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Smooth Boosting By Using An Information-Based Criterion Kohei Hatano Kyushu University, JAPAN

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Organization of this talk 1.Introduction 2.Preliminaries 3.Our booster 4.Experiments 5.Summary

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Boosting Methodology to combine prediction rules into a more accurate one. E.g. learning rule to classify web pages on “Drew Barrymore” accuracy 80% Barrymore? yn No Yes Set of pred. rules = words Labeled training data (web pages) yn Barrymore? NO ＹＥＳ yn Drew? NO ＹＥＳ yn Charlie’s engels? NO ＹＥＳ ＋ ＋ combination of prediction rules (say, majority vote) accuracy 51%! John Drew Barrymore (her father) “The Barrymore family” of Hollywood Jaid Barrymore (her mother) John Barrymore (her grandpa) Lionel Barrymore (her granduncle) Diana Barrymore (her aunt)

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Boosting by filtering [Schapire 90], [Freund 95], Advantage 2: smaller space complexity (for sample) accept reject Boosting scheme that uses random sampling from data (Huge) databoosting algorithm sample randomly Advantage 1: can determine sample size adaptively batch learning: O(1/ ) boosting by filtering : polylog(1/ ) ( : desired error)

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Some known results Boosting algorithms by filtering –Schapire’s first boosting alg. [Schapire 90],Boost-by-Majority [Freund 95], MadaBoost [Domingo&Watanabe 00], AdaFlat [Gavinsky 03]. –Criterion for choosing prediction rules ： accuracy Are there any better criteria? A candidate: information-based criterion –Real AdaBoost [Schapire&Singer 99], InfoBoost [Aslam 00] (a simple version of Real AdaBoost) –Criterion for choosing prediction rules ： mutual information –sometimes faster than those using accuracy-based criterion Experimental: [Schapire&Singer 99], Theoretical: [Hatano&Warmuth 03], [Hatano&Watanabe 04] –However, no boosting algorithm by filtering known

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Our work Boosting by filtering Information-based criterion efficient boosting by filtering using an information-based criterion lower space complexityfaster convergence our work

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1.Introduction 2.Preliminaries 3.Our booster 4.Experiments 5.Summary

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Illustration of general boosting Train. data (x 1,+1)(x 2,+1)(x 3,-1)(x 4,-1)(x 5,+1) Distribution D １ 0.2 1. Choose a pred. rule h 1 maximizing some criterion w. r. t. D 1. 2. Assign a coefficient to h 1 based on its quality. h1h1 +1 0.25 3. Update the distribution. Pred. of h １ +1 +1 lowerhigher : correct : wrong

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Pred. of h 2 +1 Illustration of general boosting(2) Train. data (x 1,+1)(x 2,+1)(x 3,-1)(x 4,-1)(x 5,+1) Distribution D 2 0.16 0.21 0.26 h2h2 +1 0.28 lowerhigher : correct : wrong 1. Choose a pred. rule h 2 maximizing some criterion w. r. t. D 2. 2. Assign a coefficient to h 1 based on its weighted error. 3. Update the distribution. Repeat these procedure for T times

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Illustration of general boosting(3) h2h2 +1 0.28 h3h3 +1 0.05 h1h1 +1 0.25 ++ Final pred. rule = weighted majority vote of chosen pred. rules. instance x predict +1, if H(x) >0 predict -1, otherwise

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Example: AdaBoost [Freund&Schapire 97] -y i H t (x i ) wrongcorrect Difficult examples (possibly noisy) may have too much weights Criterion for choosing pred. rules Coefficient Update (edge)

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Smooth boosting Keeping the distribution “smooth” makes boosting algorithms –noise-tolerant (statistical query model) MadaBoost [Domingo&Watanabe00] (malicious noise model ) SmoothBoost [Servedio01] (agnostic boosting model) AdaFlat [Gavinsky 03], –sampling from D t can be simulated efficiently via sampling from D 1 (e.g., by rejection sampling). applicable in the boosting by filtering framework D 1 (original distribution, e.g. uniform) D t (distribution costructed by the booster) sup x D t (x)/D 1 (x) is poly-bounded poly D 1

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Example: MadaBoost [Domingo & Watanabe 00] -y i H t (x i ) D t is 1/ bounded ( : error of H t ) Criterion for choosing pred. rules Coefficient Update (edge) l( -y i H t (x i ))

