Sparse methods for machine learning:
Theory and algorithms
NIPS 2009 Tutorial
(INRIA - Ecole Normale Supérieure, Paris)
Slides (6.5 Mb)
Slides (low-resolution images - 1.9 Mb)
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  or basis pursuit . 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 , multiple kernel learning  and
sparse coding .
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)
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)
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