In high-dimensional statistical inference, sparsity regularizations have shown advantages in consistency and convergence rates for coefficient estimation. We consider a generalized version of Sparse-Group Lasso which captures both element-wise sparsity and group-wise sparsity simultaneously. We state one universal theorem which is proved to obtain results on consistency and convergence rates for different forms of double sparsity regularization. The universality of the results lies in an generalization of various convergence rates for single regularization cases such as LASSO and group LASSO and also double regularization cases such as sparse-group LASSO. Our analysis identifies a generalized norm of $ε$-norm, which provides a dual formulation for our double sparsity regularization.
In this paper, we introduce structured sparsity estimators in Generalized Linear Models. Structured sparsity estimators in the least squares loss are introduced by Stucky and van de Geer (2018) recently for fixed design and normal errors. We extend their results to debiased structured sparsity estimators with Generalized Linear Model based loss. Structured sparsity estimation means penalized loss functions with a possible sparsity structure used in the chosen norm. These include weighted group lasso, lasso and norms generated from convex cones. The significant difficulty is that it is not clea…
Sparsity-constrained optimization is an important and challenging problem that has wide applicability in data mining, machine learning, and statistics. In this paper, we focus on sparsity-constrained optimization in cases where the cost function is a general nonlinear function and, in particular, the sparsity constraint is defined by a graph-structured sparsity model. Existing methods explore this problem in the context of sparse estimation in linear models. To the best of our knowledge, this is the first work to present an efficient approximation algorithm, namely, Graph-structured Matching P…
We consider high-dimensional multiclass classification by sparse multinomial logistic regression. Unlike binary classification, in the multiclass setup one can think about an entire spectrum of possible notions of sparsity associated with different structural assumptions on the regression coefficients matrix. We propose a computationally feasible feature selection procedure based on penalized maximum likelihood with convex penalties capturing a specific type of sparsity at hand. In particular, we consider global sparsity, double row-wise sparsity, and low-rank sparsity, and show that with the …
We study asymptotically normal estimation and confidence regions for low-dimensional parameters in high-dimensional sparse models. Our approach is based on the $\ell_1$-penalized M-estimator which is used for construction of a bias corrected estimator. We show that the proposed estimator is asymptotically normal, under a sparsity assumption on the high-dimensional parameter, smoothness conditions on the expected loss and an entropy condition. This leads to uniformly valid confidence regions and hypothesis testing for low-dimensional parameters. The present approach is different in that it allo…
Boosting as gradient descent algorithms is one popular method in machine learning. In this paper a novel Boosting-type algorithm is proposed based on restricted gradient descent with structural sparsity control whose underlying dynamics are governed by differential inclusions. In particular, we present an iterative regularization path with structural sparsity where the parameter is sparse under some linear transforms, based on variable splitting and the Linearized Bregman Iteration. Hence it is called \emph{Split LBI}. Despite its simplicity, Split LBI outperforms the popular generalized Lasso…
We consider covariance estimation in the multivariate generalized Gaussian distribution (MGGD) and elliptically symmetric (ES) distribution. The maximum likelihood optimization associated with this problem is non-convex, yet it has been proved that its global solution can be often computed via simple fixed point iterations. Our first contribution is a new analysis of this likelihood based on geodesic convexity that requires weaker assumptions. Our second contribution is a generalized framework for structured covariance estimation under sparsity constraints. We show that the optimizations can b…
We study a generalized framework for structured sparsity. It extends the well-known methods of Lasso and Group Lasso by incorporating additional constraints on the variables as part of a convex optimization problem. This framework provides a straightforward way of favouring prescribed sparsity patterns, such as orderings, contiguous regions and overlapping groups, among others. Existing optimization methods are limited to specific constraint sets and tend to not scale well with sample size and dimensionality. We propose a novel first order proximal method, which builds upon results on fixed po…
Quantifying the uncertainty in penalized regression under group sparsity is an important open question. We establish, under a high-dimensional scaling, the asymptotic validity of a modified parametric bootstrap method for the group lasso, assuming a Gaussian error model and mild conditions on the design matrix and the true coefficients. Simulation of bootstrap samples provides simultaneous inferences on large groups of coefficients. Through extensive numerical comparisons, we demonstrate that our bootstrap method performs much better than popular competitors, highlighting its practical utility…
