Stephen J. Wright

A. S. Lewis, S. J. Wright

We consider minimization of functions that are compositions of convex or prox-regular functions (possibly extended-valued) with smooth vector functions. A wide variety of important optimization problems fall into this framework. We describe an algorithmic framework based on a subproblem constructed from a linearized approximation to the objective and a regularization term. Properties of local solutions of this subproblem underlie both a global convergence result and an identification property of the active manifold containing the solution of the original problem. Preliminary computational results on both convex and nonconvex examples are promising.

Rudy Geelen, Laura Balzano, Stephen Wright et al.

We present a novel method for learning reduced-order models of dynamical systems using nonlinear manifolds. First, we learn the manifold by identifying nonlinear structure in the data through a general representation learning problem. The proposed approach is driven by embeddings of low-order polynomial form. A projection onto the nonlinear manifold reveals the algebraic structure of the reduced-space system that governs the problem of interest. The matrix operators of the reduced-order model are then inferred from the data using operator inference. Numerical experiments on a number of nonlinear problems demonstrate the generalizability of the methodology and the increase in accuracy that can be obtained over reduced-order modeling methods that employ a linear subspace approximation.

Adrian S. Lewis, Stephen J. Wright

Identification of active constraints in constrained optimization is of interest from both practical and theoretical viewpoints, as it holds the promise of reducing an inequality-constrained problem to an equality-constrained problem, in a neighborhood of a solution. We study this issue in the more general setting of composite nonsmooth minimization, in which the objective is a composition of a smooth vector function c with a lower semicontinuous function h, typically nonsmooth but structured. In this setting, the graph of the generalized gradient of h can often be decomposed into a union (nondisjoint) of simpler subsets. "Identification" amounts to deciding which subsets of the graph are "active" in the criticality conditions at a given solution. We give conditions under which any convergent sequence of approximate critical points finitely identifies the activity. Prominent among these properties is a condition akin to the Mangasarian-Fromovitz constraint qualification, which ensures boundedness of the set of multiplier vectors that satisfy the optimality conditions at the solution.

Ke Chen, Qin Li, Jianfeng Lu et al.

In the framework of generalized finite element methods for elliptic equations with rough coefficients, efficiency and accuracy of the numerical method depend critically on the use of appropriate basis functions. This work explores several random sampling strategies that construct approximations to the optimal set of basis functions of a given dimension, and proposes a quantitative criterion to analyze and compare these sampling strategies. Numerical evidence shows that the best results are achieved by two strategies, Random Gaussian and Smooth boundary sampling.

Changyu Gao, Stephen J. Wright

We develop simple differentially private optimization algorithms that move along directions of (expected) descent to find an approximate second-order solution for nonconvex ERM. We use line search, mini-batching, and a two-phase strategy to improve the speed and practicality of the algorithm. Numerical experiments demonstrate the effectiveness of these approaches.

Shubham Kumar Bharti, Stephen Wright, Adish Singla et al.

We study optimal teaching of Linear Behavior Cloning (LBC) learners. In this setup, the teacher can select which states to demonstrate to an LBC learner. The learner maintains a version space of infinite linear hypotheses consistent with the demonstration. The goal of the teacher is to teach a realizable target policy to the learner using minimum number of state demonstrations. This number is known as the Teaching Dimension(TD). We present a teaching algorithm called ``Teach using Iterative Elimination(TIE)" that achieves instance optimal TD. However, we also show that finding optimal teaching set computationally is NP-hard. We further provide an approximation algorithm that guarantees an approximation ratio of $\log(|A|-1)$ on the teaching dimension. Finally, we provide experimental results to validate the efficiency and effectiveness of our algorithm.

Shi Chen, Qin Li, Oliver Tse et al.

The acceleration of gradient-based optimization methods is a subject of significant practical and theoretical importance, particularly within machine learning applications. While much attention has been directed towards optimizing within Euclidean space, the need to optimize over spaces of probability measures in machine learning motivates exploration of accelerated gradient methods in this context too. To this end, we introduce a Hamiltonian-flow approach analogous to momentum-based approaches in Euclidean space. We demonstrate that, in the continuous-time setting, algorithms based on this approach can achieve convergence rates of arbitrarily high order. We complement our findings with numerical examples.

