The optimization problem behind neural networks is highly non-convex. Training with stochastic gradient descent and variants requires careful parameter tuning and provides no guarantee to achieve the global optimum. In contrast we show under quite weak assumptions on the data that a particular class of feedforward neural networks can be trained globally optimal with a linear convergence rate. Up to our knowledge this is the first practically feasible method which achieves such a guarantee. While the method can in principle be applied to deep networks, we restrict ourselves for simplicity in this paper to one- and two hidden layer networks. Our experiments confirms that these models are already rich enough to achieve good performance on a series of real-world datasets.

Stochastic gradient hard thresholding methods have recently been shown to work favorably in solving large-scale empirical risk minimization problems under sparsity or rank constraint. Despite the improved iteration complexity over full gradient methods, the gradient evaluation and hard thresholding complexity of the existing stochastic algorithms usually scales linearly with data size, which could still be expensive when data is huge and the hard thresholding step could be as expensive as singular value decomposition in rank-constrained problems. To address these deficiencies, we propose an efficient hybrid stochastic gradient hard thresholding (HSG-HT) method that can be provably shown to have sample-size-independent gradient evaluation and hard thresholding complexity bounds. Specifically, we prove that the stochastic gradient evaluation complexity of HSG-HT scales linearly with inverse of sub-optimality and its hard thresholding complexity scales logarithmically. By applying the heavy ball acceleration technique, we further propose an accelerated variant of HSG-HT which can be shown to have improved factor dependence on restricted condition number.

Sketching is a powerful dimensionality reduction tool for accelerating statistical learning algorithms. However, its applicability has been limited to a certain extent since the crucial ingredient, the so-called oblivious subspace embedding, can only be applied to data spaces with an explicit representation as the column span or row span of a matrix, while in many settings learning is done in a high-dimensional space implicitly defined by the data matrix via a kernel transformation. We propose the first {\em fast} oblivious subspace embeddings that are able to embed a space induced by a non-linear kernel {\em without} explicitly mapping the data to the high-dimensional space. In particular, we propose an embedding for mappings induced by the polynomial kernel. Using the subspace embeddings, we obtain the fastest known algorithms for computing an implicit low rank approximation of the higher-dimension mapping of the data matrix, and for computing an approximate kernel PCA of the data, as well as doing approximate kernel principal component regression. Papers published at the Neural Information Processing Systems Conference.

Based on non-local prior distributions, we propose a Bayesian model selection (BMS) procedure for boundary detection in a sequence of data with multiple systematic mean changes. The BMS method can effectively suppress the non-boundary spike points with large instantaneous changes. We establish the consistency of the estimated number and locations of the change points under various prior distributions. Extensive simulation studies are conducted to compare the BMS with existing methods, and our approach is illustrated with application to the magnetic resonance imaging guided radiation therapy data. Papers published at the Neural Information Processing Systems Conference.

The amount of data available in the world is growing faster than our ability to deal with it. However, if we take advantage of the internal structure, data may become much smaller for machine learning purposes. In this paper we focus on one of the fundamental machine learning tasks, empirical risk minimization (ERM), and provide faster algorithms with the help from the clustering structure of the data. We introduce a simple notion of raw clustering that can be efficiently computed from the data, and propose two algorithms based on clustering information. Our accelerated algorithm ClusterACDM is built on a novel Haar transformation applied to the dual space of the ERM problem, and our variance-reduction based algorithm ClusterSVRG introduces a new gradient estimator using clustering.

In this paper we study the problem of learning Rectified Linear Units (ReLUs) which are functions of the form $\vct{x}\mapsto \max(0,\langle \vct{w},\vct{x}\rangle)$ with $\vct{w}\in\R d$ denoting the weight vector. We study this problem in the high-dimensional regime where the number of observations are fewer than the dimension of the weight vector. We assume that the weight vector belongs to some closed set (convex or nonconvex) which captures known side-information about its structure. We focus on the realizable model where the inputs are chosen i.i.d. We show that projected gradient descent, when initialized at $\vct{0}$, converges at a linear rate to the planted model with a number of samples that is optimal up to numerical constants.

The dictionary learning problem is to factorize vector samples $y {1},y {2},\ldots, y {n}$ into an appropriate basis (dictionary) $A *$ and sparse vectors $x {1*},\ldots,x {n*}$. Our algorithm is a simple alternating minimization procedure that switches between $\ell_1$ minimization and gradient descent in alternate steps. Dictionary learning and specifically alternating minimization algorithms for dictionary learning are well studied both theoretically and empirically. However, in contrast to previous theoretical analyses for this problem, we replace a condition on the operator norm (that is, the largest magnitude singular value) of the true underlying dictionary $A *$ with a condition on the matrix infinity norm (that is, the largest magnitude term). This not only allows us to get convergence rates for the error of the estimated dictionary measured in the matrix infinity norm, but also ensures that a random initialization will provably converge to the global optimum. Our guarantees are under a reasonable generative model that allows for dictionaries with growing operator norms, and can handle an arbitrary level of overcompleteness, while having sparsity that is information theoretically optimal.

We show that asymptotically, completely asynchronous stochastic gradient procedures achieve optimal (even to constant factors) convergence rates for the solution of convex optimization problems under nearly the same conditions required for asymptotic optimality of standard stochastic gradient procedures. Roughly, the noise inherent to the stochastic approximation scheme dominates any noise from asynchrony. We also give empirical evidence demonstrating the strong performance of asynchronous, parallel stochastic optimization schemes, demonstrating that the robustness inherent to stochastic approximation problems allows substantially faster parallel and asynchronous solution methods. In short, we show that for many stochastic approximation problems, as Freddie Mercury sings in Queen's \emph{Bohemian Rhapsody}, Nothing really matters.'' Papers published at the Neural Information Processing Systems Conference.

We study the estimation of the latent variable Gaussian graphical model (LVGGM), where the precision matrix is the superposition of a sparse matrix and a low-rank matrix. In order to speed up the estimation of the sparse plus low-rank components, we propose a sparsity constrained maximum likelihood estimator based on matrix factorization and an efficient alternating gradient descent algorithm with hard thresholding to solve it. Our algorithm is orders of magnitude faster than the convex relaxation based methods for LVGGM. In addition, we prove that our algorithm is guaranteed to linearly converge to the unknown sparse and low-rank components up to the optimal statistical precision. Experiments on both synthetic and genomic data demonstrate the superiority of our algorithm over the state-of-the-art algorithms and corroborate our theory.

Can we effectively learn a nonlinear representation in time comparable to linear learning? We describe a new algorithm that explicitly and adaptively expands higher-order interaction features over base linear representations. The algorithm is designed for extreme computational efficiency, and an extensive experimental study shows that its computation/prediction tradeoff ability compares very favorably against strong baselines. Papers published at the Neural Information Processing Systems Conference.