INRIA - Département d’Informatique de l’Ecole Normale Supérieure PSL Research University（法国国家信息与自动化研究所 - 巴黎高等师范学院信息学系 巴黎文理研究大学）
The conditional gradient method (CGM) has been widely used for fast sparse approximation, having a low per iteration computational cost for structured sparse regularizers. We explore the sparsity acquiring properties of a generalized CGM (gCGM), where the constraint is replaced by a penalty function based on a gauge penalty; this can be done without significantly increasing the per-iteration computation, and applies to general notions of sparsity. Without assuming bounded iterates, we show $O(1/t)$ convergence of the function values and gap of gCGM. We couple this with a safe screening rule, a…
Georgia Institute of Technology（佐治亚理工学院）；Mitsubishi Electric Research Labs（三菱电机研究实验室）；Carnegie Mellon University（卡内基梅隆大学）
Several convex formulation methods have been proposed previously for statistical estimation with structured sparsity as the prior. These methods often require a carefully tuned regularization parameter, often a cumbersome or heuristic exercise. Furthermore, the estimate that these methods produce might not belong to the desired sparsity model, albeit accurately approximating the true parameter. Therefore, greedy-type algorithms could often be more desirable in estimating structured-sparse parameters. So far, these greedy methods have mostly focused on linear statistical models. In this paper w…
Texas A&M University（德克萨斯农工大学）；New York University（纽约大学）
We study the implicit regularization of gradient descent towards structured sparsity via a novel neural reparameterization, which we call a diagonally grouped linear neural network. We show the following intriguing property of our reparameterization: gradient descent over the squared regression loss, without any explicit regularization, biases towards solutions with a group sparsity structure. In contrast to many existing works in understanding implicit regularization, we prove that our training trajectory cannot be simulated by mirror descent. We analyze the gradient dynamics of the correspon…
Mila（米拉人工智能研究所）；DIRO, Université de Montréal（蒙特利尔大学计算机科学与运筹学系）
The performance of trained neural networks is robust to harsh levels of pruning. Coupled with the ever-growing size of deep learning models, this observation has motivated extensive research on learning sparse models. In this work, we focus on the task of controlling the level of sparsity when performing sparse learning. Existing methods based on sparsity-inducing penalties involve expensive trial-and-error tuning of the penalty factor, thus lacking direct control of the resulting model sparsity. In response, we adopt a constrained formulation: using the gate mechanism proposed by Louizos et a…
De-biased lasso has emerged as a popular tool to draw statistical inference for high-dimensional regression models. However, simulations indicate that for generalized linear models (GLMs), de-biased lasso inadequately removes biases and yields unreliable confidence intervals. This motivates us to scrutinize the application of de-biased lasso in high-dimensional GLMs. When $p >n$, we detect that a key sparsity condition on the inverse information matrix generally does not hold in a GLM setting, which likely explains the subpar performance of de-biased lasso. Even in a less challenging "large $n…
Leveraging Sparsity for Sample-Efficient Preference Learning: A Theoretical Perspective
LINX, EPFL, Lausanne, Switzerland（日内瓦EPFL实验室）；School of Computing and Artificial Intelligence, Shanghai University of Finance and Economics, Shanghai, China（上海金融学院计算机与人工智能学院）
Vector autoregressive models characterize a variety of time series in which linear combinations of current and past observations can be used to accurately predict future observations. For instance, each element of an observation vector could correspond to a different node in a network, and the parameters of an autoregressive model would correspond to the impact of the network structure on the time series evolution. Often these models are used successfully in practice to learn the structure of social, epidemiological, financial, or biological neural networks. However, little is known about stat…
Consider the problem of simultaneous testing for the means of independent normal observations. In this paper, we study some asymptotic optimality properties of certain multiple testing rules induced by a general class of one-group shrinkage priors in a Bayesian decision theoretic framework, where the overall loss is taken as the number of misclassified hypotheses. We assume a two-groups normal mixture model for the data and consider the asymptotic framework adopted in Bogdan et al. (2011) who introduced the notion of asymptotic Bayes optimality under sparsity in the context of multiple testing…
Graduate School of Information Sciences and Technology（信息科学技术研究生院）；The University of Tokyo（东京大学）
In dictionary selection, several atoms are selected from finite candidates that successfully approximate given data points in the sparse representation. We propose a novel efficient greedy algorithm for dictionary selection. Not only does our algorithm work much faster than the known methods, but it can also handle more complex sparsity constraints, such as average sparsity. Using numerical experiments, we show that our algorithm outperforms the known methods for dictionary selection, achieving competitive performances with dictionary learning algorithms in a smaller running time.