Shuyao Li, Stephen J. Wright

We consider minimization of a smooth nonconvex function with inexact oracle access to gradient and Hessian (without assuming access to the function value) to achieve approximate second-order optimality. A novel feature of our method is that if an approximate direction of negative curvature is chosen as the step, we choose its sense to be positive or negative with equal probability. We allow gradients to be inexact in a relative sense and relax the coupling between inexactness thresholds for the first- and second-order optimality conditions. Our convergence analysis includes both an expectation bound based on martingale analysis and a high-probability bound based on concentration inequalities. We apply our algorithm to empirical risk minimization problems and obtain improved gradient sample complexity over existing works.

Andrew Lowy, Jonathan Ullman, Stephen J. Wright

We provide a simple and flexible framework for designing differentially private algorithms to find approximate stationary points of non-convex loss functions. Our framework is based on using a private approximate risk minimizer to "warm start" another private algorithm for finding stationary points. We use this framework to obtain improved, and sometimes optimal, rates for several classes of non-convex loss functions. First, we obtain improved rates for finding stationary points of smooth non-convex empirical loss functions. Second, we specialize to quasar-convex functions, which generalize star-convex functions and arise in learning dynamical systems and training some neural nets. We achieve the optimal rate for this class. Third, we give an optimal algorithm for finding stationary points of functions satisfying the Kurdyka-Lojasiewicz (KL) condition. For example, over-parameterized neural networks often satisfy this condition. Fourth, we provide new state-of-the-art rates for stationary points of non-convex population loss functions. Fifth, we obtain improved rates for non-convex generalized linear models. A modification of our algorithm achieves nearly the same rates for second-order stationary points of functions with Lipschitz Hessian, improving over the previous state-of-the-art for each of the above problems.

Shuyao Li, Yu Cheng, Ilias Diakonikolas et al.

Finding an approximate second-order stationary point (SOSP) is a well-studied and fundamental problem in stochastic nonconvex optimization with many applications in machine learning. However, this problem is poorly understood in the presence of outliers, limiting the use of existing nonconvex algorithms in adversarial settings. In this paper, we study the problem of finding SOSPs in the strong contamination model, where a constant fraction of datapoints are arbitrarily corrupted. We introduce a general framework for efficiently finding an approximate SOSP with \emph{dimension-independent} accuracy guarantees, using $\widetilde{O}({D^2}/ε)$ samples where $D$ is the ambient dimension and $ε$ is the fraction of corrupted datapoints. As a concrete application of our framework, we apply it to the problem of low rank matrix sensing, developing efficient and provably robust algorithms that can tolerate corruptions in both the sensing matrices and the measurements. In addition, we establish a Statistical Query lower bound providing evidence that the quadratic dependence on $D$ in the sample complexity is necessary for computationally efficient algorithms.

Changyu Gao, Andrew Lowy, Xingyu Zhou et al.

Machine learning algorithms in high-dimensional settings are highly susceptible to the influence of even a small fraction of structured outliers, making robust optimization techniques essential. In particular, within the $ε$-contamination model, where an adversary can inspect and replace up to an $ε$-fraction of the samples, a fundamental open problem is determining the optimal rates for robust stochastic convex optimization (SCO) under such contamination. We develop novel algorithms that achieve minimax-optimal excess risk (up to logarithmic factors) under the $ε$-contamination model. Our approach improves over existing algorithms, which are not only suboptimal but also require stringent assumptions, including Lipschitz continuity and smoothness of individual sample functions. By contrast, our optimal algorithms do not require these stringent assumptions, assuming only population-level smoothness of the loss. Moreover, our algorithms can be adapted to handle the case in which the covariance parameter is unknown, and can be extended to nonsmooth population risks via convolutional smoothing. We complement our algorithmic developments with a tight information-theoretic lower bound for robust SCO.

Chaobing Song, Cheuk Yin Lin, Stephen J. Wright et al.

