non-convex problems that has received renewed interest over the last decade. Despite its popularity, very little is known in terms…

The graduated optimization approach is a method for finding global optimal solutions for nonconvex functions by using a function smoothing operation with stochastic noise. This paper makes three contributions regarding graduated…

Random Reshuffling (RR) is an algorithm for minimizing finite-sum functions that utilizes iterative gradient descent steps in conjunction with data reshuffling. Often contrasted with its sibling Stochastic Gradient Descent (SGD), RR is…

A very popular approach for solving stochastic optimization problems is the stochastic gradient descent method (SGD). Although the SGD iteration is computationally cheap and the practical performance of this method may be satisfactory under…

In federated distributed learning, the goal is to optimize a global training objective defined over distributed devices, where the data shard at each device is sampled from a possibly different distribution (a.k.a., heterogeneous or non…

The stochastic gradient descent (SGD) algorithm is the algorithm we use to train neural networks. However, it remains poorly understood how the SGD navigates the highly nonlinear and degenerate loss landscape of a neural network. In this…

Neural networks are trained by optimizing multi-dimensional sets of fitting parameters on non-convex loss landscapes. Low-loss regions of the landscapes correspond to the parameter sets that perform well on the training data. A key issue in…

We study convergence properties of Stochastic Gradient Descent (SGD) for convex objectives without assumptions on smoothness or strict convexity. We consider the question of establishing that with high probability the objective evaluated at…

Various optimal gradient-based algorithms have been developed for smooth nonconvex optimization. However, many nonconvex machine learning problems do not belong to the class of smooth functions and therefore the existing algorithms are…

Variance reduction (VR) methods employ stochastic gradients with decreasing variance, and they have been widely applied to solve large-scale optimization problems in machine learning because of their efficiency. Existing theoretical studies…

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…

Stochastic Gradient Descent (SGD) is a central tool in machine learning. We prove that SGD converges to zero loss, even with a fixed (non-vanishing) learning rate - in the special case of homogeneous linear classifiers with smooth monotone…

We investigate the stochastic gradient descent (SGD) method where the step size lies within a banded region instead of being given by a fixed formula. The optimal convergence rate under mild conditions and large initial step size is proved.…

Stochastic gradient descent (SGD) is the workhorse of large-scale learning, yet classical analyses rely on assumptions that can be either too strong (bounded variance) or too coarse (uniform noise). The expected smoothness (ES) condition…

Despite the non-convex optimization landscape, over-parametrized shallow networks are able to achieve global convergence under gradient descent. The picture can be radically different for narrow networks, which tend to get stuck in…

We prove convergence of a single time-scale stochastic subgradient method with subgradient averaging for constrained problems with a nonsmooth and nonconvex objective function having the property of generalized differentiability. As a tool…

Large-scale optimization problems require algorithms both effective and efficient. One such popular and proven algorithm is Stochastic Gradient Descent which uses first-order gradient information to solve these problems. This paper studies…

Stochastic Gradient Descent (SGD) is a cornerstone of large-scale optimization, yet its theoretical behavior under heavy-tailed noise -- common in modern machine learning and reinforcement learning -- remains poorly understood. In this…

This paper is concerned with convergence of stochastic gradient algorithms with momentum terms in the nonconvex setting. A class of stochastic momentum methods, including stochastic gradient descent, heavy ball, and Nesterov's accelerated…