2004 TheEntireRegulPathForTheSVM

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Subject Headings: SVM Classification Algorithm, Regularization Path, Algorithm Tuning Parameter, Regularization Cost Parameter, Kernel Function Parameter, Cost Parameter.


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The support vector machine (SVM) is a widely used tool for classification. Many efficient implementations exist for fitting a two-class SVM model. The user has to supply values for the tuning parameters: the regularization cost parameter, and the kernel parameters. It seems a common practice is to use a default value for the cost parameter, often leading to the least restrictive model. In this paper we argue that the choice of the cost parameter can be critical. We then derive an algorithm that can fit the entire path of SVM solutions for every value of the cost parameter, with essentially the same computational cost as fitting one SVM model. We illustrate our algorithm on some examples, and use our representation to give further insight into the range of SVM solutions.

1. Introduction

In this paper we study the support vector machine (SVM) (Vapnik, 1996; Schölkopf and Smola, 2001) for two-class classification. We have a set of [math]\displaystyle{ n }[/math] training pairs [math]\displaystyle{ x_i,y_i }[/math], where [math]\displaystyle{ x_i \in \Re^p }[/math] is a p-vector of real-valued predictors (attributes) for the ith observation, and yi in {−1,+1} codes its binary response. We start off with the simple case of a linear classifier, where our goal is to estimate a linear decision function

[math]\displaystyle{ f(x) = \beta_0 + \beta^T x (1) }[/math],

and its associated classifier

[math]\displaystyle{ \text{Class}(x) = \text{sign}[f(x)] (2) }[/math].

There are many ways to fit such a linear classifier, including linear regression, Fisher’s linear discriminant analysis, and logistic regression (Hastie et al., 2001, Chapter 4). If the training data are linearly separable, an appealing approach is to ask for the decision boundary {x : ƒ(x) = 0} that maximizes the margin between the two classes (Vapnik, 1996). Solving such a problem is an exercise in convex optimization;

Figure 3: Simulated data illustrate the need for regularization. The 200 data points are generated from a pair of mixture densities. The two SVM models used radial kernels with the scale and cost parameters as indicated at the top of the plots. … Figure 3 shows the results of fitting two SVM models to the same simulated data set. The data are generated from a pair of mixture densities, described in detail in Hastie et al. (2001, Chapter 2). The actual training data and test distribution are available from http:// www-stat.stanford.edu/ElemStatLearn.

It seems that the regularization parameter C (or l) is often regarded as a genuine “nuisance” in the community of SVM users. Software packages, such as the widely used SVMlight (Joachims, 1999), provide default settings for C, which are then used without much further exploration. A recent introductory document (Hsu et al., 2003) supporting the LIBSVM package does encourage grid search for C.

7. Discussion

Our work on the SVM path algorithm was inspired by earlier work on exact path algorithms in other settings. “Least Angle Regression” (Efron et al., 2002) shows that the coefficient path for the sequence of “lasso” coefficients (Tibshirani, 1996) is piecewise linear. The lasso solves the following regularized linear regression problem, .... is the L1 norm of the coefficient vector. This L1 constraint delivers a sparse solution vector bl; the larger l, the more elements of bl are zero, the remainder shrunk toward zero. In fact, any model with an L1 constraint and a quadratic, piecewise quadratic, piecewise linear, or mixed quadratic and linear loss function, will have piecewise linear coefficient paths, which can be calculated exactly and efficiently for all values of l (Rosset and Zhu, 2003). These models include, among others,

The SVM model has a quadratic constraint and a piecewise linear (“hinge”) loss function. This leads to a piecewise linear path in the dual space, hence the Lagrange coefficients ai are piecewise linear.

Other models that would share this property include

  • The e-insensitive SVM regression model
  • Quadratically regularized L1 regression, including flexible models based on kernels or smoothing splines.

Of course, quadratic criterion + quadratic constraints also lead to exact path solutions, as in the classic case of ridge regression, since a closed form solution is obtained via the SVD. However, these paths are nonlinear in the regularization parameter.

For general non-quadratic loss functions and [math]\displaystyle{ L_1 }[/math] constraints, the solution paths are typically piecewise non-linear. Logistic regression is a leading example. In this case, approximate pathfollowing algorithms are possible (Rosset, 2005).

The general techniques employed in this paper are known as parametric programming via active sets in the convex optimization literature (Allgower and Georg, 1992). The closest we have seen to our work in the literature employ similar techniques in incremental learning for SVMs (Fine and Scheinberg, 2002; Cauwenberghs and Poggio, 2001; DeCoste and Wagstaff, 2000). These authors do not, however, construct exact paths as we do, but rather focus on updating and downdating the solutions as more (or less) data arises. Diehl and Cauwenberghs (2003) allow for updating the parameters as well, but again do not construct entire solution paths. The work of Pontil and Verri (1998) recently came to our notice, who also observed that the lagrange multipliers for the margin vectors change in a piece-wise linear fashion, while the others remain constant.

The SvmPath has been implemented in the R computing environment (contributed library svmpath at CRAN), and is available from the first author’s website.


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 AuthorvolumeDate ValuetitletypejournaltitleUrldoinoteyear
2004 TheEntireRegulPathForTheSVMTrevor Hastie
Saharon Rosset
Ji Zhu
Robert Tibshirani
The Entire Regularization Path for the Support Vector MachineThe Journal of Machine Learning Researchhttp://www.jmlr.org/papers/volume5/hastie04a/hastie04a.pdf2004