We study a class of generalized linear programs (GLP) in a large-scale setting, which includes simple, possibly nonsmooth convex regularizer and simple convex set constraints. By reformulating (GLP) as an equivalent convex-concave min-max problem, we show that the linear structure in the problem can be used to design an efficient, scalable first-order algorithm, to which we give the name \emph{Coordinate Linear Variance Reduction} (\textsc{clvr}; pronounced "clever"). \textsc{clvr} yields improved complexity results for (GLP) that depend on the max row norm of the linear constraint matrix in (GLP) rather than the spectral norm. When the regularization terms and constraints are separable, \textsc{clvr} admits an efficient lazy update strategy that makes its complexity bounds scale with the number of nonzero elements of the linear constraint matrix in (GLP) rather than the matrix dimensions. On the other hand, for the special case of linear programs, by exploiting sharpness, we propose a restart scheme for \textsc{clvr} to obtain empirical linear convergence. Then we show that Distributionally Robust Optimization (DRO) problems with ambiguity sets based on both $f$-divergence and Wasserstein metrics can be reformulated as (GLPs) by introducing sparsely connected auxiliary variables. We complement our theoretical guarantees with numerical experiments that verify our algorithm's practical effectiveness, in terms of wall-clock time and number of data passes.

Zhiyan Ding, Shi Chen, Qin Li et al.

Finding the optimal configuration of parameters in ResNet is a nonconvex minimization problem, but first-order methods nevertheless find the global optimum in the overparameterized regime. We study this phenomenon with mean-field analysis, by translating the training process of ResNet to a gradient-flow partial differential equation (PDE) and examining the convergence properties of this limiting process. The activation function is assumed to be $2$-homogeneous or partially $1$-homogeneous; the regularized ReLU satisfies the latter condition. We show that if the ResNet is sufficiently large, with depth and width depending algebraically on the accuracy and confidence levels, first-order optimization methods can find global minimizers that fit the training data.

Zhiyan Ding, Shi Chen, Qin Li et al.

Finding parameters in a deep neural network (NN) that fit training data is a nonconvex optimization problem, but a basic first-order optimization method (gradient descent) finds a global optimizer with perfect fit (zero-loss) in many practical situations. We examine this phenomenon for the case of Residual Neural Networks (ResNet) with smooth activation functions in a limiting regime in which both the number of layers (depth) and the number of weights in each layer (width) go to infinity. First, we use a mean-field-limit argument to prove that the gradient descent for parameter training becomes a gradient flow for a probability distribution that is characterized by a partial differential equation (PDE) in the large-NN limit. Next, we show that under certain assumptions, the solution to the PDE converges in the training time to a zero-loss solution. Together, these results suggest that the training of the ResNet gives a near-zero loss if the ResNet is large enough. We give estimates of the depth and width needed to reduce the loss below a given threshold, with high probability.

Chaobing Song, Stephen J. Wright, Jelena Diakonikolas

We study structured nonsmooth convex finite-sum optimization that appears widely in machine learning applications, including support vector machines and least absolute deviation. For the primal-dual formulation of this problem, we propose a novel algorithm called \emph{Variance Reduction via Primal-Dual Accelerated Dual Averaging (\vrpda)}. In the nonsmooth and general convex setting, \vrpda~has the overall complexity $O(nd\log\min \{1/ε, n\} + d/ε)$ in terms of the primal-dual gap, where $n$ denotes the number of samples, $d$ the dimension of the primal variables, and $ε$ the desired accuracy. In the nonsmooth and strongly convex setting, the overall complexity of \vrpda~becomes $O(nd\log\min\{1/ε, n\} + d/\sqrtε)$ in terms of both the primal-dual gap and the distance between iterate and optimal solution. Both these results for \vrpda~improve significantly on state-of-the-art complexity estimates, which are $O(nd\log \min\{1/ε, n\} + \sqrt{n}d/ε)$ for the nonsmooth and general convex setting and $O(nd\log \min\{1/ε, n\} + \sqrt{n}d/\sqrtε)$ for the nonsmooth and strongly convex setting, in a much more simple and straightforward way. Moreover, both complexities are better than \emph{lower} bounds for general convex finite sums that lack the particular (common) structure that we consider. Our theoretical results are supported by numerical experiments, which confirm the competitive performance of \vrpda~compared to state-of-the-art.

