Can one reduce the size of a graph without significantly altering its basic properties? The graph reduction problem is hereby approached from the perspective of restricted spectral approximation, a modification of the spectral similarity measure used for graph sparsification. This choice is motivated by the observation that restricted approximation carries strong spectral and cut guarantees, and that it implies approximation results for unsupervised learning problems relying on spectral embeddings. The article then focuses on coarsening - the most common type of graph reduction. Sufficient conditions are derived for a small graph to approximate a larger one in the sense of restricted approximation. These findings give rise to algorithms that, compared to both standard and advanced graph reduction methods, find coarse graphs of improved quality, often by a large margin, without sacrificing speed.

We present a new algorithm for domain adaptation improving upon a discrepancy minimization algorithm, (DM), previously shown to outperform a number of algorithms for this problem. Unlike many previously proposed solutions for domain adaptation, our algorithm does not consist of a fixed reweighting of the losses over the training sample. Instead, the reweighting depends on the hypothesis sought. The algorithm is derived from a less conservative notion of discrepancy than the DM algorithm called generalized discrepancy. We present a detailed description of our algorithm and show that it can be formulated as a convex optimization problem. We also give a detailed theoretical analysis of its learning guarantees which helps us select its parameters. Finally, we report the results of experiments demonstrating that it improves upon discrepancy minimization.

State-of-the-art first- and second-order optimization methods are able to achieve either fast global linear convergence rates or quadratic convergence, but not both of them. In this work, we propose an interpolation between first- and second-order methods for regularized empirical risk minimization that exploits the problem structure to efficiently combine multiple update directions. Our method attains both optimal global linear convergence rate for first-order methods, and local quadratic convergence. Experimental results show that our method outperforms state-of-the-art first- and second-order optimization methods in terms of the number of data accesses, while is competitive in training time.

Despite their attractiveness, popular perception is that techniques for nonparametric function approximation do not scale to streaming data due to an intractable growth in the amount of storage they require. To solve this problem in a memory-affordable way, we propose an online technique based on functional stochastic gradient descent in tandem with supervised sparsification based on greedy function subspace projections. The method, called parsimonious online learning with kernels (POLK), provides a controllable tradeoff between its solution accuracy and the amount of memory it requires. We derive conditions under which the generated function sequence converges almost surely to the optimal function, and we establish that the memory requirement remains finite. We evaluate POLK for kernel multi-class logistic regression and kernel hinge-loss classification on three canonical data sets: a synthetic Gaussian mixture model, the MNIST hand-written digits, and the Brodatz texture database. On all three tasks, we observe a favorable trade-off of objective function evaluation, classification performance, and complexity of the nonparametric regressor extracted by the proposed method.

We study the performance of kernel methods on the acoustic modeling task for automatic speech recognition, and compare their performance to deep neural networks (DNNs). To scale the kernel methods to large data sets, we use the random Fourier feature method of Rahimi and Recht (2007). We propose two novel techniques for improving the performance of kernel acoustic models. First, we propose a simple but effective feature selection method which reduces the number of random features required to attain a fixed level of performance. Second, we present a number of metrics which correlate strongly with speech recognition performance when computed on the heldout set; we attain improved performance by using these metrics to decide when to stop training. Additionally, we show that the linear bottleneck method of Sainath et al. (2013a) improves the performance of our kernel models significantly, in addition to speeding up training and making the models more compact. Leveraging these three methods, the kernel methods attain token error rates between $0.5\%$ better and $0.1\%$ worse than fully-connected DNNs across four speech recognition data sets, including the TIMIT and Broadcast News benchmark tasks.

In this paper, we consider the problem of learning high-dimensional tensor regression problems with low-rank structure. One of the core challenges associated with learning high-dimensional models is computation since the underlying optimization problems are often non-convex. While convex relaxations could lead to polynomial-time algorithms they are often slow in practice. On the other hand, limited theoretical guarantees exist for non-convex methods. In this paper we provide a general framework that provides theoretical guarantees for learning high-dimensional tensor regression models under different low-rank structural assumptions using the projected gradient descent algorithm applied to a potentially non-convex constraint set $\Theta$ in terms of its localized Gaussian width (due to Gaussian design). We juxtapose our theoretical results for non-convex projected gradient descent algorithms with previous results on regularized convex approaches. The two main differences between the convex and non-convex approach are: (i) from a computational perspective whether the non-convex projection operator is computable and whether the projection has desirable contraction properties and (ii) from a statistical error bound perspective, the non-convex approach has a superior rate for a number of examples. We provide three concrete examples of low-dimensional structure which address these issues and explain the pros and cons for the non-convex and convex approaches. We supplement our theoretical results with simulations which show that, under several common settings of generalized low rank tensor regression, the projected gradient descent approach is superior both in terms of statistical error and run-time provided the step-sizes of the projected descent algorithm are suitably chosen.

