In this paper, we propose a novel sparse learning based feature selection method that directly optimizes a large margin linear classification model sparsity with l_(2,p)-norm (0 < p < 1)subject to data-fitting constraints, rather than using the sparsity as a regularization term. To solve the direct sparsity optimization problem that is non-smooth and non-convex when 0<p<1, we provide an efficient iterative algorithm with proved convergence by converting it to a convex and smooth optimization problem at every iteration step. The proposed algorithm has been evaluated based on publicly available datasets, and extensive comparison experiments have demonstrated that our algorithm could achieve feature selection performance competitive to state-of-the-art algorithms.
In this paper we discuss Bayesian nonconvex penalization for sparse learning problems. We explore a nonparametric formulation for latent shrinkage parameters using subordinators which are one-dimensional L\'{e}vy processes. We particularly study a family of continuous compound Poisson subordinators and a family of discrete compound Poisson subordinators. We exemplify four specific subordinators: Gamma, Poisson, negative binomial and squared Bessel subordinators. The Laplace exponents of the subordinators are Bernstein functions, so they can be used as sparsity-inducing nonconvex penalty functions. We exploit these subordinators in regression problems, yielding a hierarchical model with multiple regularization parameters. We devise ECME (Expectation/Conditional Maximization Either) algorithms to simultaneously estimate regression coefficients and regularization parameters. The empirical evaluation of simulated data shows that our approach is feasible and effective in high-dimensional data analysis.
Taking into account high-order interactions among covariates is valuable in many practical regression problems. This is, however, computationally challenging task because the number of high-order interaction features to be considered would be extremely large unless the number of covariates is sufficiently small. In this paper, we propose a novel efficient algorithm for LASSO-based sparse learning of such high-order interaction models. Our basic strategy for reducing the number of features is to employ the idea of recently proposed safe feature screening (SFS) rule. An SFS rule has a property that, if a feature satisfies the rule, then the feature is guaranteed to be non-active in the LASSO solution, meaning that it can be safely screened-out prior to the LASSO training process. If a large number of features can be screened-out before training the LASSO, the computational cost and the memory requirment can be dramatically reduced. However, applying such an SFS rule to each of the extremely large number of high-order interaction features would be computationally infeasible. Our key idea for solving this computational issue is to exploit the underlying tree structure among high-order interaction features. Specifically, we introduce a pruning condition called safe feature pruning (SFP) rule which has a property that, if the rule is satisfied in a certain node of the tree, then all the high-order interaction features corresponding to its descendant nodes can be guaranteed to be non-active at the optimal solution. Our algorithm is extremely efficient, making it possible to work, e.g., with 3rd order interactions of 10,000 original covariates, where the number of possible high-order interaction features is greater than 10^{12}.
In the last twenty-five years (1990-2014), algorithmic advances in integer optimization combined with hardware improvements have resulted in an astonishing 200 billion factor speedup in solving Mixed Integer Optimization (MIO) problems. We present a MIO approach for solving the classical best subset selection problem of choosing $k$ out of $p$ features in linear regression given $n$ observations. We develop a discrete extension of modern first order continuous optimization methods to find high quality feasible solutions that we use as warm starts to a MIO solver that finds provably optimal solutions. The resulting algorithm (a) provides a solution with a guarantee on its suboptimality even if we terminate the algorithm early, (b) can accommodate side constraints on the coefficients of the linear regression and (c) extends to finding best subset solutions for the least absolute deviation loss function. Using a wide variety of synthetic and real datasets, we demonstrate that our approach solves problems with $n$ in the 1000s and $p$ in the 100s in minutes to provable optimality, and finds near optimal solutions for $n$ in the 100s and $p$ in the 1000s in minutes. We also establish via numerical experiments that the MIO approach performs better than {\texttt {Lasso}} and other popularly used sparse learning procedures, in terms of achieving sparse solutions with good predictive power.
The proximal problem for structured penalties obtained via convex relaxations of submodular functions is known to be equivalent to minimizing separable convex functions over the corresponding submodular polyhedra. In this paper, we reveal a comprehensive class of structured penalties for which penalties this problem can be solved via an efficiently solvable class of parametric maxflow optimization. We then show that the parametric maxflow algorithm proposed by Gallo et al. and its variants, which runs, in the worst-case, at the cost of only a constant factor of a single computation of the corresponding maxflow optimization, can be adapted to solve the proximal problems for those penalties. Several existing structured penalties satisfy these conditions; thus, regularized learning with these penalties is solvable quickly using the parametric maxflow algorithm. We also investigate the empirical runtime performance of the proposed framework.
