Learning binary relations using weighted majority voting

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256 GOLDMAN AND WARMUTH Slow-Learn-Relation(0 _ No then partition i predicts that Mr i = 1 Else partition i predicts that M~j = 0 Let Ro be the set of all partitions that predict Mr 9 = 0 Let R1 be the set of all partitions that predict M w = 1 Let Wo = ~ieRo wi + ~ieR-(RoUR~) Wi/2 Let W1 = ~ieR~ w~ + ~ieR-(RoUR~) wi/2 If W1 > Wo predict 1 Else predict 0 3. Receive correct value for M~~ 4. If the prediction of the algorithm was wrong then update the weights as follows For 1 < i < kn/k! If partition i made a prediction of 1/2 then let ws e- wi/2 +/3 • w~/2 If partition i made an incorrect prediction (of 0 or 1) then let w~ ~- ws •/3 Figure 4. Our algorithm that uses kn/k! weights to obtaln an nearly optimal algorithm for learning binary relations. We would obtain the same mistake bound if Step 4 is performed at each step regardless of whether a mistake occurred.
LEARNING BINARY RELATIONS 257 Solving for tt gives the bound given in Equation (1). An interesting modification of our first algorithm would be to consider all rows in the same group as row r and then predict with the number of l's already known in column j divided by the total number of known entries in column j (i.e. a prediction of N1/(No + N1) using the notation of Figure 4). We did not use this rule because it is harder obtain a lower bound on the final weight in the system. Finally, by applying the results of Cesa-Bianchi, Freund, Helmbold, Haussler, Schapire, and Warmuth (1993) we can tune/3 as a function of an upper bound c~ on the noise. LEMMA 1 (Cesa-Bianchi, Freund, Helmbold, Haussler, Schapire & Warmuth, 1993) For any real value z > 0 or z = oc, z 2 + in a z2 g(z)
Page 1 and 2: Machine Leaming, 20, 245-271 (1995)
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Page 5 and 6: LEARNING BINARY RELATIONS 249 ~,~,.
Page 7 and 8: LEARNING BINARY RELATIONS 251 l J i
Page 9 and 10: LEARNING BINARY RELATIONS 253 Learn
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Page 15 and 16: LEARNING BINARY RELATIONS 259 In th
Page 17 and 18: LEARNING BINARY RELATIONS 261 depen
Page 19 and 20: LEARNING BINARY RELATIONS 263 m V S
Page 21 and 22: LEARNING BINARY RELATIONS 265 Final
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Page 25 and 26: LEARNING BINARY RELATIONS 269 mance
Page 27: LEARNING BINARY RELATIONS 271 Kivin

Learning binary relations using weighted majority voting

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