Gradient-following learning methods can encounter problems of implementation in many applications, and stochastic variants are sometimes used to overcome these difficulties. We analyze three online training methods used with a linear perceptron: direct gradient descent, node perturbation, and weight perturbation. Learning speed is defined as the rate of exponential decay in the learning curves....

We present a novel per-dimension learning rate method for gradient descent called ADADELTA. The method dynamically adapts over time using only first order information and has minimal computational overhead beyond vanilla stochastic gradient descent. The method requires no manual tuning of a learning rate and appears robust to noisy gradient information, different model architecture choices, var...

With the recent proliferation of large-scale learning problems, there have been a lot of interest on distributed machine learning algorithms, particularly those that are based on stochastic gradient descent (SGD) and its variants. However, existing algorithms either suffer from slow convergence due to the inherent variance of stochastic gradients, or have a fast linear convergence rate but at t...

Momentum based stochastic gradient methods such as heavy ball (HB) and Nesterov’s accelerated gradient descent (NAG) method are widely used in practice for training deep networks and other supervised learning models, as they often provide significant improvements over stochastic gradient descent (SGD). Theoretically, these “fast gradient” methods have provable improvements over gradient descent...

The success of deep learning has led to a rising interest in the generalization property of the stochastic gradient descent (SGD) method, and stability is one popular approach to study it. Existing works based on stability have studied nonconvex loss functions, but only considered the generalization error of the SGD in expectation. In this paper, we establish various generalization error bounds...

Momentum based stochastic gradient methods such as heavy ball (HB) and Nesterov’s accelerated gradient descent (NAG) method are widely used in practice for training deep networks and other supervised learning models, as they often provide significant improvements over stochastic gradient descent (SGD). In general, “fast gradient” methods have provable improvements over gradient descent only for...

Stochastic Gradient Descent (SGD) and its variants are the most important optimization algorithms used in large scale machine learning. Mini-batch version of stochastic gradient is often used in practice for taking advantage of hardware parallelism. In this work, we analyze the effect of mini-batch size over SGD convergence for the case of general non-convex objective functions. Building on the...

Pairwise learning to rank methods such as RankSVM give good performance, but suffer from the computational burden of optimizing an objective defined over O(n) possible pairs for data sets with n examples. In this paper, we remove this super-linear dependence on training set size by sampling pairs from an implicit pairwise expansion and applying efficient stochastic gradient descent learners for...

Stochastic gradient descent (SGD) is a ubiquitous algorithm for a variety of machine learning problems. Researchers and industry have developed several techniques to optimize SGD's runtime performance, including asynchronous execution and reduced precision. Our main result is a martingale-based analysis that enables us to capture the rich noise models that may arise from such techniques. Specif...

Stochastic algorithms are efficient approaches to solving machine learning and optimization problems. In this paper, we propose a general framework called Splash for parallelizing stochastic algorithms on multi-node distributed systems. Splash consists of a programming interface and an execution engine. Using the programming interface, the user develops sequential stochastic algorithms without ...