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Computational Learning Theory and Kernel Methods

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Computational Learning Theory and Kernel Methods

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  1. Computational Learning Theoryand Kernel Methods Tianyi Jiang March 8, 2004

  2. General Research Question “Under what conditions is successful learning possible and impossible?” “Under what conditions is a particular learning algorithm assured of learning successfully?” -Mitchell, ‘97

  3. Computational Learning Theory • Sample Complexity • Computational Complexity • Mistake Bound • -Mitchell, ‘97

  4. Problem Setting Instance Space: X, with a stable distribution D Concept Class: C, s.t. c: X  {0,1} Hypothesis Space: H General Learner: L

  5. h - - - c + + Where c and h disagree Error of a Hypothesis

  6. PAC Learnability True Error: • Difficulties in getting 0 error: • Multiple hypothesis consistent with training examples • Training examples can mislead the Learner

  7. PAC-Learnable Learner L will output a hypothesis h with probability (1-) s.t. in time that is polynomial in where n = size of a training example size(c) = encoding length of c in C

  8. Consistent Learner & Version Space Consistent Learner – Outputs hypotheses that perfectly fit the training data whenever possible Version Space: VSH,E is -exhausted with respect to c and Dif:

  9. Version Space Hypothesis space H ( =.21) . error=.3 r=.4 . error=.1 r=.2 . error=.2 r=0 VSH,E . error=.1 r=0 . error=.3 r=.1 . error=.2 r=.3

  10. Sample Complexity for Finite Hypothesis Spaces Theorem - -exhausting the version space: If H is finite, the probability that VSH,D is NOT -exhausted (with respect to c) is: |H|e- m where m1, sequence of i.r.d. examples of some target concept c; 0   1

  11. Upper bound on sufficient number of training examples If we set probability of failure below some level,  then… … however, too loose of a bound due to |H|

  12. Agnostic Learning What if concept c  H? Agnostic Learner: simply finds the h with min. training error Find upper bound on m s.t. Where hbest = h with lowest training error

  13. Upper bound on sufficient number of training examples - errorE(hbest) 0 From Chernoff Bounds, we have: then… thus…

  14. Example: Given a consistent learner and a target concept of conjunctions of up to 10 Boolean literals, how many training examples are needed to learn a hypothesis with error < .1 95% of the time? |H|=? =? =?

  15. Example: Given a consistent learner and a target concept of conjunctions of up to 10 Boolean literals, how many training examples are needed to learn a hypothesis with error < .1 95% of the time? |H|=310 =.1 =.05

  16. Sample Complexity for Infinite Hypothesis Spaces Consider subset of instances: S  X, and h  H s.t. h imposed dichotomy on S: 2 subsets: {x  S | h(x)=1 } & {x  S | h(x)=0 } Thus for any instance set S, there are 2|S| possible dichotomies. Definition: A set of instance S is shattered by hypothesis space H iff for every dichotomy of S there exist some h  H consistent with this dichotomy

  17. 3 Instances Shattered by 8 Hypotheses Instance Space X

  18. Vapnik-Chervonenkis Dimension Definition: VC(H), is the size of the largest finite subset of X shattered by H. If arbitrarily large finite sets of X can be shattered by H, then VC(H)= For any finite H, VC(H)  log2|H|

  19. Example of VC Dimension Along a line… In a plane…

  20. VC Dimension Example 2

  21. VC dimensions in Rn Theorem: Consider some set of m points in Rn. Choose any one of the points as origin. Then the m points can be shattered by oriented hyperplanes iff the position vectors of the remaining points are linearly independent. So VC dimension of the set of oriented hyperplanes in R10 is ?

