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Probably Approximately Correct Model (PAC)

Probably Approximately Correct Model (PAC). Example (PAC). Concept: Average body-size person Inputs: for each person: height weight Sample: labeled examples of persons label + : average body-size label - : not average body-size Two dimensional inputs. Example (PAC).

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Probably Approximately Correct Model (PAC)

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  1. Probably Approximately Correct Model (PAC)

  2. Example (PAC) • Concept: Average body-size person • Inputs: for each person: • height • weight • Sample: labeled examples of persons • label + : average body-size • label - : not average body-size • Two dimensional inputs

  3. Example (PAC) • Assumption: target concept is a rectangle. • Goal: • Find a rectangle that “approximate” the target. • Formally: • With high probability • output a rectangle such that • its error is low.

  4. Example (Modeling) • Assume: • Fixed distribution over persons. • Goal: • Low error with respect to THIS distribution!!! • How does the distribution look like? • Highly complex. • Each parameter is not uniform. • Highly correlated.

  5. Model Based approach • First try to model the distribution. • Given a model of the distribution: • find an optimal decision rule. • Bayesian Learning

  6. PAC approach • Assume that the distribution is fixed. • Samples are drawn are i.i.d. • independent • identical • Concentrate on the decision rule rather than distribution.

  7. PAC Learning • Task: learn a rectangle from examples. • Input: point (x,y) and classification + or - • classifies by a rectangle R • Goal: • in the fewest examples • compute R’ • R’ is a good approximation for R

  8. PAC Learning: Accuracy • Testing the accuracy of a hypothesis: • using the distribution D of examples. • Error = R D R’ • Pr[Error] = D(Error) = D(R D R’) • We would like Pr[Error] to be controllable. • Given a parameter e: • Find R’ such that Pr[Error] < e.

  9. PAC Learning: Hypothesis • Which Rectangle should we choose?

  10. Setting up the Analysis: • Choose smallest rectangle. • Need to show: • For any distribution D and Rectangle R • input parameters: e and d • Select m(e,d) examples. • Let R’ be the smallest consistent rectangle. • With probability 1-d: • D(R D R’) < e

  11. T’u Tu Analysis • Note that R’  R, therefore R D R’ = R - R’ R’ R

  12. Analysis (cont.) • By Definition: D(Tu)= e/4 • Compute the probability that:T’u  Tu • PrD[(x,y) in Tu]= e/4 • Probability of NO example in Tu • For m examples: (1-e/4)m < e-e m/4 • Failure probability: 4 e-e m/4 <d • Sample bound: m > (4/e) ln (4/d)

  13. PAC: comments • We only assumed that examples are i.i.d. • We have two independent parameters: • Accuracy e • Confidence d • No assumption about the likelihood of rectangles. • Hypothesis is tested on the same distribution as the sample.

  14. PAC model: Setting • A distribution: D (unknown) • Target function: ct from C • ct : X  {0,1} • Hypothesis: h from H • h: X  {0,1} • Error probability: • error(h) = ProbD[h(x) ct(x)] • Oracle: EX(ct,D)

  15. PAC Learning: Definition • C and H are concept classes over X. • C is PAC learnable by H if • There Exist an Algorithm A such that: • For any distribution D over X and ct in C • for every input eand d: • outputs a hypothesis h in H, • while having access to EX(ct,D) • with probability 1-d we have error(h) < e • Complexities.

  16. Finite Concept class • Assume C=H and finite. • h is e-bad if error(h)> e. • Algorithm: • Sample a set S of m(e,d) examples. • Find h in H which is consistent. • Algorithm fails if h is e-bad.

  17. Analysis • Assume hypothesis g is e-bad. • The probability that g is consistent: • Pr[g consistent]  (1-e)m < e- em • The probability thatthere exists: • g is e-bad and consistent: • |H| Pr[g consistent and e-bad]  |H| e- em • Sample size: • m > (1/e) ln (|H|/d)

  18. PAC: non-feasible case • What happens if ct not in H • Needs to redefine the goal. • Let h* in H minimize the error b=error(h*) • Goal: find h in H such that • error(h)  error(h*) +e = b+e

  19. Analysis • For each h in H: • let obs-error(h) be the error on the sample S. • Compute the probability that: • |obs-error(h) - error(h) | < e/2 • Chernoff bound: exp(-(e/2)2m) • Consider entire H : |H| exp(-(e/2)2m) • Sample size • m > (4/e2) ln (|H|/d)

  20. Correctness • Assume that for all h in H: • |obs-error(h) - error(h) | < e/2 • In particular: • obs-error(h*) < error(h*) + e/2 • error(h) -e/2 < obs-error(h) • For the output h: • obs-error(h) < obs-error(h*) • Conclusion: error(h) < error(h*)+e

