VC theory, Support vectors and Hedged prediction technology

Overfitting in classification

• Assume a family C of classifiers of points infeature space F. A family of classifiers is a map from CF to {0,1} (Negative and positive class).

• For each subset X of F and each c in C, c(X) defines a partitioning of X into two classes.

• C shatters X if every partitioning of X is accomplished by some c in C

• If every point set X of size d is shattered by C, then the VC dimension is at least d.

• If a point set of d+1 elements cannot be shattered by C, then the VC-dimension is at most d.

VC-dimension of hyperplanes

• The set of points on the line shatters any two points, but not three

• The set of lines in the plane shatters any three non-collinear points, but no four points.

• Any d+2 points in E^d can be partitioned into two blocks whose convex hulls intersect.

• VC-dimension of hyperplanes in E^d is thus d+1.

Why VC-dimension?

• Elegant and pedagogical, not very useful.• Bounds future error of classifier, PAC-learning.• Exchangeable distribution of (xi, yi).• For first N points, training error for c is

observed error rate for c.• Goodness of selecting from C a classifier with

best performance on training set depends on VC-dimension h:

Why VC-dimension?

Classify with hyperplanes

Frank Rosenblatt (1928 – 1971)

Pioneering work in classifying byhyperplanes in high-dimensional spaces.

Criticized by Minsky-Papert, sincereal classes are not normallylinearly separable.ANN research taken up again in1980:s, with non-linear mappingsto get improved separation.Predecessor to SVM/kernel methods

Find parallel hyperplanes

• Separate examples by wide margin hyperplanes (classifications).

• Enclose examples between hyperplanes (regression).

• If necessary, non-linearly map examples to high-dimensional space where they are better separated.

Find parallel hyperplanes

ClassificationRed true separatingplane.Blue: wide marginseparation in sampleClassify by planebetween blue planes

Find parallel hyperplanes

RegressionRed: true central plane.Blue: narrowest margin enclosing sample

New xk : predict ykso (xk, yk) lies on mid-plane (dotted).

From vector to scalar product

Soft Margins

Soft MarginsQuadratic programming goes through also with soft margins.Specification of softness constant C is part of most packages.

However, no prior rule for setting C is established, and experimentation is necessary for each application. Choice is between narrowing margin, allowing more outliers, and using a more liberal kernel (to be described).

SVM packages

• Inputs xi, yi, and KERNEL and SOFTNESS information

• Only output is , non-zero coefficients indicatesupport vectors.

• Hyperplane obtained by

Kernel Trick

Kernel Trick

Kernel TrickExample: 2D space (x1,x2). Map to 5D space(c1*x1, c2*x2, c3*x1^2, c4*x1*x2, c5*x2^2).

K(x,y)=(xy+1)^2 =2*x1*y1+2*x2*y2+x1^2*y1^2+x2^2*y2^2+2*x1*x2*y1*y2+1 =(x)(y),

Where (x)= ((x1,x2)) = (√2x1, √2x2, x1^2, √2x1*x2, x2^2).

Hyperplanes in R^5 are mapped back to conic sections in R^2!!

Kernel TrickGaussian Kernel:K(x,y) = exp(-||x-y||^2/2