CS 281B / Stat 241B, Spring 2006:

Statistical Learning Theory


Pattern Classification and Analysis of the Perceptron Algorithm

The argument giving the minimax lower bound for linear threshold functions is similar to the proof of the main result in the following paper.
`A General Lower Bound on the Number of Examples Needed for Learning.' A. Ehrenfeucht, D. Haussler, M. Kearns and L. Valiant.

The following book is an old (1987) revision of an older (1969) book on linear threshold functions and the perceptron algorithm.
Perceptrons: An Introduction to Computational Geometry Marvin L. Minsky, Seymour A. Papert, MIT Press, 1987.

That book and the following book discuss the perceptron convergence theorem.
Nilsson, N., The Mathematical Foundations of Learning Machines, San Francisco: Morgan Kaufmann, 1990. (Reprint of N.J. Nilsson, Learning machines, McGraw-Hill, New York, 1965.)

The upper bound on risk for the perceptron algorithm that we saw in lectures follows from the perceptron convergence theorem and results converting mistake bounded algorithms to average risk bounds. The following paper reviews these results.
Large margin classification using the perceptron algorithm. Yoav Freund and Robert E. Schapire.

`Pattern classification and learning theory.' G. Lugosi
(See also the text book, L. Devroye, L. Gyorfi, G. Lugosi, `A Probabilistic Theory of Pattern Recognition,' Springer, New York, 1996.)
`Learning Pattern Classification---A Survey.' S. Kulkarni, G. Lugosi, and S. Venkatesh

Kernel Methods, Support Vector Machines

`An introduction to kernel-based learning algorithms.' K.-R. Mueller, S. Mika, G. Raetsch, K. Tsuda, and B. Schoelkopf.

`A Tutorial on Support Vector Machines for Pattern Recognition.' C. J. C. Burges.

A Tutorial on Support Vector Regression.' A. J. Smola and B. Schoelkopf.

See also the text books:

`An Introduction to Support Vector Machines.' Nello Cristianini and John Shawe-Taylor. Cambridge University Press, Cambridge, UK, 2000. (see the web page)

`Kernel Methods for Pattern Analysis.' John Shawe-Taylor and Nello Cristianini. Cambridge University Press, Cambridge, UK, 2004. (see the web page)

`Learning with Kernels.' Bernhard Schoelkopf and Alex Smola. MIT Press, Cambridge, MA, 2002. (see the web page)

The following text book (in draft form) is available on the web. Chapter 5 gives a good treatment of constrained optimization problems and Lagrangian duality.
`Convex Optimization.' Stephen Boyd and Lieven Vandenberghe.

`Support Vector Networks.' Corinna Cortes and Vladimir Vapnik.

The following papers describe a geometric view of SVM optimization problems.
`A fast iterative nearest point algorithm for support vector machine classifier design.' S.S. Keerthi, S.K. Shevade, C. Bhattacharyya and K.R.K. Murthy.
`Duality and Geometry in SVM Classifiers.' Kristin P. Bennett and Erin J. Bredensteiner.
`A Geometric Interpretation of nu-SVM Classifiers.' D.J. Crisp and C.J.C. Burges

The following papers present relationships between convex cost functions and discrete loss (for two-class pattern classification). The first paper generalizes and simplifies results of the second. The third paper considers a general situations, including weighted classification, regression, quantile estimation and density estimation.
`Convexity, classification, and risk bounds.' Peter Bartlett, Mike Jordan and Jon McAuliffe.
`Statistical behavior and consistency of classification methods based on convex risk minimization.' Tong Zhang.
`How to compare different loss functions and their risks.' Ingo Steinwart.

These papers investigate the RKHS of Gaussian kernels.
`Consistency and convergence rates of one-class SVM and related algorithms.' (See Section 3.) R. Vert and J.-P. Vert.
`An Explicit Description of the Reproducing Kernel Hilbert Spaces of Gaussian RBF Kernels .' I. Steinwart, D. Hush, and C. Scovel.

