Safe Screening Rules for $\ell_0$-Regression

Screening Data Points in Empirical Risk Minimization via Ellipsoidal Regions and Safe Loss Functions

We design simple screening tests to automatically discard data samples in empirical risk minimization without losing optimization guarantees. We derive loss functions that produce dual objectives with a sparse solution. We also show how to regularize convex losses to ensure such a dual sparsity-inducing property, and propose a general method to design screening tests for classification or regression based on ellipsoidal approximations of the optimal set. In addition to producing computational gains, our approach also allows us to compress a dataset into a subset of representative points.

Safe Triplet Screening for Distance Metric Learning

We study safe screening for metric learning. Distance metric learning can optimize a metric over a set of triplets, each one of which is defined by a pair of same class instances and an instance in a different class. However, the number of possible triplets is quite huge even for a small dataset. Our safe triplet screening identifies triplets which can be safely removed from the optimization problem without losing the optimality. Compared with existing safe screening studies, triplet screening is particularly significant because of (1) the huge number of possible triplets, and (2) the semi-def…

Simultaneous Safe Screening of Features and Samples in Doubly Sparse Modeling

The problem of learning a sparse model is conceptually interpreted as the process of identifying active features/samples and then optimizing the model over them. Recently introduced safe screening allows us to identify a part of non-active features/samples. So far, safe screening has been individually studied either for feature screening or for sample screening. In this paper, we introduce a new approach for safely screening features and samples simultaneously by alternatively iterating feature and sample screening steps. A significant advantage of considering them simultaneously rather than i…

Safe Screening for Logistic Regression with $\ell_0$-$\ell_2$ Regularization

In logistic regression, it is often desirable to utilize regularization to promote sparse solutions, particularly for problems with a large number of features compared to available labels. In this paper, we present screening rules that safely remove features from logistic regression with $\ell_0-\ell_2$ regularization before solving the problem. The proposed safe screening rules are based on lower bounds from the Fenchel dual of strong conic relaxations of the logistic regression problem. Numerical experiments with real and synthetic data suggest that a high percentage of the features can be e…

Group OWL 模型的安全筛选规则

Safe Screening Rules for Group OWL Models

本文首次针对 Group OWL 正则化模型提出安全筛选规则，通过处理结构化不可分离惩罚项快速识别并移除所有任务中系数为零的非活跃特征，从而在批处理和随机优化中实现显著计算加速和内存节省，且不损失精度。

Safe Active Feature Selection for Sparse Learning

We present safe active incremental feature selection~(SAIF) to scale up the computation of LASSO solutions. SAIF does not require a solution from a heavier penalty parameter as in sequential screening or updating the full model for each iteration as in dynamic screening. Different from these existing screening methods, SAIF starts from a small number of features and incrementally recruits active features and updates the significantly reduced model. Hence, it is much more computationally efficient and scalable with the number of features. More critically, SAIF has the safe guarantee as it has t…

Distributed Dynamic Safe Screening Algorithms for Sparse Regularization

Distributed optimization has been widely used as one of the most efficient approaches for model training with massive samples. However, large-scale learning problems with both massive samples and high-dimensional features widely exist in the era of big data. Safe screening is a popular technique to speed up high-dimensional models by discarding the inactive features with zero coefficients. Nevertheless, existing safe screening methods are limited to the sequential setting. In this paper, we propose a new distributed dynamic safe screening (DDSS) method for sparsity regularized models and apply…

Safe Element Screening for Submodular Function Minimization

Submodular functions are discrete analogs of convex functions, which have applications in various fields, including machine learning and computer vision. However, in large-scale applications, solving Submodular Function Minimization (SFM) problems remains challenging. In this paper, we make the first attempt to extend the emerging technique named screening in large-scale sparse learning to SFM for accelerating its optimization process. We first conduct a careful studying of the relationships between SFM and the corresponding convex proximal problems, as well as the accurate primal optimum esti…

Distributionally Robust Safe Screening（分布鲁棒安全筛选）

Distributionally Robust Safe Screening

本文提出 DRSS 方法，将分布鲁棒学习与安全筛选结合，通过将 DR 协变量偏移转化为权重具有不确定性的加权经验风险最小化问题，在训练前可靠识别冗余样本与特征，并提供理论保证与实验验证。

Group SLOPE 的安全筛选规则

Safe Screening Rules for Group SLOPE

针对 Group SLOPE 因块不可分离分组效应导致的高计算与内存成本，提出可识别零系数非活跃组的安全筛选规则，兼容批处理与随机求解器并保持结果和精度不变。

Stable safe screening and structured dictionaries for faster L1 regularization

In this paper, we propose a way to combine two acceleration techniques for the $\ell\_{1}$-regularized least squares problem: safe screening tests, which allow to eliminate useless dictionary atoms; and the use of fast structured approximations of the dictionary matrix. To do so, we introduce a new family of screening tests, termed stable screening, which can cope with approximation errors on the dictionary atoms while keeping the safety of the test (i.e. zero risk of rejecting atoms belonging to the solution support). Some of the main existing screening tests are extended to this new framewor…

