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Structured sparsityinducing norms through submodular functions
 IN ADVANCES IN NEURAL INFORMATION PROCESSING SYSTEMS
, 2010
"... Sparse methods for supervised learning aim at finding good linear predictors from as few variables as possible, i.e., with small cardinality of their supports. This combinatorial selection problem is often turnedinto a convex optimization problem byreplacing the cardinality function by its convex en ..."
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Cited by 30 (10 self)
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Sparse methods for supervised learning aim at finding good linear predictors from as few variables as possible, i.e., with small cardinality of their supports. This combinatorial selection problem is often turnedinto a convex optimization problem byreplacing the cardinality function by its convex envelope (tightest convex lower bound), in this case the ℓ1norm. In this paper, we investigate more general setfunctions than the cardinality, that may incorporate prior knowledge or structural constraints which are common in many applications: namely, we show that for nonincreasing submodular setfunctions, the corresponding convex envelope can be obtained from its Lovász extension, a common tool in submodular analysis. This defines a family of polyhedral norms, for which we provide generic algorithmic tools (subgradients and proximal operators) and theoretical results (conditions for support recovery or highdimensional inference). By selecting specific submodular functions, we can give a new interpretation to known norms, such as those based on rankstatistics or grouped norms with potentially overlapping groups; we also define new norms, in particular ones that can be used as nonfactorial priors for supervised learning.
Accelerated and inexact forwardbackward algorithms
 Optimization Online, EPrint
"... Abstract. We propose a convergence analysis of accelerated forwardbackward splitting methods for composite function minimization, when the proximity operator is not available in closed form, and can only be computed up to a certain precision. We prove that the 1/k 2 convergence rate for the functio ..."
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Abstract. We propose a convergence analysis of accelerated forwardbackward splitting methods for composite function minimization, when the proximity operator is not available in closed form, and can only be computed up to a certain precision. We prove that the 1/k 2 convergence rate for the function values can be achieved if the admissible errors are of a certain type and satisfy a sufficiently fast decay condition. Our analysis is based on the machinery of estimate sequences first introduced by Nesterov for the study of accelerated gradient descent algorithms. Furthermore, we give a global complexity analysis, taking into account the cost of computing admissible approximations of the proximal point. An experimental analysis is also presented.
Shaping Level Sets with Submodular Functions
"... We consider a class of sparsityinducing regularization terms based on submodular functions. While previous work has focused on nondecreasing functions, we explore symmetric submodular functions and their Lovász extensions. We show that the Lovász extension may be seen as the convex envelope of a f ..."
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We consider a class of sparsityinducing regularization terms based on submodular functions. While previous work has focused on nondecreasing functions, we explore symmetric submodular functions and their Lovász extensions. We show that the Lovász extension may be seen as the convex envelope of a function that depends on level sets (i.e., the set of indices whose corresponding components of the underlying predictor are greater than a given constant): this leads to a class of convex structured regularization terms that impose prior knowledge on the level sets, and not only on the supports of the underlying predictors. We provide unified optimization algorithms, such as proximal operators, and theoretical guarantees (allowed level sets and recovery conditions). By selecting specific submodular functions, we give a new interpretation to known norms, such as the total variation; we also define new norms, in particular ones that are based on order statistics with application to clustering and outlier detection, and on noisy cuts in graphs with application to change point detection in the presence of outliers. 1
A latent factor model for highly multirelational data
"... Many data such as social networks, movie preferences or knowledge bases are multirelational, in that they describe multiple relations between entities. While there is a large body of work focused on modeling these data, modeling these multiple types of relations jointly remains challenging. Further ..."
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Many data such as social networks, movie preferences or knowledge bases are multirelational, in that they describe multiple relations between entities. While there is a large body of work focused on modeling these data, modeling these multiple types of relations jointly remains challenging. Further, existing approaches tend to breakdown when the number of these types grows. In this paper, we propose a method for modeling large multirelational datasets, with possibly thousands of relations. Our model is based on a bilinear structure, which captures various orders of interaction of the data, and also shares sparse latent factors across different relations. We illustrate the performance of our approach on standard tensorfactorization datasets where we attain, or outperform, stateoftheart results. Finally, a NLP application demonstrates our scalability and the ability of our model to learn efficient and semantically meaningful verb representations. 1
Structured Sparsity through Convex Optimization
"... Abstract. Sparse estimation methods are aimed at using or obtaining parsimonious representations of data or models. While naturally cast as a combinatorial optimization problem, variable or feature selection admits a convex relaxation through the regularization by the ℓ1norm. In this paper, we cons ..."
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Abstract. Sparse estimation methods are aimed at using or obtaining parsimonious representations of data or models. While naturally cast as a combinatorial optimization problem, variable or feature selection admits a convex relaxation through the regularization by the ℓ1norm. In this paper, we consider situations where we are not only interested in sparsity, but where some structural prior knowledge is available as well. We show that the ℓ1norm can then be extended to structured norms built on either disjoint or overlapping groups of variables, leading to a flexible framework that can deal with various structures. We present applications to unsupervised learning, for structured sparse principal component analysis and hierarchical dictionary learning, and to supervised learning in the context of nonlinear variable selection. Key words and phrases: Sparsity, convex optimization. 1.
Nonparametric Group Orthogonal Matching Pursuit for Sparse Learning with Multiple Kernels
"... We consider regularized risk minimization in a large dictionary of Reproducing kernel Hilbert Spaces (RKHSs) over which the target function has a sparse representation. This setting, commonly referred to as Sparse Multiple Kernel Learning (MKL), may be viewed as the nonparametric extension of group ..."
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We consider regularized risk minimization in a large dictionary of Reproducing kernel Hilbert Spaces (RKHSs) over which the target function has a sparse representation. This setting, commonly referred to as Sparse Multiple Kernel Learning (MKL), may be viewed as the nonparametric extension of group sparsity in linear models. While the two dominant algorithmic strands of sparse learning, namely convex relaxations using l1 norm (e.g., Lasso) and greedy methods (e.g., OMP), have both been rigorously extended for group sparsity, the sparse MKL literature has so far mainly adopted the former with mild empirical success. In this paper, we close this gap by proposing a GroupOMP based framework for sparse MKL. Unlike l1MKL, our approach decouples the sparsity regularizer (via a direct l0 constraint) from the smoothness regularizer (via RKHS norms), which leads to better empirical performance and a simpler optimization procedure that only requires a blackbox singlekernel solver. The algorithmic development and empirical studies are complemented by theoretical analyses in terms of Rademacher generalization bounds and sparse recovery conditions analogous to those for OMP [27] and GroupOMP [16]. 1
Incremental majorizationminimization optimization with application to largescale machine learning
 INRIA Grenoble RhôneAlpes / LJK Laboratoire Jean Kuntzmann
, 2014
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Structured Penalties for Loglinear Language Models
"... Language models can be formalized as loglinear regression models where the input features represent previously observed contexts up to a certain length m. The complexity of existing algorithms to learn the parameters by maximum likelihood scale linearly in nd, where n is the length of the training c ..."
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Language models can be formalized as loglinear regression models where the input features represent previously observed contexts up to a certain length m. The complexity of existing algorithms to learn the parameters by maximum likelihood scale linearly in nd, where n is the length of the training corpus and d is the number of observed features. We present a model that grows logarithmically in d, making it possible to efficiently leverage longer contexts. We account for the sequential structure of natural language using treestructured penalized objectives to avoid overfitting and achieve better generalization. 1