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Agnostically Learning Decision Trees

0. 1. X 2. X 3. 0. 1. 0. 1. 0. 1. 1. 0. Parikshit Gopalan MSR-Silicon Valley , IITB’00. Adam Tauman Kalai MSR-New England Adam R. Klivans UT Austin. Agnostically Learning Decision Trees. 1. 0. X 1. 0. 1. 1. 0. 0. 1. Computational Learning. Computational Learning.

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Agnostically Learning Decision Trees

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  1. 0 1 X2 X3 0 1 0 1 0 1 1 0 Parikshit GopalanMSR-Silicon Valley, IITB’00.Adam Tauman Kalai MSR-New EnglandAdam R. KlivansUT Austin Agnostically Learning Decision Trees 1 0 X1 0 1 1 0 0 1

  2. Computational Learning

  3. Computational Learning

  4. Computational Learning f:{0,1}n! {0,1} x, f(x) Learning: Predict f from examples.

  5. Valiant’s Model f:{0,1}n! {0,1} Halfspaces: + + + + + - + x, f(x) - + - - - + - - - - - - Assumption: f comes from a nice concept class.

  6. 0 1 X2 X3 0 1 0 1 0 1 1 0 Valiant’s Model f:{0,1}n! {0,1} Decision Trees: X1 x, f(x) Assumption: f comes from a nice concept class.

  7. X1 0 1 X2 X3 0 1 0 1 0 1 1 0 The Agnostic Model [Kearns-Schapire-Sellie’94] f:{0,1}n! {0,1} Decision Trees: x, f(x) No assumptions about f. Learner should do as well as best decision tree.

  8. X1 0 1 X2 X3 0 1 0 1 0 1 1 0 The Agnostic Model [Kearns-Schapire-Sellie’94] Decision Trees: x, f(x) No assumptions about f. Learner should do as well as best decision tree.

  9. X1 0 1 X2 X3 0 1 0 1 0 1 1 0 Agnostic Model = Noisy Learning f:{0,1}n! {0,1} + = • Concept: Message • Truth table: Encoding • Function f: Received word. • Coding: Recover the Message. • Learning:Predict f.

  10. Uniform Distribution Learning for Decision Trees • Noiseless Setting: • No queries:nlog n[Ehrenfeucht-Haussler’89]. • With queries:poly(n). [Kushilevitz-Mansour’91] Agnostic Setting: Polynomial time, uses queries. [G.-Kalai-Klivans’08] Reconstruction for sparse real polynomials in the l1 norm.

  11. The Fourier Transform Method • Powerful tool for uniform distribution learning. • Introduced by Linial-Mansour-Nisan. • Small depth circuits[Linial-Mansour-Nisan’89] • DNFs[Jackson’95] • Decision trees[Kushilevitz-Mansour’94, O’Donnell-Servedio’06, G.-Kalai-Klivans’08] • Halfspaces, Intersections[Klivans-O’Donnell-Servedio’03, Kalai-Klivans-Mansour-Servedio’05] • Juntas[Mossel-O’Donnell-Servedio’03] • Parities[Feldman-G.-Khot-Ponnsuswami’06]

  12. The Fourier Polynomial • Let f:{-1,1}n! {-1,1}. • Write f as a polynomial. • AND:½ + ½X1 + ½X2 - ½X1X2 • Parity:X1X2 • Parity of ½ [n]: (x) = i 2Xi • Write f(x) =  c()(x) •  c()2 =1. Standard Basis Function f Parities

  13. The Fourier Polynomial • Let f:{-1,1}n! {-1,1}. • Write f as a polynomial. • AND:½ + ½X1 + ½X2 - ½X1X2 • Parity:X1X2 • Parity of ½ [n]: (x) = i 2Xi • Write f(x) =  c()(x) •  c()2 =1. c()2: Weight of . 

  14. Low Degree Functions • Sparse Functions: Most of the weight lies on small subsets. • Halfspaces, Small-depth circuits. • Low-degree algorithm. [Linial-Mansour-Nisan] • Finds the low-degree Fourier coefficients. Least Squares Regression: Find low-degree P minimizing Ex[ |P(x) – f(x)|2 ].

  15. Sparse Functions • Sparse Functions: Most of the weight lies on a few subsets. • Decision trees. t leaves )O(t) subsets • Sparse Algorithm. • [Kushilevitz-Mansour’91] Sparse l2 Regression: Find t-sparse P minimizing Ex[ |P(x) – f(x)|2 ].

  16. Sparse l2 Regression • Sparse Functions: Most of the weight lies on a few subsets. • Decision trees. t leaves )O(t) subsets • Sparse Algorithm. • [Kushilevitz-Mansour’91] Sparse l2 Regression: Find t-sparse P minimizing Ex[ |P(x) – f(x)|2 ]. Finding large coefficients: Hadamard decoding. [Kushilevitz-Mansour’91, Goldreich-Levin’89]

  17. f:{-1,1}n! {-1,1} +1 -1 Agnostic Learning via l2 Regression?

  18. X1 0 1 X2 X3 +1 0 1 0 1 0 1 1 0 -1 Agnostic Learning via l2 Regression?

  19. +1 -1 Agnostic Learning via l2 Regression? Target f Best Tree • l2 Regression: Loss |P(x) –f(x)|2 • Pay 1 for indecision. • Pay 4 for a mistake. • l1 Regression: [KKMS’05] • Loss |P(x) –f(x)| • Pay 1 for indecision. • Pay 2 for a mistake.

  20. +1 -1 Agnostic Learning via l1 Regression? • l2 Regression: Loss |P(x) –f(x)|2 • Pay 1 for indecision. • Pay 4 for a mistake. • l1 Regression: [KKMS’05] • Loss |P(x) –f(x)| • Pay 1 for indecision. • Pay 2 for a mistake.

