VC theory, Support vectors and Hedged prediction technology

1. VC theory, Support vectors and Hedged prediction technology

2. Overfitting in classification • Assume a family C of classifiers of points infeature space F. A family of classifiers is a map from CF 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.

3. 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.

4. 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 isobserved error rate for c. • Goodness of selecting from C a classifier with best performance on training set depends on VC-dimension h:

5. Why VC-dimension?

6. 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

7. 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.

8. Find parallel hyperplanes Classification Red true separatingplane. Blue: wide marginseparation in sample Classify by planebetween blue planes

9. Find parallel hyperplanes Regression Red: true central plane. Blue: narrowest margin enclosing sample New xk : predict ykso (xk, yk) lies on mid-plane (dotted).

15. From vector to scalar product

16. Soft Margins

17. Soft Margins Quadratic 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).

18. SVM packages • Inputs xi, yi, and KERNEL and SOFTNESS information • Only output is , non-zero coefficients indicatesupport vectors. • Hyperplane obtained by

19. Kernel Trick

20. Kernel Trick

21. Kernel Trick Example: 2D space (x1,x2). Map to 5D space (c1*x1, c2*x2, c3*x1^2, c4*x1*x2, c5*x2^2). K(x,y)=(xy+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!!

22. Kernel Trick Gaussian Kernel: K(x,y) = exp(-||x-y||^2/2