We study minimax rates for high-dimensional linear regression with additive errors under the $\ell_p\ (1\leq p<\infty)$-losses, where the regression parameter is of weak sparsity. Our lower and upper bounds agree up to constant factors, implying that the proposed estimator is minimax optimal.
The restricted isometry property (RIP) is an integral tool in the analysis of various inverse problems with sparsity models. Motivated by the applications of compressed sensing and dimensionality reduction of low-rank tensors, we propose generalized notions of sparsity and provide a unified framework for the corresponding RIP, in particular when combined with isotropic group actions. Our results extend an approach by Rudelson and Vershynin to a much broader context including commutative and noncommutative function spaces. Moreover, our Banach space notion of sparsity applies to affine group ac…
The restricted isometry property (RIP) is a universal tool for data recovery. We explore the implication of the RIP in the framework of generalized sparsity and group measurements introduced in the Part I paper. It turns out that for a given measurement instrument the number of measurements for RIP can be improved by optimizing over families of Banach spaces. Second, we investigate the preservation of difference of two sparse vectors, which is not trivial in generalized models. Third, we extend the RIP of partial Fourier measurements at optimal scaling of number of measurements with random sig…
École Normale Supérieure, PSL Research University（巴黎高等师范学院，PSL研究大学）；Université Pierre et Marie Curie, Sorbonne Universités（皮埃尔和玛丽·居里大学，索邦大学）；INRIA Paris（法国国家信息与自动化研究所巴黎中心）
In this work, we consider compressed sensing reconstruction from $M$ measurements of $K$-sparse structured signals which do not possess a writable correlation model. Assuming that a generative statistical model, such as a Boltzmann machine, can be trained in an unsupervised manner on example signals, we demonstrate how this signal model can be used within a Bayesian framework of signal reconstruction. By deriving a message-passing inference for general distribution restricted Boltzmann machines, we are able to integrate these inferred signal models into approximate message passing for compress…
We show that a new design criterion, i.e., the least squares on subband errors regularized by a weighted norm, can be used to generalize the proportionate-type normalized subband adaptive filtering (PtNSAF) framework. The new criterion directly penalizes subband errors and includes a sparsity penalty term which is minimized using the damped regularized Newton's method. The impact of the proposed generalized PtNSAF (GPtNSAF) is studied for the system identification problem via computer simulations. Specifically, we study the effects of using different numbers of subbands and various sparsity pe…
Samsung Research Institute（三星研究院）；Indian Institute of Technology Madras（印度理工学院马德拉斯分校）
Simultaneous orthogonal matching pursuit (SOMP) and block OMP (BOMP) are two widely used techniques for sparse support recovery in multiple measurement vector (MMV) and block sparse (BS) models respectively. For optimal performance, both SOMP and BOMP require \textit{a priori} knowledge of signal sparsity or noise variance. However, sparsity and noise variance are unavailable in most practical applications. This letter presents a novel technique called generalized residual ratio thresholding (GRRT) for operating SOMP and BOMP without the \textit{a priori} knowledge of signal sparsity and noise…
In compressed sensing, in order to recover a sparse or nearly sparse vector from possibly noisy measurements, the most popular approach is $\ell_1$-norm minimization. Upper bounds for the $\ell_2$- norm of the error between the true and estimated vectors are given in [1] and reviewed in [2], while bounds for the $\ell_1$-norm are given in [3]. When the unknown vector is not conventionally sparse but is "group sparse" instead, a variety of alternatives to the $\ell_1$-norm have been proposed in the literature, including the group LASSO, sparse group LASSO, and group LASSO with tree structured o…
In this paper, we provide a Banach-space formulation of supervised learning with generalized total-variation (gTV) regularization. We identify the class of kernel functions that are admissible in this framework. Then, we propose a variation of supervised learning in a continuous-domain hybrid search space with gTV regularization. We show that the solution admits a multi-kernel expansion with adaptive positions. In this representation, the number of active kernels is upper-bounded by the number of data points while the gTV regularization imposes an $\ell_1$ penalty on the kernel coefficients. F…
Zhejiang University（浙江大学）；The University of Hong Kong（香港大学）；Tongji University（同济大学）；Shenzhen Research Institute of Big Data（深圳大数据研究院）；National and Kapodistrian University of Athens（雅典国立卡波迪斯特里安大学）；Aalborg University（奥尔堡大学）