  21. Example: Learning OR of literals • Inputs: x1, … , xn • Literals : x1, • OR functions: • Number of functions? 3n

  22. ELIM: Algorithm for learning OR • Keep a list of all literals • For every example whose classification is 0: • Erase all the literals that are 1. • Example • Correctness: • Our hypothesis h: An OR of our set of literals. • Our set of literals includes the target OR literals. • Every time h predicts zero: we are correct. • Sample size: m > (1/e) ln (3n/d)

  23. Learning parity • Functions: x1  x7 x9 • Number of functions: 2n • Algorithm: • Sample set of examples • Solve linear equations • Sample size: m > (1/e) ln (2n/d)

  24. max min Infinite Concept class • X=[0,1] and H={cq | q in [0,1]} • cq(x) = 0 iff x < q • Assume C=H: • Which cq should we choose in [min,max]?

  25. Proof I • Show that the probability that • Pr[ D([min,max]) > e ] < d • Proof: By Contradiction. • The probability that x in [min,max] at least e • The probability we do not sample from [min,max] Is (1-e)m • Needs m > (1/e) ln (1/d) What’s WRONG ?!

  26. Proof II (correct): • Let max’ be : D([q,max’])=e/2 • Let min’ be : D([q,min’])=e/2 • Goal: Show that with high probability • X+ in [max’,q] and • X- in [q,min’] • In such a case any value in [x-,x+] is good. • Compute sample size!

  27. Non-Feasible case • Suppose we sample: • Algorithm: • Find the function h with lowest error!

  28. Analysis • Define: zi as a e/4 - net (w.r.t. D) • For the optimal h* and our h there are • zj : |error(h[zj]) - error(h*)| < e/4 • zk : |error(h[zk]) - error(h)| < e/4 • Show that with high probability: • |obs-error(h[zi]) -error(h[zi])| < e/4 • Completing the proof. • Computing the sample size.

  29. General e-net approach • Given a class H define a class G • For every h in H • There exist a g in G such that • D(g D h) < e/4 • Algorithm: Find the best h in H. • Computing the confidence and sample size.

  30. Occam Razor • Finding the shortest consistent hypothesis. • Definition: (a,b)-Occam algorithm • a >0 and b <1 • Input: a sample S of size m • Output: hypothesis h • for every (x,b) in S: h(x)=b • size(h) < sizea(ct) mb • Efficiency.

  31. Occam algorithm and compression A B S (xi,bi) x1, … , xm

  32. compression • Option 1: • A sends B the values b1 , … , bm • m bits of information • Option 2: • A sends B the hypothesis h • Occam: large enough m has size(h) < m • Option 3 (MDL): • A sends B a hypothesis h and “corrections” • complexity: size(h) + size(errors)

  33. Occam Razor Theorem • A: (a,b)-Occam algorithm for C using H • D distribution over inputs X • ct in C the target function, n=size(ct) • Sample size: • with probability 1-d A(S)=h has error(h) < e

  34. Occam Razor Theorem • Use the bound for finite hypothesis class. • Effective hypothesis class size 2size(h) • size(h) < na mb • Sample size:

  35. Learning OR with few attributes • Target function: OR of k literals • Goal: learn in time: • polynomial in k and log n • e and d constant • ELIM makes “slow” progress • disqualifies one literal per round • May remain with O(n) literals

  36. Set Cover - Definition • Input: S1 , … , St and Si U • Output: Si1, … , Sik and j Sjk=U • Question: Are there k sets that cover U? • NP-complete

  37. Set Cover Greedy algorithm • j=0 ; Uj=U; C= • While Uj  • Let Si be arg max |Si Uj| • Add Si to C • Let Uj+1 = Uj – Si • j = j+1

  38. Set Cover: Greedy Analysis • At termination, C is a cover. • Assume there is a cover C’ of size k. • C’ is a cover for every Uj • Some S in C’ covers Uj/k elements of Uj • Analysis of Uj: |Uj+1|  |Uj| - |Uj|/k • Solving the recursion. • Number of sets j < k ln |U|

  39. Building an Occam algorithm • Given a sample S of size m • Run ELIM on S • Let LIT be the set of literals • There exists k literals in LIT that classify correctly all S • Negative examples: • any subset of LIT classifies theme correctly

  40. Building an Occam algorithm • Positive examples: • Search for a small subset of LIT • Which classifies S+ correctly • For a literal z build Tz={x | z satisfies x} • There are k sets that cover S+ • Find k ln m sets that cover S+ • Output h = the OR of the k ln m literals • Size (h) < k ln m log 2n • Sample size m =O( k log n log (k log n))

  41. Summary • PAC model • Confidence and accuracy • Sample size • Finite (and infinite) concept class • Occam Razor

  42. Learning algorithms • OR function • Parity function • OR of a few literals • Open problems • OR in the non-feasible case • Parity of a few literals

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