Applying standard Markov chain Monte Carlo (MCMC) algorithms to large data sets is computationally infeasible. The recently proposed stochastic gradient Langevin dynamics (SGLD) method circumvents this problem in three ways: it generates proposed moves using only a subset of the data, it skips the Metropolis-Hastings accept-reject step, and it uses sequences of decreasing step sizes. In \cite{TehThierryVollmerSGLD2014}, we provided the mathematical foundations for the decreasing step size SGLD, including consistency and a central limit theorem. However, in practice the SGLD is run for a relatively small number of iterations, and its step size is not decreased to zero. The present article investigates the behaviour of the SGLD with fixed step size. In particular we characterise the asymptotic bias explicitly, along with its dependence on the step size and the variance of the stochastic gradient. On that basis a modified SGLD which removes the asymptotic bias due to the variance of the stochastic gradients up to first order in the step size is derived. Moreover, we are able to obtain bounds on the finite-time bias, variance and mean squared error (MSE). The theory is illustrated with a Gaussian toy model for which the bias and the MSE for the estimation of moments can be obtained explicitly. For this toy model we study the gain of the SGLD over the standard Euler method in the limit of large data sets.

Abstract--In the co-sparse analysis model a set of filters is applied to a signal out of the signal class of interest yielding sparse filter responses. As such, it may serve as a prior in inverse problems, or for structural analysis of signals that are known to belong to the signal class. The more the model is adapted to the class, the more reliable it is for these purposes. The task of learning such operators for a given class is therefore a crucial problem. In many applications, it is also required that the filter responses are obtained in a timely manner, which can be achieved by filters with a separable structure. Not only can operators of this sort be efficiently used for computing the filter responses, but they also have the advantage that less training samples are required to obtain a reliable estimate of the operator . The first contribution of this work is to give theoretical evidence for this claim by providing an upper bound for the sample complexity of the learning process. The second is a stochastic gradient descent (SGD) method designed to learn an analysis operator with separable structures, which includes a novel and efficient step size selection rule. Numerical experiments are provided that link the sample complexity to the convergence speed of the SGD algorithm. HE ability to sparsely represent signals has become standard practice in signal processing over the last decade. The commonly used synthesis approach has been extensively investigated and has proven its validity in many applications. Its closely related counterpart, the co-sparse analysis approach, was at first not treated with as much interest. In recent years this has changed and more and more work regarding the application and the theoretical validity of the co-sparse analysis model has been published. Both models assume that the signalss of a certain class are (approximately) contained in a union of subspaces. In the synthesis model, this reads as s Dx, x is sparse. Personal use of this material is permitted.

Optimization problems with rank constraints arise in many applications, including matrix regression, structured PCA, matrix completion and matrix decomposition problems. An attractive heuristic for solving such problems is to factorize the low-rank matrix, and to run projected gradient descent on the nonconvex factorized optimization problem. The goal of this problem is to provide a general theoretical framework for understanding when such methods work well, and to characterize the nature of the resulting fixed point. We provide a simple set of conditions under which projected gradient descent, when given a suitable initialization, converges geometrically to a statistically useful solution. Our results are applicable even when the initial solution is outside any region of local convexity, and even when the problem is globally concave. Working in a non-asymptotic framework, we show that our conditions are satisfied for a wide range of concrete models, including matrix regression, structured PCA, matrix completion with real and quantized observations, matrix decomposition, and graph clustering problems. Simulation results show excellent agreement with the theoretical predictions.

The general perception is that kernel methods are not scalable, and neural nets are the methods of choice for nonlinear learning problems. Or have we simply not tried hard enough for kernel methods? Here we propose an approach that scales up kernel methods using a novel concept called "doubly stochastic functional gradients". Our approach relies on the fact that many kernel methods can be expressed as convex optimization problems, and we solve the problems by making two unbiased stochastic approximations to the functional gradient, one using random training points and another using random functions associated with the kernel, and then descending using this noisy functional gradient. We show that a function produced by this procedure after $t$ iterations converges to the optimal function in the reproducing kernel Hilbert space in rate $O(1/t)$, and achieves a generalization performance of $O(1/\sqrt{t})$. This doubly stochasticity also allows us to avoid keeping the support vectors and to implement the algorithm in a small memory footprint, which is linear in number of iterations and independent of data dimension. Our approach can readily scale kernel methods up to the regimes which are dominated by neural nets. We show that our method can achieve competitive performance to neural nets in datasets such as 8 million handwritten digits from MNIST, 2.3 million energy materials from MolecularSpace, and 1 million photos from ImageNet.