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Examples of other smooth boosters LogitBoost [Freidman, et al 00] AdaFlat [Gavinsky 03] logistic function stepwise linear function

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1.Introduction 2.Preliminaries 3.Our booster 4.Experiments 5.Summary

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Our new booster -y i H t (x i ) Criterion for choosing pred. rules Coefficient Update Still, D t is 1/ bounded ( : error of H t ) (pseudo gain) l( -y i H t (x i ))

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Pseudo gain Relation to edge Property: 2 (by convexity of of the square function)

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Interpretation of pseudo gain but, ・・・ the entropy function is NOT defined with Shannon’s entropy, but defined with Gini index min h (conditional entropy of labels given h t ) max h (mutual information between h and labels)

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Information-based criteria Cf. Real AdaBoost and InfoBoost choose a pred. rule that maximizes the mutual information defined with KM entropy. [Kearns & Mansour 98] Good news: Gini index can be estimated via sampling efficiently! Our booster chooses a pred. rule maximizing the mutual information defined by Gini Index (GiniBoost)

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Convergence of train. error (GiniBoost) Thm. Suppose that (train. error of H t )> for t=1,…,T. Then Coro. Further, if t (h t ) ¸ , train.err(H T ) · in T= O(1/ ) steps.

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Comparison on convergence speed booster #of iterations to get a final rule with error comments MadaBoost [Domingo&Watanabe 00] O(1/ 2 ) ○ boost by filtering ○ adaptive (don’ need to know ) × needs technical assumptions SmoothBoost [Servedio 01] O(1/ 2 ) ○ boost by filtering × not adaptive AdaFlat [Gavinsky 03] O(1/ 2 2 ) ○ boost by filtering ○ adaptive GiniBoost (our result) O(1/ ) 1/ 2 ) ○ boost by filtering ○ adaptive AdaBoost [Schapire& Freund 97] O(log(1/ ) / 2 ) ○ adaptive ×boost by filtering : minimum pseudo gain : minimum edge

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Boosting- by- filtering version of GiniBoost (outline) Multiplicative bounds for pseudo gain (and more practical bounds using the central limit approximation). Adaptive pred. rule selector. Boosting alg. in the PAC learning sense.

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1.Introduction 2.Preliminaries 3.Our booster 4.Experiments 5.Summary

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Experiments Topic classification of Reuters news (Reuters-21578) Binary classification for each 5 topics (Results are averaged). 10,000 examples. 30,000 words used as base pred. rules. Run algorithms until they sample 1,000,000 examples in total. 10-fold CV.

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Test error over Reuters Note ： GiniBoost2 doubles coefficients t [+1], t [-1] used in GiniBoost

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Execution time test error(%)time (sec.) AdaBoost (w/o sampling ， run in 100 step) 5.61349 MadaBoost6.7493 GiniBoost5.8408 GiniBoost25.5359 faster by about 4 times! (Cf. similar result w/o sampling RealAdaBoost [Schapire & Singer 99] )

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1.Introduction 2.Preliminaries 3.Our booster 4.Experiments 5.Summary

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Summary/Open problem Summary GiniBoost: uses pseudo gain (Gini index) to choose base prediction rules shows faster convergence in the filtering scheme. Open problem Theoretical analysis on noise-tolerance

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Comparison on sample size # of sampling# of accepted examples time （ sec. ） AdaBoost (w/o sampling, run in 100 steps) N/A 1349 MadaBoost1,032,219157,320493 GiniBoost11,039,943156,856408 GiniBoost21,027,874140,916359 Observation ： smaller accepted examples → faster selection of pred. rules

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Extension to boosting-by-filtering Instance space train. sample distribution Ｄｔ distribution Ｄ boosting alg. Instance space random example filtering distribution Ｄ random example distribution Ｄｔ Choose a pred. rule that maximazes pseudo-gain w.r.t. Ｄ ｔ boosting alg. Choose a pred. rule that approximately maximazes pseudo-gain w.r.t. Ｄｔ Batch learning Learn over train. sample Boosting-by-filtering Learn over instance space directly via random sampling

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Chernoff-type bound for pseudo gain

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complexity of GiniBoost filt query complexity space complexity

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Derivation of GiniBoost(1)

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Derivation of GiniBoost(2)

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