Margins, support vectors, and linear programming

1. Margins, support vectors, and linear programming Thanks to Terran Lane and S. Dreiseitl

2. Exercise • Derive the vector derivative expressions: • Find an expression for the minimum squared error weight vector, w, in the loss function:

3. Solution to LSE regression

4. The LSE method • The quantity is called a Gram matrix and is positive semidefinite and symmetric • The quantity is the pseudoinverse of X • May not exist if Gram matrix is not invertible • The complete “learning algorithm” is 2 whole lines of Matlab code

5. LSE example w=[6.72, -0.36]

6. LSE example t=y(x1,w)

7. 6.72 LSE example t=y(x1,w) [0.36, 1]

8. The LSE method • So far, we have a regressor -- estimates a real valued ti for each xi • Can convert to a classifier by assigning t=+1 or -1 to binary class training data

9. Multiclass trouble ?

10. Handling non-binary data • All against all: • Train O(c2) classifiers, one for each pair of classes • Run every test point through all classifiers • Majority vote for final classifier • More stable than 1 vs many • Lot more overhead, esp for large c • Data may be more balanced • Each classifier trained on very small part of data

11. Support Vector Machines

12. Linear separators are nice • ... but what if your data looks like this:

13. Linearly nonseparable data • 2 possibilities: • Use nonlinear separators (diff hypothesis space) • Possibly intersection of multiple linear separators, etc. (E.g., decision tree)

14. Linearly nonseparable data • 2 possibilities: • Use nonlinear separators (diff hypothesis space) • Possibly intersection of multiple linear separators, etc. (E.g., decision tree) • Change the data • Nonlinear projection of data • These turn out to be flip sides of each other • Easier to think about (do math for) 2nd case

15. Nonlinear data projection • Suppose you have a “projection function”: • Original feature space • “Projected” space • Usually • Do learning w/ linear model in • Ex:

16. Nonlinear data projection

17. Nonlinear data projection

18. Nonlinear data projection

19. Common projections • Degree-k polynomials: • Fourier expansions:

20. Example nonlinear surfaces SVM images from lecture notes by S. Dreiseitl

21. Example nonlinear surfaces SVM images from lecture notes by S. Dreiseitl

22. Example nonlinear surfaces SVM images from lecture notes by S. Dreiseitl

23. Example nonlinear surfaces SVM images from lecture notes by S. Dreiseitl

24. The catch... • How many dimensions does have? • For degree-k polynomial expansions: • E.g., for k=4, d=256 (16x16 images), • Yike! • For “radial basis functions”,

25. Linear surfaces for cheap • Can’t directly find linear surfaces in • Have to find a clever “method” for finding them indirectly • It’ll take (quite) a bit of work to get there... • Will need different criterion than • We’ll look for the “maximum margin” classifier • Surface s.t. class 1 (“true”) data falls as possible on one side; class -1 (“false”) falls as far as possible on the other

26. Max margin hyperplanes Hyperplane Margin

27. Max margin is unique Hyperplane Margin

28. Back to SVMs & margins • The margins are parallel to hyperplane, so are defined by same w, plus constant offsets b b w

29. Back to SVMs & margins • The margins are parallel to hyperplane, so are defined by same w, plus constant offsets • Want to ensure that all data points are “outside” the margins b b w

30. Maximizing the margin • So now we have a learning criterion function: • Pick w to maximize b s.t. all points still satisfy • Note: w.l.o.g. can rescale w arbitrarily (why?)

31. Maximizing the margin • So now we have a learning criterion function: • Pick w to maximize b s.t. all points still satisfy • Note: w.l.o.g. can rescale w arbitrarily (why?) • So can formulate full problem as: • Minimize: • Subject to: • But how do you do that? And how does this help?

32. Quadratic programming • Problems of the form • Minimize: • Subject to: • are called “quadratic programming” problems • There are off-the-shelf methods to solve them • Actually solving this is way, way beyond the scope of this class • Consider it a black box • If a solution exists, it will be found & be unique • Expensive, but not intractably so

33. w Nonseparable data • What if the data isn’t linearly separable? • Project into higher dim space (we’ll get there) • Allow some “slop” in the system • Allow margins to be violated “a little”

34. The new “slackful” QP • The are “slack variables” • Allow margins to be violated a little • Still want to minimize margin violations, so add them to QP instance: • Minimize: • Subject to:

35. You promised nonlinearity! • Where did the nonlinear transform go in all this? • Another clever trick • With a little algebra (& help from Lagrange multipliers), can rewrite our QP in the form: • Maximize: • Subject to:

36. Kernel functions • So??? It’s still the same linear system • Note, though, that appears in the system only as a dot product:

37. Kernel functions • So??? It’s still the same linear system • Note, though, that appears in the system only as a dot product: • Can replace with : • The inner product • is called a “kernel function”

38. Why are kernel fns cool? • The cool trick is that many useful projections can be written as kernel functions in closed form • I.e., can work with K() rather than • If you know K(xi,xj) for every (i,j) pair, then you can construct the maximum margin hyperplane between the projected data without ever explicitly doing the projection!

39. Example kernels • Homogeneous degree-k polynomial:

40. Example kernels • Homogeneous degree-k polynomial: • Inhomogeneous degree-k polynomial:

41. Example kernels • Homogeneous degree-k polynomial: • Inhomogeneous degree-k polynomial: • Gaussian radial basis function:

42. Example kernels • Homogeneous degree-k polynomial: • Inhomogeneous degree-k polynomial: • Gaussian radial basis function: • Sigmoidal (neural network):

43. Side note on kernels • What precisely do kernel functions mean? • Metric functions take two points and return a (generalized) distance between them • What is the equivalent interpretation for kernels? • Hint: think about what kernel function replaces in the max margin QP formulation

44. Side note on kernels • Kernel functions are generalized inner products • Essentially, give you the cosine of the angle between vectors • Recall the law of cosines:

45. Side note on kernels • Replace traditional dot product with “generalized inner product” and get:

46. Side note on kernels • Replace traditional dot product with “generalized inner product” and get: • Kernel (essentially) represents: • Angle between vectors in the projected, high-dimensional space

47. Side note on kernels • Replace traditional dot product with “generalized inner product” and get: • Kernel (essentially) represents: • Angle between vectors in the projected, high-dimensional space • Alternatively: • Nonlinear distribution of angles in low-dim space

48. Example of Kernel nonlin.

49. Example of Kernel nonlin.

50. Using the classifier • Solution of the QP gives back a set of • Data points for which are called “support vectors” • Turns out that we can write w as