Sparse methods for machine learning:

Theory and algorithms

NIPS 2009 Tutorial

Francis Bach

(INRIA - Ecole Normale Supérieure, Paris)

Theory and algorithms

NIPS 2009 Tutorial

Francis Bach

(INRIA - Ecole Normale Supérieure, Paris)

Slides (6.5 Mb)

Slides (low-resolution images - 1.9 Mb)

Abstract

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 leads 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 works (e.g., [7, 8]) have 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] and sparse coding [14].

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 between non convex sparse methods (Bayesian, greedy)

2. Structured sparse methods on vectors

• Learning with groups of features (group Lasso)

• Multiple kernel learning (non-linear sparse methods)

• Extensions

3. Sparse methods on matrices

• Multi-task learning (joint variable selection)

• Multi-category classification

• Sparse principal component analysis

• Matrix factorization (sparse coding, low-rank decomposition, NMF)

References

[1] R. Tibshirani. Regression shrinkage and selection via the lasso. Journal of The 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.

[5] M. R. Osborne, B. Presnell, and B. A. Turlach. On the lasso and its dual. Journal of Computational and Graphical Statistics, 9(2):319–337, 2000.

[6] B. Efron, T. Hastie, I. Johnstone, and R. Tibshirani. Least angle regression. Annals of Statistics, 32:407, 2004.

[7] P. Zhao and B. Yu. On model selection consistency of Lasso. Journal of Machine Learning Research, 7:2541–2563, 2006.

[8] M. J. Wainwright. Sharp thresholds for noisy and high-dimensional recovery 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, 2009. To appear.

[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] 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.