For supervised and unsupervised learning, positive definite kernels allow to use large and potentially infinite dimensional feature spaces with a computational cost that only depends on the number of observations. This is usually done through the penalization of predictor functions by Euclidean or Hilbertian norms. In this paper, we explore penalizing by sparsity-inducing norms such as the ℓ 1-norm or the block ℓ 1-norm. We assume that the kernel decomposes into a large sum of individual basis kernels which can be embedded in a directed acyclic graph; we show that it is then possible to perform kernel selection through a hierarchical multiple kernel learning framework, in polynomial time in the number of selected kernels. This framework is naturally applied to non linear variable selection; our extensive simulations on synthetic datasets and datasets from the UCI repository show that efficiently exploring the large feature space through sparsity-inducing norms leads to state-of-the-art predictive performance. 1

We consider the problem of high-dimensional non-linear variable selection for supervised learning. Our approach is based on performing linear selection among exponentially many appropriately defined positive definite kernels that characterize non-linear interactions between the original variables. To select efficiently from these many kernels, we use the natural hierarchical structure of the problem to extend the multiple kernel learning framework to kernels that can be embedded in a directed acyclic graph; we show that it is then possible to perform kernel selection through a graph-adapted sparsity-inducing norm, in polynomial time in the number of selected kernels. Moreover, we study the consistency of variable selection in high-dimensional settings, showing that under certain assumptions, our regularization framework allows a number of irrelevant variables which is exponential in the number of observations. Our simulations on synthetic datasets and datasets from the UCI repository show state-of-the-art predictive performance for non-linear regression problems. 1

Motivated from real world problems, like object categorization, we study a particular mixed-norm regularization for Multiple Kernel Learning (MKL). It is assumed that the given set of kernels are grouped into distinct components where each component is crucial for the learning task at hand. The formulation hence employs l ∞ regularization for promoting combinations at the component level and l1 regularization for promoting sparsity among kernels in each component. While previous attempts have formulated this as a non-convex problem, the formulation given here is an instance of non-smooth convex optimization problem which admits an efficient Mirror-Descent (MD) based procedure. The MD procedure optimizes over product of simplexes, which is not a well-studied case in literature. Results on real-world datasets show that the new MKL formulation is well-suited for object categorization tasks and that the MD based algorithm outperforms stateof-the-art MKL solvers like simpleMKL in terms of computational effort. 1

We consider the problem of how to improve the efficiency of Multiple Kernel Learning (MKL). In literature, MKL is often solved by an alternating approach: (1) the minimization of the kernel weights is solved by complicated techniques, such as Semi-infinite Linear Programming, Gradient Descent, or Level method; (2) the maximization of SVM dual variables can be solved by standard SVM solvers. However, the minimization step in these methods is usually dependent on its solving techniques or commercial softwares, which therefore limits the efficiency and applicability. In this paper, we formulate a closed-form solution for optimizing the kernel weights based on the equivalence between group-lasso and MKL. Although this equivalence is not our invention, our derived variant equivalence not only leads to an efficient algorithm for MKL, but also generalizes to the case for Lp-MKL (p ≥ 1 and denoting the Lp-norm of kernel weights). Therefore, our proposed algorithm provides a unified solution for the entire family of Lp-MKL models. Experiments on multiple data sets show the promising performance of the proposed technique compared with other competitive methods. 1.

Abstract—Recently, there has been a lot of interest around multi-task learning (MTL) problem with the constraints that tasks should share a common sparsity profile. Such a problem can be addressed through a regularization framework where the regularizer induces a joint-sparsity pattern between task decision functions. We follow this principled framework and focus on ℓp−ℓq (with 0 ≤ p ≤ 1 and 1 ≤ q ≤ 2) mixed-norms as sparsityinducing penalties. Our motivation for addressing such a larger class of penalty is to adapt the penalty to a problem at hand leading thus to better performances and better sparsity pattern. For solving the problem in the general multiple kernel case, we first derive a variational formulation of the ℓ1 − ℓq penalty which helps up in proposing an alternate optimization algorithm. Although very simple, the latter algorithm provably converges to the global minimum of the ℓ1 − ℓq penalized problem. For the linear case, we extend existing works considering accelerated proximal gradient to this penalty. Our contribution in this context is to provide an efficient scheme for computing the ℓ1−ℓq proximal operator. Then, for the more general case when 0 < p < 1, we solve the resulting non-convex problem through a majorization-minimization approach. The resulting algorithm is an iterative scheme which, at each iteration, solves a weighted ℓ1 − ℓq sparse MTL problem. Empirical evidences from toy dataset and real-word datasets dealing with BCI single trial EEG classification and protein subcellular localization show the benefit of the proposed approaches and algorithms. Index Terms—Multi-task learning, multiple kernel learning, sparsity, mixed-norm, Support Vector Machines I.

Learning linear combinations of multiple kernels is an appealing strategy when the right choice of features is unknown. Previous approaches to multiple kernel learning (MKL) promote sparse kernel combinations to support interpretability. Unfortunately, ℓ1-norm MKL is hardly observed to outperform trivial baselines in practical applications. To allow for robust kernel mixtures, we generalize MKL to arbitrary ℓp-norms. We devise new insights on the connection between several existing MKL formulations and develop two efficient interleaved optimization strategies for arbitrary p> 1. Empirically, we demonstrate that the interleaved optimization strategies are much faster compared to the traditionally used wrapper approaches. Finally, we apply ℓp-norm MKL to real-world problems from computational biology, showing that non-sparse MKL achieves accuracies that go beyond the state-of-the-art. 1

This paperaddressesthe problem ofoptimal featureextractionfrom a waveletrepresentation. Our workaims at building features by selecting waveletcoefficients resulting from signalorimage decomposition on an adapted wavelet basis. For this purpose, we jointly learn in a kernelized large-margin context the wavelet shape as well as the appropriate scale and translation of the wavelets, hence the name “wavelet kernel learning”. This problem is posed as a multiple kernel learning problem where the number of kernels can be very large. For solving such a problem, we introduce a novel multiple kernel learning algorithm based on active constraints methods. We furthermore propose some variants of this algorithm that can produce approximate solutions more efficiently. Empirical analysis show that our active constraint MKL algorithm achieves state-of-the art efficiency. When used for wavelet kernel learning, our experimental results show that the approaches we propose are competitive with respect to the state of the art on Brain-Computer Interface and Brodatz texture datasets.

This paper addresses the problem of Rule Ensemble Learning (REL), where the goal is simultaneous discovery of a small set of simple rules and their optimal weights that lead to good generalization. Rules are assumed to be conjunctions of basic propositions concerning the values taken by the input features. From the perspectives of interpretability as well as generalization, it is highly desirable to construct rule ensembles with low training error, having rules that are i) simple, i.e., involve few conjunctions and ii) few in number. We propose to explore the (exponentially) large feature space of all possible conjunctions optimally and efficiently by employing the recently introduced Hierarchical Kernel Learning (HKL) framework. The regularizer employed in the HKL formulation can be interpreted as a potential for discouraging selection of rules involving large number of conjunctions – justifying its suitability for constructing rule ensembles. Simulation results show that, in case of many benchmark datasets, the proposed approach improves over state-of-the-art REL algorithms in terms of generalization and indeed learns simple rules. Unfortunately, HKL selects a conjunction only if all its subsets are selected. We propose a novel convex formulation which alleviates this problem and generalizes the HKL framework. The main technical contribution of this paper is an efficient mirrordescent based active set algorithm for solving the new formulation. Empirical evaluations on REL problems illustrate the utility of generalized