Deep structured output learning shows great promise in tasks like semantic image segmentation. We proffer a new, efficient deep structured model learning scheme, in which we show how deep Convolutional Neural Networks (CNNs) can be used to estimate the messages in message passing inference for structured prediction with Conditional Random Fields (CRFs). With such CNN message estimators, we obviate the need to learn or evaluate potential functions for message calculation. This confers significant efficiency for learning, since otherwise when performing structured learning for a CRF with CNN potentials it is necessary to undertake expensive inference for every stochastic gradient iteration. The network output dimension for message estimation is the same as the number of classes, in contrast to the network output for general CNN potential functions in CRFs, which is exponential in the order of the potentials. Hence CNN message learning has fewer network parameters and is more scalable for cases that a large number of classes are involved. We apply our method to semantic image segmentation on the PASCAL VOC 2012 dataset. We achieve an intersection-over-union score of 73.4 on its test set, which is the best reported result for methods using the VOC training images alone. This impressive performance demonstrates the effectiveness and usefulness of our CNN message learning method.

This text investigates relations between two well-known family of algorithms, matrix factorisations and recursive linear filters, by describing a probabilistic model in which approximate inference corresponds to a matrix factorisation algorithm. Using the probabilistic model, we derive a matrix factorisation algorithm as a recursive linear filter. More precisely, we derive a matrix-variate recursive linear filter in order to perform efficient inference in high dimensions. We also show that it is possible to interpret our algorithm as a nontrivial stochastic gradient algorithm. Demonstrations and comparisons on an image restoration task are given.

While stochastic variational inference is relatively well known for scaling inference in Bayesian probabilistic models, related methods also offer ways to circumnavigate the approximation of analytically intractable expectations. The key challenge in either setting is controlling the variance of gradient estimates: recent work has shown that for continuous latent variables, particularly multivariate Gaussians, this can be achieved by using the gradient of the log posterior. In this paper we apply the same idea to gamma distributed latent variables given gamma variational distributions, enabling straightforward "black box" variational inference in models where sparsity and non-negativity are appropriate. We demonstrate the method on a recently proposed gamma process model for network data, as well as a novel sparse factor analysis. We outperform generic sampling algorithms and the approach of using Gaussian variational distributions on transformed variables.

In applications of Gaussian processes where quantification of uncertainty is of primary interest, it is necessary to accurately characterize the posterior distribution over covariance parameters. This paper proposes an adaptation of the Stochastic Gradient Langevin Dynamics algorithm to draw samples from the posterior distribution over covariance parameters with negligible bias and without the need to compute the marginal likelihood. In Gaussian process regression, this has the enormous advantage that stochastic gradients can be computed by solving linear systems only. A novel unbiased linear systems solver based on parallelizable covariance matrix-vector products is developed to accelerate the unbiased estimation of gradients. The results demonstrate the possibility to enable scalable and exact (in a Monte Carlo sense) quantification of uncertainty in Gaussian processes without imposing any special structure on the covariance or reducing the number of input vectors.

Stochastic gradient descent~(SGD) and its variants have become more and more popular in machine learning due to their efficiency and effectiveness. To handle large-scale problems, researchers have recently proposed several parallel SGD methods for multicore systems. However, existing parallel SGD methods cannot achieve satisfactory performance in real applications. In this paper, we propose a fast asynchronous parallel SGD method, called AsySVRG, by designing an asynchronous strategy to parallelize the recently proposed SGD variant called stochastic variance reduced gradient~(SVRG). Both theoretical and empirical results show that AsySVRG can outperform existing state-of-the-art parallel SGD methods like Hogwild! in terms of convergence rate and computation cost.

We study distributed stochastic convex optimization under the delayed gradient model where the server nodes perform parameter updates, while the worker nodes compute stochastic gradients. We discuss, analyze, and experiment with a setup motivated by the behavior of real-world distributed computation networks, where the machines are differently slow at different time. Therefore, we allow the parameter updates to be sensitive to the actual delays experienced, rather than to worst-case bounds on the maximum delay. This sensitivity leads to larger stepsizes, that can help gain rapid initial convergence without having to wait too long for slower machines, while maintaining the same asymptotic complexity. We obtain encouraging improvements to overall convergence for distributed experiments on real datasets with up to billions of examples and features.