Zhiyan Ding, Qin Li, Jianfeng Lu et al.

The Underdamped Langevin Monte Carlo (ULMC) is a popular Markov chain Monte Carlo sampling method. It requires the computation of the full gradient of the log-density at each iteration, an expensive operation if the dimension of the problem is high. We propose a sampling method called Random Coordinate ULMC (RC-ULMC), which selects a single coordinate at each iteration to be updated and leaves the other coordinates untouched. We investigate the computational complexity of RC-ULMC and compare it with the classical ULMC for strongly log-concave probability distributions. We show that RC-ULMC is always cheaper than the classical ULMC, with a significant cost reduction when the problem is highly skewed and high dimensional. Our complexity bound for RC-ULMC is also tight in terms of dimension dependence.

Zhiyan Ding, Qin Li, Jianfeng Lu et al.

Langevin Monte Carlo (LMC) is a popular Markov chain Monte Carlo sampling method. One drawback is that it requires the computation of the full gradient at each iteration, an expensive operation if the dimension of the problem is high. We propose a new sampling method: Random Coordinate LMC (RC-LMC). At each iteration, a single coordinate is randomly selected to be updated by a multiple of the partial derivative along this direction plus noise, and all other coordinates remain untouched. We investigate the total complexity of RC-LMC and compare it with the classical LMC for log-concave probability distributions. When the gradient of the log-density is Lipschitz, RC-LMC is less expensive than the classical LMC if the log-density is highly skewed for high dimensional problems, and when both the gradient and the Hessian of the log-density are Lipschitz, RC-LMC is always cheaper than the classical LMC, by a factor proportional to the square root of the problem dimension. In the latter case, our estimate of complexity is sharp with respect to the dimension.

Nam Ho-Nguyen, Stephen J. Wright

We study a model for adversarial classification based on distributionally robust chance constraints. We show that under Wasserstein ambiguity, the model aims to minimize the conditional value-at-risk of the distance to misclassification, and we explore links to adversarial classification models proposed earlier and to maximum-margin classifiers. We also provide a reformulation of the distributionally robust model for linear classification, and show it is equivalent to minimizing a regularized ramp loss objective. Numerical experiments show that, despite the nonconvexity of this formulation, standard descent methods appear to converge to the global minimizer for this problem. Inspired by this observation, we show that, for a certain class of distributions, the only stationary point of the regularized ramp loss minimization problem is the global minimizer.

Sinong Geng, Zhaobin Kuang, Jie Liu et al.

We study the $L_1$-regularized maximum likelihood estimator/estimation (MLE) problem for discrete Markov random fields (MRFs), where efficient and scalable learning requires both sparse regularization and approximate inference. To address these challenges, we consider a stochastic learning framework called stochastic proximal gradient (SPG; Honorio 2012a, Atchade et al. 2014,Miasojedow and Rejchel 2016). SPG is an inexact proximal gradient algorithm [Schmidtet al., 2011], whose inexactness stems from the stochastic oracle (Gibbs sampling) for gradient approximation - exact gradient evaluation is infeasible in general due to the NP-hard inference problem for discrete MRFs [Koller and Friedman, 2009]. Theoretically, we provide novel verifiable bounds to inspect and control the quality of gradient approximation. Empirically, we propose the tighten asymptotically (TAY) learning strategy based on the verifiable bounds to boost the performance of SPG.

Soroosh Khoram, Stephen J Wright, Jing Li

Similarity search retrieves the nearest neighbors of a query vector from a dataset of high-dimensional vectors. As the size of the dataset grows, the cost of performing the distance computations needed to implement a query can become prohibitive. A method often used to reduce this computational cost is quantization of the vector space and location-based encoding of the dataset vectors. These encodings can be used during query processing to find approximate nearest neighbors of the query point quickly. Search speed can be improved by using shorter codes, but shorter codes have higher quantization error, leading to degraded precision. In this work, we propose the Interleaved Composite Quantization (ICQ) which achieves fast similarity search without using shorter codes. In ICQ, a small subset of the code is used to approximate the distances, with complete codes being used only when necessary. Our method effectively reduces both code length and quantization error. Furthermore, ICQ is compatible with several recently proposed techniques for reducing quantization error and can be used in conjunction with these other techniques to improve results. We confirm these claims and show strong empirical performance of ICQ using several synthetic and real-word datasets.