A Safe Screening Rule for Sparse Logistic Regression

The l1-regularized logistic regression (or sparse logistic regression) is a widely used method for simultaneous classification and feature selection. Although many recent efforts have been devoted to its efficient implementation, its application to high dimensional data still poses significant challenges. In this paper, we present a fast and effective sparse logistic regression screening rule (Slores) to identify the 0 components in the solution vector, which may lead to a substantial reduction in the number of features to be entered to the optimization. An appealing feature of Slores is that …

Beyond GAP screening for Lasso by exploiting new dual cutting half-spaces with supplementary material

In this paper, we propose a novel safe screening test for Lasso. Our procedure is based on a safe region with a dome geometry and exploits a canonical representation of the set of half-spaces (referred to as "dual cutting half-spaces" in this paper) containing the dual feasible set. The proposed safe region is shown to be always included in the state-of-the-art "GAP Sphere" and "GAP Dome" proposed by Fercoq et al. (and strictly so under very mild conditions) while involving the same computational burden. Numerical experiments confirm that our new dome enables to devise more powerful screening …

Safe Screening for Sparse Conditional Random Fields

Sparse Conditional Random Field (CRF) is a powerful technique in computer vision and natural language processing for structured prediction. However, solving sparse CRFs in large-scale applications remains challenging. In this paper, we propose a novel safe dynamic screening method that exploits an accurate dual optimum estimation to identify and remove the irrelevant features during the training process. Thus, the problem size can be reduced continuously, leading to great savings in the computational cost without sacrificing any accuracy on the finally learned model. To the best of our knowled…

Fast OSCAR and OWL Regression via Safe Screening Rules

Ordered Weighted $L_{1}$ (OWL) regularized regression is a new regression analysis for high-dimensional sparse learning. Proximal gradient methods are used as standard approaches to solve OWL regression. However, it is still a burning issue to solve OWL regression due to considerable computational cost and memory usage when the feature or sample size is large. In this paper, we propose the first safe screening rule for OWL regression by exploring the order of the primal solution with the unknown order structure via an iterative strategy, which overcomes the difficulties of tackling the non-sep…

GAP Safe Screening Rules for Sparse-Group-Lasso

In high dimensional settings, sparse structures are crucial for efficiency, either in term of memory, computation or performance. In some contexts, it is natural to handle more refined structures than pure sparsity, such as for instance group sparsity. Sparse-Group Lasso has recently been introduced in the context of linear regression to enforce sparsity both at the feature level and at the group level. We adapt to the case of Sparse-Group Lasso recent safe screening rules that discard early in the solver irrelevant features/groups. Such rules have led to important speed-ups for a wide range o…

GAP Safe screening rules for sparse multi-task and multi-class models

High dimensional regression benefits from sparsity promoting regularizations. Screening rules leverage the known sparsity of the solution by ignoring some variables in the optimization, hence speeding up solvers. When the procedure is proven not to discard features wrongly the rules are said to be \emph{safe}. In this paper we derive new safe rules for generalized linear models regularized with $\ell_1$ and $\ell_1/\ell_2$ norms. The rules are based on duality gap computations and spherical safe regions whose diameters converge to zero. This allows to discard safely more variables, in particul…

Gap Safe screening rules for sparsity enforcing penalties

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…

Safe Sample Screening for Support Vector Machines

Sparse classifiers such as the support vector machines (SVM) are efficient in test-phases because the classifier is characterized only by a subset of the samples called support vectors (SVs), and the rest of the samples (non SVs) have no influence on the classification result. However, the advantage of the sparsity has not been fully exploited in training phases because it is generally difficult to know which sample turns out to be SV beforehand. In this paper, we introduce a new approach called safe sample screening that enables us to identify a subset of the non-SVs and screen them out prior…

Safe Screening for the Generalized Conditional Gradient Method

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…

Efficient Mixed-Norm Regularization: Algorithms and Safe Screening Methods

Sparse learning has recently received increasing attention in many areas including machine learning, statistics, and applied mathematics. The mixed-norm regularization based on the l1q norm with q>1 is attractive in many applications of regression and classification in that it facilitates group sparsity in the model. The resulting optimization problem is, however, challenging to solve due to the inherent structure of the mixed-norm regularization. Existing work deals with special cases with q=1, 2, infinity, and they cannot be easily extended to the general case. In this paper, we propose an e…

Joint Screening Tests for LASSO

This paper focusses on "safe" screening techniques for the LASSO problem. Motivated by the need for low-complexity algorithms, we propose a new approach, dubbed "joint" screening test, allowing to screen a set of atoms by carrying out one single test. The approach is particularized to two different sets of atoms, respectively expressed as sphere and dome regions. After presenting the mathematical derivations of the tests, we elaborate on their relative effectiveness and discuss the practical use of such procedures.

