We study the stochastic Riemannian gradient algorithm for matrix eigen-decomposition. The state-of-the-art stochastic Riemannian algorithm requires the learning rate to decay to zero and thus suffers from slow convergence and sub-optimal solutions. In this paper, we address this issue by deploying the variance reduction (VR) technique of stochastic gradient descent (SGD). The technique was originally developed to solve convex problems in the Euclidean space. We generalize it to Riemannian manifolds and realize it to solve the non-convex eigen-decomposition problem. We are the first to propose and analyze the generalization of SVRG to Riemannian manifolds. Specifically, we propose the general variance reduction form, SVRRG, in the framework of the stochastic Riemannian gradient optimization. It's then specialized to the problem with eigensolvers and induces the SVRRG-EIGS algorithm. We provide a novel and elegant theoretical analysis on this algorithm. The theory shows that a fixed learning rate can be used in the Riemannian setting with an exponential global convergence rate guaranteed. The theoretical results make a significant improvement over existing studies, with the effectiveness empirically verified.

Motivated by an application of eliciting users' preferences, we investigate the problem of learning hemimetrics, i.e., pairwise distances among a set of $n$ items that satisfy triangle inequalities and non-negativity constraints. In our application, the (asymmetric) distances quantify private costs a user incurs when substituting one item by another. We aim to learn these distances (costs) by asking the users whether they are willing to switch from one item to another for a given incentive offer. Without exploiting structural constraints of the hemimetric polytope, learning the distances between each pair of items requires $\Theta(n^2)$ queries. We propose an active learning algorithm that substantially reduces this sample complexity by exploiting the structural constraints on the version space of hemimetrics. Our proposed algorithm achieves provably-optimal sample complexity for various instances of the task. For example, when the items are embedded into $K$ tight clusters, the sample complexity of our algorithm reduces to $O(n K)$. Extensive experiments on a restaurant recommendation data set support the conclusions of our theoretical analysis.

This paper considers the problem of canonical-correlation analysis (CCA) (Hotelling, 1936) and, more broadly, the generalized eigenvector problem for a pair of symmetric matrices. These are two fundamental problems in data analysis and scientific computing with numerous applications in machine learning and statistics (Shi and Malik, 2000; Hardoon et al., 2004; Witten et al., 2009). We provide simple iterative algorithms, with improved runtimes, for solving these problems that are globally linearly convergent with moderate dependencies on the condition numbers and eigenvalue gaps of the matrices involved. We obtain our results by reducing CCA to the top-$k$ generalized eigenvector problem. We solve this problem through a general framework that simply requires black box access to an approximate linear system solver. Instantiating this framework with accelerated gradient descent we obtain a running time of $O(\frac{z k \sqrt{\kappa}}{\rho} \log(1/\epsilon) \log \left(k\kappa/\rho\right))$ where $z$ is the total number of nonzero entries, $\kappa$ is the condition number and $\rho$ is the relative eigenvalue gap of the appropriate matrices. Our algorithm is linear in the input size and the number of components $k$ up to a $\log(k)$ factor. This is essential for handling large-scale matrices that appear in practice. To the best of our knowledge this is the first such algorithm with global linear convergence. We hope that our results prompt further research and ultimately improve the practical running time for performing these important data analysis procedures on large data sets.

High dimensional problems where the regressor belongs to a small number of groups play a critical role in many machine learning and signal processing applications, such as computational biology [9] and multitask learning [14]. In most of these cases, the groups overlap, i.e., the same feature can belong to multiple groups. For example, gene pathways overlap in computational biology applications, and parent-child pairs of wavelet transform coefficients overlap in signal processing applications. The existing state-of-the-art methods for solving such group sparsity structured regression problems can be categorized into two broad classes: a) convex relaxation based methods, b) iterative hard thresholding (IHT) or greedy methods. In practice, IHT methods tend to be significantly more scalable than the (group-)lasso style methods that solve a convex program. But, these methods require a certain projection operator which in general is NPhard to compute and often certain simple heuristics are used with relatively weak theoretical guarantees. Moreover, existing guarantees for both classes of methods require relatively restrictive assumptions on the data, like Restricted Isometry Property or variants thereof [2, 8, 18, 14], that are unlikely to hold in most common applications. In fact, even under such settings, the group sparsity based convex programs offer at most polylogarithmic gains over standard sparsity based methods [18].