  22. Bounds on m with VC Dimension Upper Bound: VC(H)  log2|H| Lower Bound:

  23. Mistake Bound Model of Learning “How many mistakes will the learner make in its predictions before it learns the target concept?” The best algorithm in worst case scenario (hardest target concept, hardest training sequence) will make Opt(C) mistakes, where

  24. Linear Support Vector Machines Consider a binary classification problem: Training data: {xi, yi}, i=1,…,; yi {-1, +1}; xi  Rd Points x lie on the separating hyperplane satisfy: wx+b=0 where w is normal to the hyperplane |b|/||w|| is the perpendicular distance to origin ||w|| is the Euclidean norm of w

  25. Linear Support Vector Machine, Definitions Let d+ (d-) be the shortest distance from the separating hyperplane to the closest positive (negative) example Margin of a separating hyperplane= d+ + d- =1/||W||+1/||W||=2/||w|| Constraints:

  26. Linear Separating Hyperplane for the Separable Case

  27. Problem of Maximizing the Margins H1 and H2 are parallel, & with no training points in between Thus we reformulate the problem as: Maximize margin by minimizing ||W||2 s.t.

  28. Ties to Least Squares y  b x Loss Function:

  29. Lagrangian Formulation • Transform constraints into Lagrange multipliers • Training data will only appear in dot products form • Let be positive Lagrange multipliers We have the Lagrangian:

  30. Transform the convex quadratic programming problem Observations: minimizing LP w.r.t. w, b, and simultaneously require that subject to is a convex quadratic programming problem that can be easily solved in its Dual form

  31. Transform the convex quadratic programming problem – the Dual LP’s Dual: Maximize LP, subject to gradients of LP w.r.t. w and b vanish, and i0

  32. Observations about the Dual • There is a Lagrangian multiplier i for every training point • In the solution, points for which i > 0 are called “support • vectors”. They lie on either H1 or H2 • Support vectors are critical elements of the training set, • they lie closest to the “boundary” • If all other points are removed or moved around (but not • crossing H1 or H2), the same separating hyperplane would • be found

  33. Prediction • Solving the SVM problem is equivalent to finding a • solution for the Karush-Kuhn-Tucker (KTT) conditions • (KTT conditions are satisfied at the solution of any • constrained optimization problem) • Once we solved for w, b, we predict x to be • sign(wx+b)

  34. Linear SVM: The Non-Separable Case We account for outliers by introducing slack conditions: We penalize outliers by changing the cost function to:

  35. Example of Linear SVM with slacks

  36. Linear SVM Classification Examples Linearly Separable Linearly Non-Separable

  37. Nonlinear SVM Observation: data appear as dot products in the training problem So we can use a mapping function , to map data into a high dimensional space where points are linearly separable: To make things easier, we define a kernel function K s.t.

  38. Nonlinear SVM (cont.) Kernel functions can compute dot products in the high dimensional space without explicitly work with  Example: Rather than computing w, we make prediction on x via:

  39. Example of  mapping Image, in , of the square [-1,1]x[-1,1]  R2 under the mapping 

  40. Example Kernel Functions Kernel functions must satisfy the Mercer’s condition, or simple, the Hessian Matrix must be positive semidefinite. (non-negative eigenvalues) Example Kernels:

  41. Nonlinear SVM Classification Examples (Degree 3 Polynomial Kernel) Linearly Separable Linearly Non-Separable

  42. Multi-Class SVM • One-against-all • One-against-one (majority vote) • One-against-one (DAGSVM)

  43. Global Solution and Uniqueness • Every local solution is also global (property of any • convex programming problem) • Solution is guaranteed unique if the objective function • is strictly convex (Hessian matrix is positive definite)

  44. Complexity and Scalability • Curse of dimensionality: • The proliferation of parameters causing • intractable complexity • The proliferation of parameters causing overfitting • SVM circumvent these via the use of • Kernel functions (trick) that computes at O(dL) • Support vectors that focus on the “boundary”

  45. Structural Risk Minimization Empirical Risk: Expected Risk:

  46. Structural Risk Minimization Nested subsets of functions, ordered by VC dimensions