From safe screening rules to working sets for faster Lasso-type solvers

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 …

Look-Ahead Screening Rules for the Lasso（套索的前瞻筛选规则）

Look-Ahead Screening Rules for the Lasso

针对高维场景下套索求解计算成本高的问题，本文提出前瞻筛选策略，利用安全筛选规则确定预测变量无法入模的惩罚值范围以沿解路径筛选变量，实验证明其性能优于Gap Safe规则的主动热启动版本。

Screening Rules for Convex Problems

We propose a new framework for deriving screening rules for convex optimization problems. Our approach covers a large class of constrained and penalized optimization formulations, and works in two steps. First, given any approximate point, the structure of the objective function and the duality gap is used to gather information on the optimal solution. In the second step, this information is used to produce screening rules, i.e. safely identifying unimportant weight variables of the optimal solution. Our general framework leads to a large variety of useful existing as well as new screening rul…

Mind the duality gap: safer rules for the Lasso

Screening rules allow to early discard irrelevant variables from the optimization in Lasso problems, or its derivatives, making solvers faster. In this paper, we propose new versions of the so-called $\textit{safe rules}$ for the Lasso. Based on duality gap considerations, our new rules create safe test regions whose diameters converge to zero, provided that one relies on a converging solver. This property helps screening out more variables, for a wider range of regularization parameter values. In addition to faster convergence, we prove that we correctly identify the active sets (supports) of…

Lasso Screening Rules via Dual Polytope Projection

Lasso is a widely used regression technique to find sparse representations. When the dimension of the feature space and the number of samples are extremely large, solving the Lasso problem remains challenging. To improve the efficiency of solving large-scale Lasso problems, El Ghaoui and his colleagues have proposed the SAFE rules which are able to quickly identify the inactive predictors, i.e., predictors that have $0$ components in the solution vector. Then, the inactive predictors or features can be removed from the optimization problem to reduce its scale. By transforming the standard Lass…

Hessian Screening Rule（黑塞筛选规则）

The Hessian Screening Rule

针对lasso等稀疏回归的求解提速问题，提出利用模型二阶信息的Hessian Screening Rule，兼具高效筛选与准确热启动优势，在模拟与真实数据集上表现均优于现有替代方案。

Safe Pattern Pruning: An Efficient Approach for Predictive Pattern Mining

In this paper we study predictive pattern mining problems where the goal is to construct a predictive model based on a subset of predictive patterns in the database. Our main contribution is to introduce a novel method called safe pattern pruning (SPP) for a class of predictive pattern mining problems. The SPP method allows us to efficiently find a superset of all the predictive patterns in the database that are needed for the optimal predictive model. The advantage of the SPP method over existing boosting-type method is that the former can find the superset by a single search over the databas…

Celer: a Fast Solver for the Lasso with Dual Extrapolation

Convex sparsity-inducing regularizations are ubiquitous in high-dimensional machine learning, but solving the resulting optimization problems can be slow. To accelerate solvers, state-of-the-art approaches consist in reducing the size of the optimization problem at hand. In the context of regression, this can be achieved either by discarding irrelevant features (screening techniques) or by prioritizing features likely to be included in the support of the solution (working set techniques). Duality comes into play at several steps in these techniques. Here, we propose an extrapolation technique …

混合安全-强规则用于lasso类型问题中的高效优化

Hybrid safe-strong rules for efficient optimization in lasso-type problems

本文提出混合安全-强规则（HSSR）以提高lasso问题的计算效率，通过整合安全筛查规则与序列强规则，改进了现有方法，在lasso、弹性网和分组lasso问题中均表现出色。

通过二次 Lasso 重构实现最优设计的快速筛选规则

Fast Screening Rules for Optimal Design via Quadratic Lasso Reformulation

本文利用平方 ℓ1 惩罚线性回归与实验最优设计的等价性，提出快速安全筛选规则及重参数化同伦算法，以高效计算 Bayes c-最优设计。

Ising Model Selection Using $\ell_{1}$-Regularized Linear Regression: A Statistical Mechanics Analysis

We theoretically analyze the typical learning performance of $\ell_{1}$-regularized linear regression ($\ell_1$-LinR) for Ising model selection using the replica method from statistical mechanics. For typical random regular graphs in the paramagnetic phase, an accurate estimate of the typical sample complexity of $\ell_1$-LinR is obtained. Remarkably, despite the model misspecification, $\ell_1$-LinR is model selection consistent with the same order of sample complexity as $\ell_{1}$-regularized logistic regression ($\ell_1$-LogR), i.e., $M=\mathcal{O}\left(\log N\right)$, where $N$ is the num…