For any positive integer $k$, there exist neural networks with $\Theta(k^3)$ layers, $\Theta(1)$ nodes per layer, and $\Theta(1)$ distinct parameters which can not be approximated by networks with $\mathcal{O}(k)$ layers unless they are exponentially large --- they must possess $\Omega(2^k)$ nodes. This result is proved here for a class of nodes termed "semi-algebraic gates" which includes the common choices of ReLU, maximum, indicator, and piecewise polynomial functions, therefore establishing benefits of depth against not just standard networks with ReLU gates, but also convolutional networks with ReLU and maximization gates, sum-product networks, and boosted decision trees (in this last case with a stronger separation: $\Omega(2^{k^3})$ total tree nodes are required).

I analyse the frequentist regret of the famous Gittins index strategy for multi-armed bandits with Gaussian noise and a finite horizon. Remarkably it turns out that this approach leads to finite-time regret guarantees comparable to those available for the popular UCB algorithm. Along the way I derive finite-time bounds on the Gittins index that are asymptotically exact and may be of independent interest. I also discuss some computational issues and present experimental results suggesting that a particular version of the Gittins index strategy is a modest improvement on existing algorithms with finite-time regret guarantees such as UCB and Thompson sampling.

Many real-world regression problems demand a measure of the uncertainty associated with each prediction. Standard decision forests deliver efficient state-of-the-art predictive performance, but high-quality uncertainty estimates are lacking. Gaussian processes (GPs) deliver uncertainty estimates, but scaling GPs to large-scale data sets comes at the cost of approximating the uncertainty estimates. We extend Mondrian forests, first proposed by Lakshminarayanan et al. (2014) for classification problems, to the large-scale non-parametric regression setting. Using a novel hierarchical Gaussian prior that dovetails with the Mondrian forest framework, we obtain principled uncertainty estimates, while still retaining the computational advantages of decision forests. Through a combination of illustrative examples, real-world large-scale datasets, and Bayesian optimization benchmarks, we demonstrate that Mondrian forests outperform approximate GPs on large-scale regression tasks and deliver better-calibrated uncertainty assessments than decision-forest-based methods.

Markov chain Monte Carlo algorithms are a popular and well-studied methodology that can be used to draw samples from posterior distributions. Over the past few years these algorithms have been extended to tackle problems where the model likelihood is intractable (Beaumont, 2003). Andrieu and Roberts (2009) showed that within the Metropolis-Hastings algorithm, if the likelihood is replaced with an unbiased estimate, then the sampler still targets the correct stationary distribution. Andrieu et al. (2010) extended this work further to create a class of 1 Markov chain algorithms that use sequential Monte Carlo methods, also known as particle filters. Current implementations of pseudo-marginal and particle Markov chain Monte Carlo use random-walk proposals to update the parameters (e.g., Golightly and Wilkinson, 2011; Knape and de Valpine, 2012) and shall be referred to herein as particle random-walk Metropolis algorithms. Random walk-based algorithms propose a new value from some symmetric density centred on the current value.

Learning in neural networks poses peculiar challenges when using discretized rather then continuous synaptic states. The choice of discrete synapses is motivated by biological reasoning and experiments, and possibly by hardware implementation considerations as well. In this paper we extend a previous large deviations analysis which unveiled the existence of peculiar dense regions in the space of synaptic states which accounts for the possibility of learning efficiently in networks with binary synapses. We extend the analysis to synapses with multiple states and generally more plausible biological features. The results clearly indicate that the overall qualitative picture is unchanged with respect to the binary case, and very robust to variation of the details of the model. We also provide quantitative results which suggest that the advantages of increasing the synaptic precision (i.e.~the number of internal synaptic states) rapidly vanish after the first few bits, and therefore that, for practical applications, only few bits may be needed for near-optimal performance, consistently with recent biological findings. Finally, we demonstrate how the theoretical analysis can be exploited to design efficient algorithmic search strategies.

Classification is one of fundamental problems in the fields of machine learning and signal processing. The commonly used classifier assigns a sample or a signal to the class with maximum posterior probability, which usually requires probability density function (PDF) estimation in an either model-driven or data-driven manner [1] [2] [3]. For high-dimensional data sets, it is necessary to perform feature reduction to estimate the PDFs robustly in a lowdimensional feature subspace. However, feature reduction may lose pertinent information for discrimination. For example, data samples from different classes that could be well separated in the raw data space may be overlapped in the feature subspace, causing classification errors. The PDF reconstruction approach provides a solution to address this information loss issue in feature reduction by reconstructing the PDF on raw data and making classification in raw data space, which could improve classification performance. Several approaches have been developed along this track.