Traditional machine learning methods usually minimize a simple loss function to learn a predictive model, and then use a complex performance measure to measure the prediction performance. However, minimizing a simple loss function cannot guarantee that an optimal performance. In this paper, we study the problem of optimizing the complex performance measure directly to obtain a predictive model. We proposed to construct a maximum likelihood model for this problem, and to learn the model parameter, we minimize a com- plex loss function corresponding to the desired complex performance measure. To optimize the loss function, we approximate the upper bound of the complex loss. We also propose impose the sparsity to the model parameter to obtain a sparse model. An objective is constructed by combining the upper bound of the loss function and the sparsity of the model parameter, and we develop an iterative algorithm to minimize it by using the fast iterative shrinkage- thresholding algorithm framework. The experiments on optimization on three different complex performance measures, including F-score, receiver operating characteristic curve, and recall precision curve break even point, over three real-world applications, aircraft event recognition of civil aviation safety, in- trusion detection in wireless mesh networks, and image classification, show the advantages of the proposed method over state-of-the-art methods.
In this note we compare two recently proposed semidefinite relaxations for the sparse linear regression problem by Pilanci, Wainwright and El Ghaoui (Sparse learning via boolean relaxations, 2015) and Dong, Chen and Linderoth (Relaxation vs. Regularization A conic optimization perspective of statistical variable selection, 2015). We focus on the cardinality constrained formulation, and prove that the relaxation proposed by Dong, etc. is theoretically no weaker than the one proposed by Pilanci, etc. Therefore any sufficient condition of exact recovery derived by Pilanci can be readily applied to the other relaxation, including their results on high probability recovery for Gaussian ensemble. Finally we provide empirical evidence that the relaxation by Dong, etc. requires much fewer observations to guarantee the recovery of true support.
We propose a novel, efficient approach for distributed sparse learning in high-dimensions, where observations are randomly partitioned across machines. Computationally, at each round our method only requires the master machine to solve a shifted ell_1 regularized M-estimation problem, and other workers to compute the gradient. In respect of communication, the proposed approach provably matches the estimation error bound of centralized methods within constant rounds of communications (ignoring logarithmic factors). We conduct extensive experiments on both simulated and real world datasets, and demonstrate encouraging performances on high-dimensional regression and classification tasks.
Identification of regions of interest (ROI) associated with certain disease has a great impact on public health. Imposing sparsity of pixel values and extracting active regions simultaneously greatly complicate the image analysis. We address these challenges by introducing a novel region-selection penalty in the framework of image-on-scalar regression. Our penalty combines the Smoothly Clipped Absolute Deviation (SCAD) regularization, enforcing sparsity, and the SCAD of total variation (TV) regularization, enforcing spatial contiguity, into one group, which segments contiguous spatial regions against zero-valued background. Efficient algorithm is based on the alternative direction method of multipliers (ADMM) which decomposes the non-convex problem into two iterative optimization problems with explicit solutions. Another virtue of the proposed method is that a divide and conquer learning algorithm is developed, thereby allowing scaling to large images. Several examples are presented and the experimental results are compared with other state-of-the-art approaches.
This paper considers the problem of high dimensional signal detection in a large distributed network whose nodes can collaborate with their one-hop neighboring nodes (spatial collaboration). We assume that only a small subset of nodes communicate with the Fusion Center (FC). We design optimal collaboration strategies which are universal for a class of deterministic signals. By establishing the equivalence between the collaboration strategy design problem and sparse PCA, we solve the problem efficiently and evaluate the impact of collaboration on detection performance.
In the era of big data, it is becoming common to have data with multiple modalities or coming from multiple sources, known as "multi-view data". Multi-view data are usually unlabeled and come from high-dimensional spaces (such as language vocabularies), unsupervised multi-view feature selection is crucial to many applications. However, it is nontrivial due to the following challenges. First, there are too many instances or the feature dimensionality is too large. Thus, the data may not fit in memory. How to select useful features with limited memory space? Second, how to select features from streaming data and handles the concept drift? Third, how to leverage the consistent and complementary information from different views to improve the feature selection in the situation when the data are too big or come in as streams? To the best of our knowledge, none of the previous works can solve all the challenges simultaneously. In this paper, we propose an Online unsupervised Multi-View Feature Selection, OMVFS, which deals with large-scale/streaming multi-view data in an online fashion. OMVFS embeds unsupervised feature selection into a clustering algorithm via NMF with sparse learning. It further incorporates the graph regularization to preserve the local structure information and help select discriminative features. Instead of storing all the historical data, OMVFS processes the multi-view data chunk by chunk and aggregates all the necessary information into several small matrices. By using the buffering technique, the proposed OMVFS can reduce the computational and storage cost while taking advantage of the structure information. Furthermore, OMVFS can capture the concept drifts in the data streams. Extensive experiments on four real-world datasets show the effectiveness and efficiency of the proposed OMVFS method. More importantly, OMVFS is about 100 times faster than the off-line methods.
