Theoretical Foundations of Active Learning

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1 Theoretical Foundations of Active Learning Steve Hanneke Machine Learning Department Carnegie Mellon University

2 Passive Learning Learning Algorithm Data Source Expert / Oracle Labeled data points Algorithm outputs a classifier Steve Hanneke 2

3 Active Learning Learning Algorithm Data Source Request for the label of a data point The label of that point Request for the label of another data point The label of that point Expert / Oracle... Algorithm outputs a classifier Steve Hanneke 3

4 Active Learning (Sequential Design) Learning Algorithm Data Source Expert / Oracle How many label requests Request for the label of a data point The label of that point are required to learn? Request for the label of another data point The label of that point Label Complexity... Algorithm outputs a classifier e.g., Das04, Das05, DKM05, BBL06, Kaa06, Han07a&b, BBZ07, DHM07, BHW08 Steve Hanneke 4

5 Active Learning Sometimes Helps An Example: 1-dimensional threshold functions. - + Steve Hanneke 5

6 Active Learning Sometimes Helps An Example: 1-dimensional threshold functions. Take m unlabeled examples Repeatedly request the label of the median point between -/+ boundaries. Take any threshold consistent with the observed labels Used only log(m) label requests, but get a classifier consistent with all m examples! Exponential improvement over passive! Steve Hanneke 6

7 Outline Formal Model Analysis of Uncertainty-based Active Learning Strict Improvements Over Passive Learning Open Problems Steve Hanneke 7

8 Formal Model Steve Hanneke 8

9 Formal Model Steve Hanneke 9

10 CAL A simple idea from Cohn, Atlas & Ladner (1994). Assuming ν=0, produces a perfectly labeled data set, which we can feed into any passive algorithm! So we get a natural fallback guarantee. Can we characterize the label complexity achieved by CAL? Can we generalize it to handle label noise or non-separable data? Steve Hanneke 10

11 Disagreement Coefficient [Hanneke,07] (for our purposes, take r 0 = ε) DIS(B(f,r)) f Concepts in B(f,r) look like this Steve Hanneke 11

12 Disagreement Coefficient [Hanneke,07] (for our purposes, take r 0 = ε) Steve Hanneke 12

13 θ Characterizes CAL s Performance Steve Hanneke 13

14 What about Noise? Steve Hanneke 14

15 What about Noise? Steve Hanneke 15

16 Activized Learning Activizer Meta-algorithm Data Source Expert / Oracle Request for the label of a data point The label of that point Request for the label of another data point The label of that point Algorithm outputs a classifier Passive Learning Algorithm (Supervised / Semi-Supervised) Steve Hanneke 16

17 Activized Learning Activizer Meta-algorithm Data Source Expert / Oracle Request for the label of a data point The label of that point Request for the label of another data point The label of that point Algorithm outputs a classifier Passive Learning Algorithm (Supervised / Semi-Supervised) Are there general-purpose activizers that strictly improve the label complexity of any passive algorithm? Steve Hanneke 17

18 Formal Model Steve Hanneke 18

19 Uncertainty-based Sampling Doesn t Activize Intervals Steve Hanneke 19

20 Uncertainty-based Sampling Doesn t Activize Intervals Suppose the target labels everything -1 1 Uncertainty-based sampling requests every label. No improvements over passive. Steve Hanneke 20

21 What s Wrong? (formally) Steve Hanneke 21

22 How Can We Fix It? Steve Hanneke 22

23 A Simple Activizer So, which ever of the 2 k classifications can t be realized by V, look at the label of x and take the opposite. Steve Hanneke 23

24 This Works for Any C! [HLW94] passive algorithm has O(1/ε) sample complexity. Steve Hanneke 24

25 Dealing with Noise and Misspecification Recall passive gets O(1/ε 2 ) (minimax) Steve Hanneke 25

26 Open Questions Question: What can we activize with noise? Question: Can we give more detailed bounds on Λ a when θ>>1? Question: Is there a labeled/unlabeled trade-off under arbitrary D XY? Steve Hanneke 26

27 Thank You Steve Hanneke 27

28 A Simple Activizer Intervals revisited Again, suppose the target labels everything -1 Passive algorithm trained on Ω(n 2 ) samples. Improved label complexity. x 1 1 Steve Hanneke 28

29 Efficiency? m = # unlabeled examples used by the algorithm. Suppose can test separability of O(n) points in poly(n) time Then SimpleActivizer runs in poly(n)m time (plus the time of the passive algorithm). For most learning problems, can set a poly(n) limit on m in the algorithm without losing our guarantees. Steve Hanneke 29

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