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Stochastic gradient descent with momentum (SGDm) is one of the most popular optimization algorithms in deep learning. While there is a rich theory of SGDm for convex problems, the theory is considerably less developed in the context of deep…

Stochastic Gradient Descent (SGD) is the workhorse algorithm of deep learning technology. At each step of the training phase, a mini batch of samples is drawn from the training dataset and the weights of the neural network are adjusted…

Gradient optimization algorithms using epochs, that is those based on stochastic gradient descent without replacement (SGDo), are predominantly used to train machine learning models in practice. However, the mathematical theory of SGDo and…

Most existing approximate Thompson Sampling (TS) algorithms for multi-armed bandits use Stochastic Gradient Langevin Dynamics (SGLD) or its variants in each round to sample from the posterior, relaxing the need for conjugacy assumptions…

Understanding the behavior of stochastic gradient descent (SGD) in the context of deep neural networks has raised lots of concerns recently. Along this line, we study a general form of gradient based optimization dynamics with unbiased…

Stochastic gradient descent (SGD) gives an optimal convergence rate when minimizing convex stochastic objectives $f(x)$. However, in terms of making the gradients small, the original SGD does not give an optimal rate, even when $f(x)$ is…

Stochastic gradient algorithms are the main focus of large-scale optimization problems and led to important successes in the recent advancement of the deep learning algorithms. The convergence of SGD depends on the careful choice of…

We propose a stochastic modified equations (SME) for modeling the asynchronous stochastic gradient descent (ASGD) algorithms. The resulting SME of Langevin type extracts more information about the ASGD dynamics and elucidates the…

Stochastic Gradient Descent with a constant learning rate (constant SGD) simulates a Markov chain with a stationary distribution. With this perspective, we derive several new results. (1) We show that constant SGD can be used as an…

Under mild assumptions stochastic gradient methods asymptotically achieve an optimal rate of convergence if the arithmetic mean of all iterates is returned as an approximate optimal solution. However, in the absence of stochastic noise, the…

A framework previously introduced in [3] for solving a sequence of stochastic optimization problems with bounded changes in the minimizers is extended and applied to machine learning problems such as regression and classification. The…

Stochastic gradient Langevin dynamics (SGLD) and stochastic gradient Hamiltonian Monte Carlo (SGHMC) are two popular Markov Chain Monte Carlo (MCMC) algorithms for Bayesian inference that can scale to large datasets, allowing to sample from…

Stochastic gradient descent with momentum (SGDM) is one of the most widely used optimization algorithms in machine learning. While optimization properties of SGDM have been extensively studied in the literature, it remains insufficiently…

Gradient descent (GD) is known to converge quickly for convex objective functions, but it can be trapped at local minima. On the other hand, Langevin dynamics (LD) can explore the state space and find global minima, but in order to give…

We consider the stochastic gradient descent (SGD) algorithm driven by a general stochastic sequence, including i.i.d noise and random walk on an arbitrary graph, among others; and analyze it in the asymptotic sense. Specifically, we employ…

Modern machine learning is trained by stochastic gradient descent (SGD), whose performance critically depends on how the learning rate (LR) is adjusted and decreased over time. Yet existing LR regimes may be intricate, or need to tune one…

Classical assumptions like strong convexity and Lipschitz smoothness often fail to capture the nature of deep learning optimization problems, which are typically non-convex and non-smooth, making traditional analyses less applicable. This…

Low-precision training has emerged as a promising low-cost technique to enhance the training efficiency of deep neural networks without sacrificing much accuracy. Its Bayesian counterpart can further provide uncertainty quantification and…

We introduce a novel algorithm for gradient-based optimization of stochastic objective functions. The method may be seen as a variant of SGD with momentum equipped with an adaptive learning rate automatically adjusted by an 'energy'…

Optimizing noisy functions online, when evaluating the objective requires experiments on a deployed system, is a crucial task arising in manufacturing, robotics and many others. Often, constraints on safe inputs are unknown ahead of time,…