Feature selection, as a data preprocessing strategy, has been proven to be effective and efficient in preparing data (especially high-dimensional data) for various data mining and machine learning problems. The objectives of feature selection include: building simpler and more comprehensible models, improving data mining performance, and preparing clean, understandable data. The recent proliferation of big data has presented some substantial challenges and opportunities to feature selection. In this survey, we provide a comprehensive and structured overview of recent advances in feature selection research. Motivated by current challenges and opportunities in the era of big data, we revisit feature selection research from a data perspective and review representative feature selection algorithms for conventional data, structured data, heterogeneous data and streaming data. Methodologically, to emphasize the differences and similarities of most existing feature selection algorithms for conventional data, we categorize them into four main groups: similarity based, information theoretical based, sparse learning based and statistical based methods. To facilitate and promote the research in this community, we also present an open-source feature selection repository that consists of most of the popular feature selection algorithms (\url{http://featureselection.asu.edu/}). Also, we use it as an example to show how to evaluate feature selection algorithms. At the end of the survey, we present a discussion about some open problems and challenges that require more attention in future research.
In this paper, we develop a randomized algorithm and theory for learning a sparse model from large-scale and high-dimensional data, which is usually formulated as an empirical risk minimization problem with a sparsity-inducing regularizer. Under the assumption that there exists a (approximately) sparse solution with high classification accuracy, we argue that the dual solution is also sparse or approximately sparse. The fact that both primal and dual solutions are sparse motivates us to develop a randomized approach for a general convex-concave optimization problem. Specifically, the proposed approach combines the strength of random projection with that of sparse learning: it utilizes random projection to reduce the dimensionality, and introduces $\ell_1$-norm regularization to alleviate the approximation error caused by random projection. Theoretical analysis shows that under favored conditions, the randomized algorithm can accurately recover the optimal solutions to the convex-concave optimization problem (i.e., recover both the primal and dual solutions).
The pathwise coordinate optimization is one of the most important computational frameworks for high dimensional convex and nonconvex sparse learning problems. It differs from the classical coordinate optimization algorithms in three salient features: {\it warm start initialization}, {\it active set updating}, and {\it strong rule for coordinate preselection}. Such a complex algorithmic structure grants superior empirical performance, but also poses significant challenge to theoretical analysis. To tackle this long lasting problem, we develop a new theory showing that these three features play pivotal roles in guaranteeing the outstanding statistical and computational performance of the pathwise coordinate optimization framework. Particularly, we analyze the existing pathwise coordinate optimization algorithms and provide new theoretical insights into them. The obtained insights further motivate the development of several modifications to improve the pathwise coordinate optimization framework, which guarantees linear convergence to a unique sparse local optimum with optimal statistical properties in parameter estimation and support recovery. This is the first result on the computational and statistical guarantees of the pathwise coordinate optimization framework in high dimensions. Thorough numerical experiments are provided to support our theory.
Decentralized optimization algorithms have received much attention due to the recent advances in network information processing. However, conventional decentralized algorithms based on projected gradient descent are incapable of handling high dimensional constrained problems, as the projection step becomes computationally prohibitive to compute. To address this problem, this paper adopts a projection-free optimization approach, a.k.a.~the Frank-Wolfe (FW) or conditional gradient algorithm. We first develop a decentralized FW (DeFW) algorithm from the classical FW algorithm. The convergence of the proposed algorithm is studied by viewing the decentralized algorithm as an inexact FW algorithm. Using a diminishing step size rule and letting $t$ be the iteration number, we show that the DeFW algorithm's convergence rate is ${\cal O}(1/t)$ for convex objectives; is ${\cal O}(1/t^2)$ for strongly convex objectives with the optimal solution in the interior of the constraint set; and is ${\cal O}(1/\sqrt{t})$ towards a stationary point for smooth but non-convex objectives. We then show that a consensus-based DeFW algorithm meets the above guarantees with two communication rounds per iteration. Furthermore, we demonstrate the advantages of the proposed DeFW algorithm on low-complexity robust matrix completion and communication efficient sparse learning. Numerical results on synthetic and real data are presented to support our findings.