Ching-pei Lee, Cong Han Lim, Stephen J. Wright

We propose a communication- and computation-efficient distributed optimization algorithm using second-order information for solving empirical risk minimization (ERM) problems with a nonsmooth regularization term. Our algorithm is applicable to both the primal and the dual ERM problem. Current second-order and quasi-Newton methods for this problem either do not work well in the distributed setting or work only for specific regularizers. Our algorithm uses successive quadratic approximations of the smooth part, and we describe how to maintain an approximation of the (generalized) Hessian and solve subproblems efficiently in a distributed manner. When applied to the distributed dual ERM problem, unlike state of the art that takes only the block-diagonal part of the Hessian, our approach is able to utilize global curvature information and is thus magnitudes faster. The proposed method enjoys global linear convergence for a broad range of non-strongly convex problems that includes the most commonly used ERMs, thus requiring lower communication complexity. It also converges on non-convex problems, so has the potential to be used on applications such as deep learning. Computational results demonstrate that our method significantly improves on communication cost and running time over the current state-of-the-art methods.

Rahul Mazumder, Stephen Wright, Andrew Zheng

We consider a class of linear-programming based estimators in reconstructing a sparse signal from linear measurements. Specific formulations of the reconstruction problem considered here include Dantzig selector, basis pursuit (for the case in which the measurements contain no errors), and the fused Dantzig selector (for the case in which the underlying signal is piecewise constant). In spite of being estimators central to sparse signal processing and machine learning, solving these linear programming problems for large scale instances remains a challenging task, thereby limiting their usage in practice. We show that classic constraint- and column-generation techniques from large scale linear programming, when used in conjunction with a commercial implementation of the simplex method, and initialized with the solution from a closely-related Lasso formulation, yields solutions with high efficiency in many settings.

Zachary Charles, Shashank Rajput, Stephen Wright et al.

Adversarial training is a technique for training robust machine learning models. To encourage robustness, it iteratively computes adversarial examples for the model, and then re-trains on these examples via some update rule. This work analyzes the performance of adversarial training on linearly separable data, and provides bounds on the number of iterations required for large margin. We show that when the update rule is given by an arbitrary empirical risk minimizer, adversarial training may require exponentially many iterations to obtain large margin. However, if gradient or stochastic gradient update rules are used, only polynomially many iterations are required to find a large-margin separator. By contrast, without the use of adversarial examples, gradient methods may require exponentially many iterations to achieve large margin. Our results are derived by showing that adversarial training with gradient updates minimizes a robust version of the empirical risk at a $\mathcal{O}(\ln(t)^2/t)$ rate, despite non-smoothness. We corroborate our theory empirically.

Kwang-Sung Jun, Rebecca Willett, Stephen Wright et al.

We introduce the bilinear bandit problem with low-rank structure in which an action takes the form of a pair of arms from two different entity types, and the reward is a bilinear function of the known feature vectors of the arms. The unknown in the problem is a $d_1$ by $d_2$ matrix $\mathbfΘ^*$ that defines the reward, and has low rank $r \ll \min\{d_1,d_2\}$. Determination of $\mathbfΘ^*$ with this low-rank structure poses a significant challenge in finding the right exploration-exploitation tradeoff. In this work, we propose a new two-stage algorithm called "Explore-Subspace-Then-Refine" (ESTR). The first stage is an explicit subspace exploration, while the second stage is a linear bandit algorithm called "almost-low-dimensional OFUL" (LowOFUL) that exploits and further refines the estimated subspace via a regularization technique. We show that the regret of ESTR is $\widetilde{\mathcal{O}}((d_1+d_2)^{3/2} \sqrt{r T})$ where $\widetilde{\mathcal{O}}$ hides logarithmic factors and $T$ is the time horizon, which improves upon the regret of $\widetilde{\mathcal{O}}(d_1d_2\sqrt{T})$ attained for a naïve linear bandit reduction. We conjecture that the regret bound of ESTR is unimprovable up to polylogarithmic factors, and our preliminary experiment shows that ESTR outperforms a naïve linear bandit reduction.

