Sparse methods for machine learning:

Theory and algorithms

ECML/PKDD 2010 Tutorial

Francis Bach and Guillaume Obozinski

(INRIA - Ecole Normale Supérieure, Paris)

Theory and algorithms

ECML/PKDD 2010 Tutorial

Francis Bach and Guillaume Obozinski

(INRIA - Ecole Normale Supérieure, Paris)

Slides - part I (L1-norm)

Slides - part II (matrix and structured sparsity)

Tutorial objectives

Regularization by the L1-norm has attracted a lot of interest in recent years in statistics, machine learning and signal processing. In the context of least-square linear regression, the problem is usually referred to as the Lasso [1] or basis pursuit [2]. Much of the early effort has been dedicated to algorithms to solve the optimization problem efficiently, either through first-order methods [3, 4], or through homotopy methods that lead to the entire regularization path (i.e., the set of solutions for all values of the regularization parameters) at the cost of a single matrix inversion [5, 6]. A well-known property of the regularization by the L1-norm is the sparsity of the solutions, i.e., it leads to loading vectors with many zeros, and thus performs model selection on top of regularization. Recent work (e.g., [7, 8]) has looked precisely at the model consistency of the Lasso, i.e., if we know that the data were generated from a sparse loading vector, does the Lasso actually recover the sparsity pattern when the number of observations grows? Moreover, how many irrelevant variables could we consider while still being able to infer correctly the relevant ones?

The objective of the tutorial is to give a unified overview of the recent contributions of sparse convex methods to machine learning, both in terms of theory and algorithms. The course will be divided in three parts: in the first part, the focus will be on the regular L1-norm and variable selection, introducing key algorithms [3, 4, 5, 6] and key theoretical results [7, 8, 9]. Then, several more structured machine learning problems will be discussed, on vectors (second part) and matrices (third part), such as multi-task learning [10, 11], sparse principal component analysis [12], multiple

kernel learning [13, 14], structured sparsity [15, 16] and sparse coding [17]. Throughout the tutorial, applications to data from various domains (computer vision, image processing, bioinformatics, speech processing, recommender systems) will be considered.

Relationship with NIPS 2009 tutorial by F. Bach

The proposed tutorial will be based on the tutorial given at NIPS in 2009. Since there is more time (3 hours instead of 2 hours) and an additional presenter, more details will be given, with more focus on applications in data mining from different domains. Slides of the NIPS tutorial may be found at http://www.di.ens.fr/~fbach/nips2009tutorial/.

Target audience

Sparse methods have generated a lot of new work recently and the goal of the tutorial is to present these new advances to researchers and graduate students with a general knowledge of machine learning. In particular, we do not assume strong prior knowledge in convex optimization or statistics.

Presenters

• Francis Bach is a researcher in the Willow INRIA project-team, in the Computer Science Department of the Ecole Normale Sup´erieure, Paris, France. He graduated from the Ecole Polytechnique, Palaiseau, France, in 1997, and earned his PhD in 2005 from the Computer Science division at the University of California, Berkeley. His research interests include machine learning, statistics, optimization, graphical models, kernel methods, sparse methods and statistical signal processing. He has been awarded a starting investigator grant from the European Research Council in 2009. He has given tutorials at several conferences (ECCV 2008, ICCV 2009, NIPS 2009, CVPR 2010).

• Guillaume Obozinski is a researcher in the Willow INRIA project-team, a research group affiliated with the Computer Science department of the Ecole Normale Sup´erieure, Paris, France. A former student of the Ecole Normale Superieure de Cachan, he earned his PhD in 2009 from the Statistics department of the University of California at Berkeley. His research interests include machine learning, statistics, optimization and their applications to computer vision and computational biology.

Syllabus

1. Sparse linear estimation - variable selection

• Regularization by the L1-norm (Lasso)

• Efficient algorithms (homotopy algorithms, coordinate descent)

• Theoretical results (consistency, efficiency, exponentially many irrelevant

variables)

• Relationships with non convex sparse methods (Bayesian, greedy)

• Applications

2. Structured sparse methods on vectors

• Learning with groups of features (group Lasso)

• Multiple kernel learning (non-linear sparse methods)

• Extensions

• Applications

3. Sparse methods on matrices

• Multi-task learning (joint variable selection)

• Multi-category classification

• Sparse principal component analysis

• Matrix factorization (sparse coding, low-rank)

• Applications

References

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Royal Statistical Society Series B, 58(1):267–288, 1996.

[2] S. S. Chen, D L. Donoho, and M. A. Saunders. Atomic decomposition by basis

pursuit. SIAM Review, 43(1):129–159, 2001.

[3] W. Fu. Penalized regressions: the bridge vs. the Lasso. Journal of Computational

and Graphical Statistics, 7(3):397–416, 1998).

[4] J. Friedman, T. Hastie T, and R. Tibshirani. Pathwise coordinate optimization.

Annals of Applied Statistics, 1(2):302–332, 2007.

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of Computational and Graphical Statistics, 9(2):319–337, 2000.

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Annals of Statistics, 32:407, 2004.

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Learning Research, 7:2541–2563, 2006.

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of sparsity using ℓ1-constrained quadratic programming. Technical Report 709,

Department of Statistics, UC Berkeley, 2006.

[9] P. J. Bickel, Y. Ritov, and A. Tsybakov. Simultaneous analysis of Lasso and

Dantzig selector. Annals of Statistics, 2008. To appear.

[10] M. Pontil, A. Argyriou, and T. Evgeniou. Multi-task feature learning. In Advances

in Neural Information Processing Systems, 2007.

[11] G. Obozinski, B. Taskar, and M.I. Jordan. Joint covariate selection and joint

subspace selection for multiple classification problems. Statistics and Computing,

pages 1–22, 2009.

[12] A. D’aspremont, El L. Ghaoui, M. I. Jordan, and G. R. G. Lanckriet. A direct

formulation for sparse PCA using semidefinite programming. SIAM Review,

49(3):434–48, 2007.

[13] F. R. Bach, G. R. G. Lanckriet, and M. I. Jordan. Multiple kernel learning, conic

duality, and the SMO algorithm. In Proceedings of the International Conference

on Machine Learning (ICML), 2004.

[14] F. Bach. Exploring large feature spaces with hierarchical multiple kernel learning.

In Advances in Neural Information Processing Systems (NIPS), 2008.

[15] L. Jacob, G. Obozinski, and J.-P. Vert. Group Lasso with overlaps and graph

Lasso. In Proceedings of the 26th International Conference on Machine Learning

(ICML), 2009.

[16] R. Jenatton, G. Obozinski, and F. Bach. Structured sparse principal component

analysis. Technical report, arXiv:0909.1440, 2009.

[17] B. A. Olshausen and D. J. Field. Sparse coding with an overcomplete basis set:

A strategy employed by V1? Vision Research, 37:3311–3325, 1997.