Convex sparsity-promoting regularizations are ubiquitous in modern statistical learning. By construction, they yield solutions with few non-zero coefficients, which correspond to saturated constraints in the dual optimization formulation. Working set (WS) strategies are generic optimization techniques that consist in solving simpler problems that only consider a subset of constraints, whose indices form the WS. Working set methods therefore involve two nested iterations: the outer loop corresponds to the definition of the WS and the inner loop calls a solver for the subproblems. For the Lasso estimator a WS is a set of features, while for a Group Lasso it refers to a set of groups. In practice, WS are generally small in this context so the associated feature Gram matrix can fit in memory. Here we show that the Gauss-Southwell rule (a greedy strategy for block coordinate descent techniques) leads to fast solvers in this case. Combined with a working set strategy based on an aggressive use of so-called Gap Safe screening rules, we propose a solver achieving state-of-the-art performance on sparse learning problems. Results are presented on Lasso and multi-task Lasso estimators.
We explore a recently proposed Variational Dropout technique that provided an elegant Bayesian interpretation to Gaussian Dropout. We extend Variational Dropout to the case when dropout rates are unbounded, propose a way to reduce the variance of the gradient estimator and report first experimental results with individual dropout rates per weight. Interestingly, it leads to extremely sparse solutions both in fully-connected and convolutional layers. This effect is similar to automatic relevance determination effect in empirical Bayes but has a number of advantages. We reduce the number of parameters up to 280 times on LeNet architectures and up to 68 times on VGG-like networks with a negligible decrease of accuracy.
The cardinality constraint is an intrinsic way to restrict the solution structure in many domains, for example, sparse learning, feature selection, and compressed sensing. To solve a cardinality constrained problem, the key challenge is to solve the projection onto the cardinality constraint set, which is NP-hard in general when there exist multiple overlapped cardinality constraints. In this paper, we consider the scenario where the overlapped cardinality constraints satisfy a Three-view Cardinality Structure (TVCS), which reflects the natural restriction in many applications, such as identification of gene regulatory networks and task-worker assignment problem. We cast the projection into a linear programming, and show that for TVCS, the vertex solution of this linear programming is the solution for the original projection problem. We further prove that such solution can be found with the complexity proportional to the number of variables and constraints. We finally use synthetic experiments and two interesting applications in bioinformatics and crowdsourcing to validate the proposed TVCS model and method.
Convex regularizers are often used for sparse learning. They are easy to optimize, but can lead to inferior prediction performance. The difference of $\ell_1$ and $\ell_2$ ($\ell_{1-2}$) regularizer has been recently proposed as a nonconvex regularizer. It yields better recovery than both $\ell_0$ and $\ell_1$ regularizers on compressed sensing. However, how to efficiently optimize its learning problem is still challenging. The main difficulty is that both the $\ell_1$ and $\ell_2$ norms in $\ell_{1-2}$ are not differentiable, and existing optimization algorithms cannot be applied. In this paper, we show that a closed-form solution can be derived for the proximal step associated with this regularizer. We further extend the result for low-rank matrix learning and the total variation model. Experiments on both synthetic and real data sets show that the resultant accelerated proximal gradient algorithm is more efficient than other noncovex optimization algorithms.
The redundant features existing in high dimensional datasets always affect the performance of learning and mining algorithms. How to detect and remove them is an important research topic in machine learning and data mining research. In this paper, we propose a graph based approach to find and remove those redundant features automatically for high dimensional data. Based on the sparse learning based unsupervised feature selection framework, Sparse Feature Graph (SFG) is introduced not only to model the redundancy between two features, but also to disclose the group redundancy between two groups of features. With SFG, we can divide the whole features into different groups, and improve the intrinsic structure of data by removing detected redundant features. With accurate data structure, quality indicator vectors can be obtained to improve the learning performance of existing unsupervised feature selection algorithms such as multi-cluster feature selection (MCFS). Our experimental results on benchmark datasets show that the proposed SFG and feature redundancy remove algorithm can improve the performance of unsupervised feature selection algorithms consistently.