An Analysis of Graph Cut Size for Transductive Learning

An Analysis of Graph Cut Size for Transductive Learning Steve Hanneke Machine Learning Department Carnegie Mellon University

Outline ,[object Object],[object Object],[object Object],MACHINE LEARNING DEPARTMENT 1

Transductive Learning MACHINE LEARNING DEPARTMENT Labeled Training Data Classifier Predictions Unlabeled Test Data Inductive Learning Distribution iid iid Data Random Split Labeled Training Data Unlabeled Test Data Predictions Transductive Learning 2

Vertex Labeling in Graphs ,[object Object],[object Object],[object Object],[object Object],MACHINE LEARNING DEPARTMENT 3

Example: Data with Similarity ,[object Object],[object Object],[object Object],MACHINE LEARNING DEPARTMENT 2 2 2 1 1 3 4

Example: Social Networks ,[object Object],3 MACHINE LEARNING DEPARTMENT 2 2 2 1 1 5

Adjacency ,[object Object],MACHINE LEARNING DEPARTMENT ,[object Object],[object Object],? = 2 2 2 1 1 3 6

Cut Size ,[object Object],MACHINE LEARNING DEPARTMENT 2 2 2 1 1 3 Example: Cut Size 2 7

Learning Algorithms ,[object Object],[object Object],[object Object],[object Object],[object Object],MACHINE LEARNING DEPARTMENT 8

Mincut (Blum & Chawla, 2001) ,[object Object],[object Object],[object Object],[object Object],MACHINE LEARNING DEPARTMENT 9

Error Bounds ,[object Object],[object Object],MACHINE LEARNING DEPARTMENT 10

Bounding a Single Labeling ,[object Object],[object Object],MACHINE LEARNING DEPARTMENT 11

Bounding a Single Labeling ,[object Object],[object Object],[object Object],MACHINE LEARNING DEPARTMENT 12

The PAC-MDL Perspective ,[object Object],[object Object],[object Object],[object Object],MACHINE LEARNING DEPARTMENT 13

The Structural Risk Trick ,[object Object],MACHINE LEARNING DEPARTMENT 14 H S 0 S 1 S 2 S 3 S c S |E|   /(|E|+1)  /(|E|+1)  /(|E|+1)  /(|E|+1)  /(|E|+1)  /(|E|+1) S c =labelings with cut size c. . . . . . .

The Structural Risk Trick ,[object Object],MACHINE LEARNING DEPARTMENT 15 S c  /(|E|+1) h c1 h c2 h c3 h c4 h ci h cSc

The Structural Risk Trick ,[object Object],[object Object],[object Object],MACHINE LEARNING DEPARTMENT 16

Bounding |S c | ,[object Object],[object Object],[object Object],MACHINE LEARNING DEPARTMENT 17

Minimum k-Cut Size ,[object Object],[object Object],MACHINE LEARNING DEPARTMENT 18

A Tighter Bound on |S c | ,[object Object],[object Object],[object Object],MACHINE LEARNING DEPARTMENT 19

Error Bounds ,[object Object],[object Object],MACHINE LEARNING DEPARTMENT 20 (can be slightly improved: see the paper)

Error Bounds ,[object Object],[object Object],[object Object],[object Object],MACHINE LEARNING DEPARTMENT 21

Visualizing the Bounds ,[object Object],[object Object],MACHINE LEARNING DEPARTMENT 22

Conclusions & Open Problems ,[object Object],[object Object],MACHINE LEARNING DEPARTMENT 23

[object Object],MACHINE LEARNING DEPARTMENT An Analysis of Graph Cut Size for Transductive Learning Steve Hanneke

An Analysis of Graph Cut Size for Transductive Learning

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