We address the problem of Bayesian structure learning for domains with hundreds of variables by employing non-parametric bootstrap, recursively. We propose a method that covers both model averaging and model selection in the same framework. The proposed method deals with the main weakness of constraint-based learning---sensitivity to errors in the independence tests---by a novel way of combining bootstrap with constraint-based learning. Essentially, we provide an algorithm for learning a tree, in which each node represents a scored CPDAG for a subset of variables and the level of the node corresponds to the maximal order of conditional independencies that are encoded in the graph. As higher order independencies are tested in deeper recursive calls, they benefit from more bootstrap samples, and therefore are more resistant to the curse-of-dimensionality. Moreover, the re-use of stable low order independencies allows greater computational efficiency.

Backpropagation and the chain rule of derivatives have been prominent; however, the total derivative rule has not enjoyed the same amount of attention. In this work we show how the total derivative rule leads to an intuitive visual framework for creating gradient estimators on graphical models. In particular, previous "policy gradient theorems" are easily derived. We derive new gradient estimators based on density estimation, as well as a likelihood ratio gradient, which "jumps" to an intermediate node, not directly to the objective function. We evaluate our methods on model-based policy gradient algorithms, achieve good performance, and present evidence towards demystifying the success of the popular PILCO algorithm.

Sketching and stochastic gradient methods are arguably the most common techniques to derive efficient large scale learning algorithms. In this paper, we investigate their application in the context of nonparametric statistical learning. More precisely, we study the estimator defined by stochastic gradient with mini batches and random features. The latter can be seen as form of nonlinear sketching and used to define approximate kernel methods. The considered estimator is not explicitly penalized/constrained and regularization is implicit.

We present an algorithm STRSAGA for efficiently maintaining a machine learning model over data points that arrive over time, quickly updating the model as new training data is observed. We present a competitive analysis comparing the sub-optimality of the model maintained by STRSAGA with that of an offline algorithm that is given the entire data beforehand, and analyze the risk-competitiveness of STRSAGA under different arrival patterns. Our theoretical and experimental results show that the risk of STRSAGA is comparable to that of offline algorithms on a variety of input arrival patterns, and its experimental performance is significantly better than prior algorithms suited for streaming data, such as SGD and SSVRG. Papers published at the Neural Information Processing Systems Conference.

Stochastic gradient methods are the workhorse (algorithms) of large-scale optimization problems in machine learning, signal processing, and other computational sciences and engineering. This paper studies Markov chain gradient descent, a variant of stochastic gradient descent where the random samples are taken on the trajectory of a Markov chain. Existing results of this method assume convex objectives and a reversible Markov chain and thus have their limitations. We establish new non-ergodic convergence under wider step sizes, for nonconvex problems, and for non-reversible finite-state Markov chains. Nonconvexity makes our method applicable to broader problem classes.

Distributed model training suffers from communication overheads due to frequent gradient updates transmitted between compute nodes. To mitigate these overheads, several studies propose the use of sparsified stochastic gradients. We argue that these are facets of a general sparsification method that can operate on any possible atomic decomposition. Notable examples include element-wise, singular value, and Fourier decompositions. Given a gradient, an atomic decomposition, and a sparsity budget, ATOMO gives a random unbiased sparsification of the atoms minimizing variance.

Adaptive gradient methods that rely on scaling gradients down by the square root of exponential moving averages of past squared gradients, such RMSProp, Adam, Adadelta have found wide application in optimizing the nonconvex problems that arise in deep learning. However, it has been recently demonstrated that such methods can fail to converge even in simple convex optimization settings. In this work, we provide a new analysis of such methods applied to nonconvex stochastic optimization problems, characterizing the effect of increasing minibatch size. Our analysis shows that under this scenario such methods do converge to stationarity up to the statistical limit of variance in the stochastic gradients (scaled by a constant factor). In particular, our result implies that increasing minibatch sizes enables convergence, thus providing a way to circumvent the non-convergence issues.

Recovering high-resolution images from limited sensory data typically leads to a serious ill-posed inverse problem, demanding inversion algorithms that effectively capture the prior information. Learning a good inverse mapping from training data faces severe challenges, including: (i) scarcity of training data; (ii) need for plausible reconstructions that are physically feasible; (iii) need for fast reconstruction, especially in real-time applications. We develop a successful system solving all these challenges, using as basic architecture the repetitive application of alternating proximal and data fidelity constraints. We learn a proximal map that works well with real images based on residual networks with recurrent blocks. Extensive experiments are carried out under different settings: (a) reconstructing abdominal MRI of pediatric patients from highly undersampled k-space data and (b) super-resolving natural face images.

For models with many parameters, the covari- ance matrix they are based on becomes gigantic, making them inapplicable in their original form. This has motivated research into both simple diagonal approxima- tions and more sophisticated factored approximations such as KFAC (Heskes, 2000; Martens & Grosse, 2015; Grosse & Martens, 2016). In the present work we draw inspiration from both to propose a novel approximation that is provably better than KFAC and amendable to cheap partial updates. It consists in tracking a diagonal variance, not in parameter coordinates, but in a Kronecker-factored eigenbasis, in which the diagonal approximation is likely to be more effective. Experiments show improvements over KFAC in optimization speed for several deep network architectures.

We show that gradient descent on full-width linear convolutional networks of depth $L$ converges to a linear predictor related to the $\ell_{2/L}$ bridge penalty in the frequency domain. This is in contrast to linearly fully connected networks, where gradient descent converges to the hard margin linear SVM solution, regardless of depth. Papers published at the Neural Information Processing Systems Conference.