Tensor rank learning for canonical polyadic decomposition (CPD) has long been deemed as an essential yet challenging problem. In particular, since the tensor rank controls the complexity of the CPD model, its inaccurate learning would cause overfitting to noise or underfitting to the signal sources, and even destroy the interpretability of model parameters. However, the optimal determination of a tensor rank is known to be a non-deterministic polynomial-time hard (NP-hard) task. Rather than exhaustively searching for the best tensor rank via trial-and-error experiments, Bayesian inference unde…
In this paper, the use of the Generalized Beta Mixture (GBM) and Horseshoe distributions as priors in the Bayesian Compressive Sensing framework is proposed. The distributions are considered in a two-layer hierarchical model, making the corresponding inference problem amenable to Expectation Maximization (EM). We present an explicit, algebraic EM-update rule for the models, yielding two fast and experimentally validated algorithms for signal recovery. Experimental results show that our algorithms outperform state-of-the-art methods on a wide range of sparsity levels and amplitudes in terms of …
We propose a novel algorithm for efficiently computing a sparse directed adjacency matrix from a group of time series following a causal graph process. Our solution is scalable for both dense and sparse graphs and automatically selects the LASSO coefficient to obtain an appropriate number of edges in the adjacency matrix. Current state-of-the-art approaches rely on sparse-matrix-computation libraries to scale, and either avoid automatic selection of the LASSO penalty coefficient or rely on the prediction mean squared error, which is not directly related to the correct number of edges. Instead,…
Stochastic optimization algorithms update models with cheap per-iteration costs sequentially, which makes them amenable for large-scale data analysis. Such algorithms have been widely studied for structured sparse models where the sparsity information is very specific, e.g., convex sparsity-inducing norms or $\ell^0$-norm. However, these norms cannot be directly applied to the problem of complex (non-convex) graph-structured sparsity models, which have important application in disease outbreak and social networks, etc. In this paper, we propose a stochastic gradient-based method for solving gr…
In high dimensional regression settings, sparsity enforcing penalties have proved useful to regularize the data-fitting term. A recently introduced technique called screening rules propose to ignore some variables in the optimization leveraging the expected sparsity of the solutions and consequently leading to faster solvers. When the procedure is guaranteed not to discard variables wrongly the rules are said to be safe. In this work, we propose a unifying framework for generalized linear models regularized with standard sparsity enforcing penalties such as $\ell_1$ or $\ell_1/\ell_2$ norms. O…
In high-dimensional linear models, the sparsity assumption is typically made, stating that most of the parameters are equal to zero. Under the sparsity assumption, estimation and, recently, inference have been well studied. However, in practice, sparsity assumption is not checkable and more importantly is often violated; a large number of covariates might be expected to be associated with the response, indicating that possibly all, rather than just a few, parameters are non-zero. A natural example is a genome-wide gene expression profiling, where all genes are believed to affect a common disea…
Sparsity learning with known grouping structure has received considerable attention due to wide modern applications in high-dimensional data analysis. Although advantages of using group information have been well-studied by shrinkage-based approaches, benefits of group sparsity have not been well-documented for greedy-type methods, which much limits our understanding and use of this important class of methods. In this paper, generalizing from a popular forward-backward greedy approach, we propose a new interactive greedy algorithm for group sparsity learning and prove that the proposed greedy-…
We establish $ L^{\infty} $ and $ L^2 $ error bounds for functions of many variables that are approximated by linear combinations of ReLU (rectified linear unit) and squared ReLU ridge functions with $ \ell^1 $ and $ \ell^0 $ controls on their inner and outer parameters. With the squared ReLU ridge function, we show that the $ L^2 $ approximation error is inversely proportional to the inner layer $ \ell^0 $ sparsity and it need only be sublinear in the outer layer $ \ell^0 $ sparsity. Our constructions are obtained using a variant of the Jones-Barron probabilistic method, which can be interpre…
Nonconvex penalty methods for sparse modeling in linear regression have been a topic of fervent interest in recent years. Herein, we study a family of nonconvex penalty functions that we call the trimmed Lasso and that offers exact control over the desired level of sparsity of estimators. We analyze its structural properties and in doing so show the following: 1) Drawing parallels between robust statistics and robust optimization, we show that the trimmed-Lasso-regularized least squares problem can be viewed as a generalized form of total least squares under a specific model of uncertainty. In…