Hongyi Wang, Scott Sievert, Zachary Charles et al.

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. We present ATOMO, a general framework for atomic sparsification of stochastic gradients. Given a gradient, an atomic decomposition, and a sparsity budget, ATOMO gives a random unbiased sparsification of the atoms minimizing variance. We show that recent methods such as QSGD and TernGrad are special cases of ATOMO and that sparsifiying the singular value decomposition of neural networks gradients, rather than their coordinates, can lead to significantly faster distributed training.

Bin Hu, Stephen Wright, Laurent Lessard

Techniques for reducing the variance of gradient estimates used in stochastic programming algorithms for convex finite-sum problems have received a great deal of attention in recent years. By leveraging dissipativity theory from control, we provide a new perspective on two important variance-reduction algorithms: SVRG and its direct accelerated variant Katyusha. Our perspective provides a physically intuitive understanding of the behavior of SVRG-like methods via a principle of energy conservation. The tools discussed here allow us to automate the convergence analysis of SVRG-like methods by capturing their essential properties in small semidefinite programs amenable to standard analysis and computational techniques. Our approach recovers existing convergence results for SVRG and Katyusha and generalizes the theory to alternative parameter choices. We also discuss how our approach complements the linear coupling technique. Our combination of perspectives leads to a better understanding of accelerated variance-reduced stochastic methods for finite-sum problems.

Gábor Braun, Sebastian Pokutta, Dan Tu et al.

We present a blended conditional gradient approach for minimizing a smooth convex function over a polytope P, combining the Frank--Wolfe algorithm (also called conditional gradient) with gradient-based steps, different from away steps and pairwise steps, but still achieving linear convergence for strongly convex functions, along with good practical performance. Our approach retains all favorable properties of conditional gradient algorithms, notably avoidance of projections onto P and maintenance of iterates as sparse convex combinations of a limited number of extreme points of P. The algorithm is lazy, making use of inexpensive inexact solutions of the linear programming subproblem that characterizes the conditional gradient approach. It decreases measures of optimality (primal and dual gaps) rapidly, both in the number of iterations and in wall-clock time, outperforming even the lazy conditional gradient algorithms of [arXiv:1410.8816]. We also present a streamlined version of the algorithm for the probability simplex.

Xuezhou Zhang, Xiaojin Zhu, Stephen J. Wright

Training set bugs are flaws in the data that adversely affect machine learning. The training set is usually too large for man- ual inspection, but one may have the resources to verify a few trusted items. The set of trusted items may not by itself be adequate for learning, so we propose an algorithm that uses these items to identify bugs in the training set and thus im- proves learning. Specifically, our approach seeks the smallest set of changes to the training set labels such that the model learned from this corrected training set predicts labels of the trusted items correctly. We flag the items whose labels are changed as potential bugs, whose labels can be checked for veracity by human experts. To find the bugs in this way is a challenging combinatorial bilevel optimization problem, but it can be relaxed into a continuous optimization problem. Ex- periments on toy and real data demonstrate that our approach can identify training set bugs effectively and suggest appro- priate changes to the labels. Our algorithm is a step toward trustworthy machine learning.

Kwang-Sung Jun, Francesco Orabona, Stephen Wright et al.