  21. +1 -1 Agnostic Learning via l1 Regression Target f Best Tree Thm [KKMS’05]:l1 Regression always gives a good predictor. l1 regression for low degree polynomials via Linear Programming.

  22. Agnostically Learning Decision Trees Sparse l1 Regression: Find a t-sparse polynomial P minimizing Ex[ |P(x) – f(x)|]. • Why is this Harder: • l2 is basis independent, l1 is not. • Don’t know the support of P. • [G.-Kalai-Klivans]: Polynomial time algorithm for Sparse l1Regression.

  23. The Gradient-Projection Method L1(P,Q) =  |c() – d()| L2(P,Q) = [ (c() –d())2]1/2 f(x) P(x) =  c() (x) Q(x) =  d() (x) Variables:c()’s. Constraint: |c() | · t Minimize:Ex|P(x) – f(x)|

  24. Gradient The Gradient-Projection Method Variables:c()’s. Constraint: |c() | · t Minimize:Ex|P(x) – f(x)|

  25. Gradient The Gradient-Projection Method Projection Variables:c()’s. Constraint: |c() | · t Minimize:Ex|P(x) – f(x)|

  26. +1 -1 The Gradient • g(x) = sgn[f(x) - P(x)] • P(x) := P(x) +  g(x). f(x) P(x) Increase P(x) if low. Decrease P(x) if high.

  27. Gradient The Gradient-Projection Method Variables:c()’s. Constraint: |c() | · t Minimize:Ex|P(x) – f(x)|

  28. Gradient The Gradient-Projection Method Projection Variables:c()’s. Constraint: |c() | · t Minimize:Ex|P(x) – f(x)|

  29. Projection onto the L1 ball Currently: |c()| > t Want:|c()| · t.

  30. Projection onto the L1 ball Currently: |c()| > t Want:|c()| · t.

  31. Projection onto the L1 ball • Below cutoff:Set to 0. • Above cutoff:Subtract.

  32. Projection onto the L1 ball • Below cutoff:Set to 0. • Above cutoff:Subtract.

  33. Analysis of Gradient-Projection [Zinkevich’03] • Progress measure:Squared L2 distance from optimum P*. Key Equation: |Pt – P*|2 - |Pt+1 – P*|2¸ 2 (L(Pt) – L(P*)) • Within  of optimal in 1/2 iterations. • Good L2 approximation to Pt suffices. – 2 Progress made in this step. How suboptimal current soln is.

  34. +1 -1 Gradient f(x) • g(x) = sgn[f(x) - P(x)]. P(x) Projection

  35. +1 -1 The Gradient • g(x) = sgn[f(x) - P(x)]. f(x) P(x) • Compute sparse approximation g’ = KM(g). • Is g a good L2 approximation to g’? • No. Initially g = f. • L2(g,g’) can be as large 1.

  36. Sparse l1 Regression Approximate Gradient Variables:c()’s. Constraint: |c() | · t Minimize:Ex|P(x) – f(x)|

  37. Sparse l1 Regression Projection Compensates Variables:c()’s. Constraint: |c() | · t Minimize:Ex|P(x) – f(x)|

  38. KM as l2 Approximation The KM Algorithm: Input:g:{-1,1}n! {-1,1}, and t. Output: A t-sparse polynomial g’ minimizing Ex [|g(x) – g’(x)|2] Run Time: poly(n,t).

  39. KM as L1 Approximation The KM Algorithm: Input: A Boolean function g =  c()(x). A error bound . Output: Approximation g’ =  c’()(x) s.t |c() – c’()| · for all½ [n]. Run Time: poly(n,1/)

  40. KM as L1 Approximation Only 1/2 • Identify coefficients larger than. • Estimate via sampling, set rest to 0.

  41. KM as L1 Approximation • Identify coefficients larger than. • Estimate via sampling, set rest to 0.

  42. Projection Preserves L1 Distance L1distance at most 2after projection. Both lines stop within  of each other.

  43. Projection Preserves L1 Distance L1distance at most 2after projection. Both lines stop within  of each other. Else, Blue dominates Red.

  44. Projection Preserves L1 Distance L1distance at most 2after projection. Projecting onto the L1 ball does not increase L1 distance.

  45. Sparse l1 Regression • L1(P, P’) · 2 • L1(P, P’) · 2t • L2(P, P’)2· 4t P’ P Can take  = 1/t2. Variables:c()’s. Constraint: |c() | · t Minimize:Ex|P(x) – f(x)|

  46. Agnostically Learning Decision Trees Sparse L1 Regression: Find a sparse polynomial P minimizing Ex[ |P(x) – f(x)|]. • [G.-Kalai-Klivans’08]: • Can get within  of optimum in poly(t,1/) iterations. • Algorithm for Sparsel1Regression. • First polynomial time algorithm for Agnostically Learning Sparse Polynomials.

  47. l1 Regression from l2 Regression Functionf: D ! [-1,1],Orthonormal BasisB. Sparse l2 Regression: Find a t-sparse polynomial P minimizing Ex[ |P(x) – f(x)|2 ]. Sparse l1 Regression: Find a t-sparse polynomial P minimizing Ex[ |P(x) – f(x)|]. • [G.-Kalai-Klivans’08]:Given solution to l2 Regression, can solve l1 Regression.

  48. Agnostically Learning DNFs? • Problem: Can we agnostically learn DNFs in polynomial time? (uniform dist. with queries) • Noiseless Setting: Jackson’s Harmonic Sieve. • Implies weak learner for depth-3 circuits. • Beyond current Fourier techniques. Thank You!

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