The following papers describe kernels defined on structures, such as sequences and trees. The first describes a number of operations that can be used in constructing kernels.
`Convolution Kernels on discrete structures' D. Haussler
`Dynamic alignment kernels' C. Watkins
`Convolution kernels for natural language' Michael Collins and Nigel Duffy.
`Text classification using string kernels' H. Lodhi, C. Saunders, J. Sahwe-Taylor, N. Cristianini, and C. Watkins

Another good resource is the kernel machines web site: http://www.kernel-machines.org

Ensemble Methods

`Bagging predictors' Leo Breiman.

`The boosting approach to machine learning: An overview.'.' Robert E. Schapire.

Slides from tutorial on boosting: Part 1 and Part 2
Yoav Freund and Robert E. Schapire.

`A decision-theoretic generalization of on-line learning and an application to boosting.' Yoav Freund and Robert E. Schapire.
`Experiments with a new boosting algorithm.' Yoav Freund and Robert E. Schapire.

`Boosting the margin: A new explanation for the effectiveness of voting methods.' Robert E. Schapire, Yoav Freund, Peter Bartlett and Wee Sun Lee.

`Arcing classifiers.' Leo Breiman.

`Additive logistic regression: a statistical view of boosting.' Jerome Friedman, Trevor Hastie and Robert Tibshirani.

Risk Bounds

`Concentration-of-measure inequalities.' Gabor Lugosi.

` A few notes on Statistical Learning Theory.' Shahar Mendelson

Review paper on Rademacher averages and local Rademacher averages:
`New Approaches to Statistical Learning Theory.' Olivier Bousquet.

The following paper collects some useful properties of Rademacher averages.
`Rademacher and Gaussian complexities: risk bounds and structural results' P. L. Bartlett and S. Mendelson.

Rademacher averages for large margin classifiers:
`Empirical margin distributions and bounding the generalization error of combined classifiers.' Vladimir Koltchinskii and Dmitriy Panchenko.

The `finite lemma' (Rademacher averages of finite sets) is Lemma 5.2 in this paper, which also introduces local Rademacher averages.
`Some applications of concentration inequalities to statistics.' Pascal Massart.

The growth function, VC-dimension, and pseudodimension are described in the following text (see chapter 3). Estimates of these quantities for parameterized function classes is covered in chapters 7 and 8.
`Neural network learning: Theoretical foundations.' Martin Anthony and Peter Bartlett. Cambridge University Press. 1999.

Model Selection

`An experimental and theoretical comparison of model selection methods.' M. Kearns, Y. Mansour, A. Ng, and D. Ron

`Risk bounds for model selection via penalisation.'. Andrew Barron, Lucien Birge and Pascal Massart.

`Model selection and error estimation.' P. L. Bartlett, S. Boucheron, and G. Lugosi.

`Some applications of concentration inequalities to statistics.' Pascal Massart.

`A consistent strategy for boosting algorithms.' Gabor Lugosi and Nicolas Vayatis.

`Consistency of support vector machines and other regularized kernel classifiers.' I. Steinwart.

`The consistency of greedy algorithms for classification.' Shie Mannor, Ron Meir, Tong Zhang.

`Concentration inequalities and model selection.' Pascal Massart.

For Bernstein's inequality, and generalizations, see:
`Concentration-of-measure inequalities.' Gabor Lugosi.

`A Bennett concentration inequality and its application to suprema of empirical processes', Olivier Bousquet.

`Concentration inequalities using the entropy method.' S. Boucheron, G. Lugosi and P. Massart.
`A sharp concentration inequality with applications.' Stephane Boucheron, Gabor Lugosi and Pascal Masssart.

Bounding the variance for excess loss classes:
`Improving the sample complexity using global data.' S. Mendelson.

`Convexity, classification, and risk bounds.' P. Bartlett, M. Jordan, J. McAuliffe.

Local Rademacher averages:
`Some applications of concentration inequalities to statistics.' Pascal Massart.

`Rademacher processes and bounding the risk of function learning.' V. Koltchinskii and D. Panchenko.

`Complexity regularization via localized random penalties.' G. Lugosi and M. Wegkamp

Local Rademacher complexities.' P.L. Bartlett, O. Bousquet, S. Mendelson.

`Geometric parameters of kernel machines,' S. Mendelson.

`Complexity of convex hulls and generalization bounds' O. Bousquet, V. Koltchinskii and D. Panchenko.

Back to course home page