In nonparametric regression, the derivative estimation has attracted much attention in recent years due to its wide applications. In this paper, we propose a new method for the derivative estimation using the locally weighted least absolute deviation regression. Different from the local polynomial regression, the proposed method does not require a finite variance for the error term and so is robust to the presence of heavy-tailed errors. Meanwhile, it does not require a zero median or a positive density at zero for the error term in comparison with the local median regression. We further show that the proposed estimator with random difference is asymptotically equivalent to the (infinitely) composite quantile regression estimator. In other words, running one regression is equivalent to combining infinitely many quantile regressions. In addition, the proposed method is also extended to estimate the derivatives at the boundaries and to estimate higher-order derivatives. For the equidistant design, we derive theoretical results for the proposed estimators, including the asymptotic bias and variance, consistency, and asymptotic normality. Finally, we conduct simulation studies to demonstrate that the proposed method has better performance than the existing methods in the presence of outliers and heavy-tailed errors, and analyze the Chinese house price data for the past ten years to illustrate the usefulness of the proposed method.

In stochastic optimization, the population risk is generally approximated by the empirical risk which is in turn minimized by an iterative algorithm. However, in the large-scale setting, empirical risk minimization may be computationally restrictive. In this paper, we design an efficient algorithm to approximate the population risk minimizer in generalized linear problems such as binary classification with surrogate losses and generalized linear regression models. We focus on large-scale problems where the iterative minimization of the empirical risk is computationally intractable, i.e., the number of observations $n$ is much larger than the dimension of the parameter $p$ ($n \gg p \gg 1$). We show that under random sub-Gaussian design, the true minimizer of the population risk is approximately proportional to the corresponding ordinary least squares (OLS) estimator. Using this relation, we design an algorithm that achieves the same accuracy as the empirical risk minimizer through iterations that attain up to a quadratic convergence rate, and that are computationally cheaper than any batch optimization algorithm by at least a factor of $\mathcal{O}(p)$. We provide theoretical guarantees for our algorithm, and analyze the convergence behavior in terms of data dimensions. Finally, we demonstrate the performance of our algorithm on well-known classification and regression problems, through extensive numerical studies on large-scale datasets, and show that it achieves the highest performance compared to several other widely used optimization algorithms.

The higher order singular value decomposition (HOSVD) of tensors is a generalization of matrix SVD. The perturbation analysis of HOSVD under random noise is more delicate than its matrix counterpart. Recently, polynomial time algorithms have been proposed where statistically optimal estimates of the singular subspaces and the low rank tensors are attainable in the Euclidean norm. In this article, we analyze the sup-norm perturbation bounds of HOSVD and introduce estimators of the singular subspaces with sharp deviation bounds in the sup-norm. We also investigate a low rank tensor denoising estimator and demonstrate its fast convergence rate with respect to the entry-wise errors. The sup-norm perturbation bounds reveal unconventional phase transitions for statistical learning applications such as the exact clustering in high dimensional Gaussian mixture model and the exact support recovery in sub-tensor localizations. In addition, the bounds established for HOSVD also elaborate the one-sided sup-norm perturbation bounds for the singular subspaces of unbalanced (or fat) matrices.

We study the pricing problem faced by a firm that sells a large number of products, described via a wide range of features, to customers that arrive over time. Customers independently make purchasing decisions according to a general choice model that includes products features and customers' characteristics, encoded as $d$-dimensional numerical vectors, as well as the price offered. The parameters of the choice model are a priori unknown to the firm, but can be learned as the (binary-valued) sales data accrues over time. The firm's objective is to maximize its revenue. We benchmark the performance using the classic regret minimization framework where the regret is defined as the expected revenue loss against a clairvoyant policy that knows the parameters of the choice model in advance, and always offers the revenue-maximizing price. This setting is motivated in part by the prevalence of online marketplaces that allow for real-time pricing. We assume a structured choice model, parameters of which depend on $s_0$ out of the $d$ product features. Assuming that the market noise distribution is known, we propose a dynamic policy, called Regularized Maximum Likelihood Pricing (RMLP) that leverages the (sparsity) structure of the high-dimensional model and obtains a logarithmic regret in $T$. More specifically, the regret of our algorithm is of $O(s_0 \log d \cdot \log T)$. Furthermore, we show that no policy can obtain regret better than $O(s_0 (\log d + \log T))$. {In addition, we propose a generalization of our policy to a setting that the market noise distribution is unknown but belongs to a parametrized family of distributions. This policy obtains regret of $O(\sqrt{(\log d)T})$. We further show that no policy can obtain regret better than $\Omega(\sqrt{T})$ in such environments.}