A key challenge in online learning is that classical algorithms can be slow to adapt to changing environments. Recent studies have proposed "meta" algorithms that convert any online learning algorithm to one that is adaptive to changing environments, where the adaptivity is analyzed in a quantity called the strongly-adaptive regret. This paper describes a new meta algorithm that has a strongly-adaptive regret bound that is a factor of $\sqrt{\log(T)}$ better than other algorithms with the same time complexity, where $T$ is the time horizon. We also extend our algorithm to achieve a first-order (i.e., dependent on the observed losses) strongly-adaptive regret bound for the first time, to our knowledge. At its heart is a new parameter-free algorithm for the learning with expert advice (LEA) problem in which experts sometimes do not output advice for consecutive time steps (i.e., \emph{sleeping} experts). This algorithm is derived by a reduction from optimal algorithms for the so-called coin betting problem. Empirical results show that our algorithm outperforms state-of-the-art methods in both learning with expert advice and metric learning scenarios.

Kwang-Sung Jun, Francesco Orabona, Rebecca Willett et al.

This paper describes a new parameter-free online learning algorithm for changing environments. In comparing against algorithms with the same time complexity as ours, we obtain a strongly adaptive regret bound that is a factor of at least $\sqrt{\log(T)}$ better, where $T$ is the time horizon. Empirical results show that our algorithm outperforms state-of-the-art methods in learning with expert advice and metric learning scenarios.

Nikhil Rao, Parikshit Shah, Stephen Wright

In many signal processing applications, the aim is to reconstruct a signal that has a simple representation with respect to a certain basis or frame. Fundamental elements of the basis known as "atoms" allow us to define "atomic norms" that can be used to formulate convex regularizations for the reconstruction problem. Efficient algorithms are available to solve these formulations in certain special cases, but an approach that works well for general atomic norms, both in terms of speed and reconstruction accuracy, remains to be found. This paper describes an optimization algorithm called CoGEnT that produces solutions with succinct atomic representations for reconstruction problems, generally formulated with atomic-norm constraints. CoGEnT combines a greedy selection scheme based on the conditional gradient approach with a backward (or "truncation") step that exploits the quadratic nature of the objective to reduce the basis size. We establish convergence properties and validate the algorithm via extensive numerical experiments on a suite of signal processing applications. Our algorithm and analysis also allow for inexact forward steps and for occasional enhancements of the current representation to be performed. CoGEnT can outperform the basic conditional gradient method, and indeed many methods that are tailored to specific applications, when the enhancement and truncation steps are defined appropriately. We also introduce several novel applications that are enabled by the atomic-norm framework, including tensor completion, moment problems in signal processing, and graph deconvolution.

Ryan Kennedy, Laura Balzano, Stephen J. Wright et al.

We present a family of online algorithms for real-time factorization-based structure from motion, leveraging a relationship between incremental singular value decomposition and recently proposed methods for online matrix completion. Our methods are orders of magnitude faster than previous state of the art, can handle missing data and a variable number of feature points, and are robust to noise and sparse outliers. We demonstrate our methods on both real and synthetic sequences and show that they perform well in both online and batch settings. We also provide an implementation which is able to produce 3D models in real time using a laptop with a webcam.

Laura Balzano, Stephen J. Wright

GROUSE (Grassmannian Rank-One Update Subspace Estimation) is an incremental algorithm for identifying a subspace of Rn from a sequence of vectors in this subspace, where only a subset of components of each vector is revealed at each iteration. Recent analysis has shown that GROUSE converges locally at an expected linear rate, under certain assumptions. GROUSE has a similar flavor to the incremental singular value decomposition algorithm, which updates the SVD of a matrix following addition of a single column. In this paper, we modify the incremental SVD approach to handle missing data, and demonstrate that this modified approach is equivalent to GROUSE, for a certain choice of an algorithmic parameter.

Ji Liu, Stephen J. Wright

We consider the reconstruction problem in compressed sensing in which the observations are recorded in a finite number of bits. They may thus contain quantization errors (from being rounded to the nearest representable value) and saturation errors (from being outside the range of representable values). Our formulation has an objective of weighted $\ell_2$-$\ell_1$ type, along with constraints that account explicitly for quantization and saturation errors, and is solved with an augmented Lagrangian method. We prove a consistency result for the recovered solution, stronger than those that have appeared to date in the literature, showing in particular that asymptotic consistency can be obtained without oversampling. We present extensive computational comparisons with formulations proposed previously, and variants thereof.