MarginBased Active Learning MariaFlorina Balcan Carnegie Mellon University

Margin-Based Active Learning Maria-Florina Balcan Carnegie Mellon University Joint with Andrei Broder & Tong Zhang Yahoo! Research Maria-Florina Balcan

Incorporating Unlabeled Data in the Learning Process • OCR, Image classification • Web page, document classification • All the classification problems at Yahoo! Research. Unlabeled data cheap & easy to obtain. Labeled data much more expensive. Maria-Florina Balcan

Semi-Supervised Passive Learning • Several SSL methods developed to use unlabeled data to improve performance, e. g. : • Transductive SVM [Joachims ’ 98] • Co-training [Blum & Mitchell ’ 98], • Graph-based methods [Blum & Chawla’ 01] • Unlabeled data - allows to focus on a priori reasonable classifiers. See Avrim’s talk at the “Open Problems” session. Maria-Florina Balcan

Active Learning • The learner can choose specific examples to be labeled: - The learner works harder to use fewer labeled examples. Setting This talk: linear separators. • P distribution over X £ Y; hypothesis class C. • Get a set of unlabeled examples from PX. Interactively request labels of any of these examples. Goal: find h with small error over P. Minimize the number of label requests. Maria-Florina Balcan

Can Adaptive Querying Help? [CAL ’ 92, Dasgupta ’ 04] C = {linear separators in R 1}, realizable case. Active setting: O(log 1/ ) labels to find an -accurate threshold. Exponential improvement in sample complexity. In general, number of queries needed depends on C and P. C = {linear separators in R 2}: for some target hyp. no improvement can be achieved. Learning to accuracy requires 1/ labels. h 3 h 2 h 1 h 0 Maria-Florina Balcan

When Active Learning Helps In general, number of queries needed depends on C and P. C - homogeneous linear separators in Rd, PX - uniform distribution over unit sphere. Realizable case • O(d log 1/ ) labels to find a hypothesis with error . [Freund et al. , ’ 97; Dasgupta, Kalai, Monteleoni ’ 05] Agnostic Case • low noise, O(d 2 log 1/ ) labels to find a hypothesis with error . A 2 algorithm [Balcan, Beygelzimer, Langford ’ 06] [Hanneke ’ 07] Maria-Florina Balcan

An Overview of Our Results Analyze a class of margin based active learning algorithms for learning linear separators. • C - homogeneous linear separators in Rd, PX - uniform distrib. over unit sphere get exponential improvement in the realizable case. • Naturally extend the analysis to the bounded noise setting. • Dimension independent bounds when we have a good margin distribution. Maria-Florina Balcan

Margin Based Active-Learning, Realizable Case Algorithm Draw m 1 unlabeled examples, label them, add them to W(1). iterate k=2, …, s • find a hypothesis wk-1 consistent with W(k-1). • W(k)=W(k-1). • sample mk unlabeled samples x satisfying |wk-1 ¢ x| · k-1 ; • label them and add them to W(k). end iterate Maria-Florina Balcan

Margin Based Active-Learning, Realizable Case Draw m 1 unlabeled examples, label them, add them to W(1). iterate k = 2, …, s • find a hypothesis wk-1 consistent with W(k-1). • W(k)=W(k-1). • sample mk unlabeled samples x satisfying |wk-1 T ¢ x| · k-1 • label them and add them to W(k). 1 w 2 w 3 w 1 2 Maria-Florina Balcan

Margin Based Active-Learning, Realizable Case Theorem PX is uniform over Sd. If and then after iterations ws has error · . Fact 1 (u, v) u v Fact 2 If uv and v Maria-Florina Balcan

Margin Based Active-Learning, Realizable Case iterate k=2, … , s • find a hypothesis wk-1 consistent with W(k-1). • W(k)=W(k-1). • sample mk unlabeled samples x satisfying |wk-1 T ¢ x| · k-1 • label them and add them to W(k). w wk-1 w* Proof Idea k-1 Induction: all w consistent with W(k) have error · 1/2 k; so, wk has error · 1/2 k. For · 1/2 k+1 Maria-Florina Balcan

Proof Idea w Under the uniform distr. for wk-1 w* · 1/2 k+1 k-1 Maria-Florina Balcan

Proof Idea w Under the uniform distr. for wk-1 w* · 1/2 k+1 k-1 Enough to ensure Can do with only labels. Maria-Florina Balcan

Realizable Case, Suboptimal Alternative w Could imagine: wk-1 zero w* Suboptimal Need and so need labels to find a hyp. with error . Similar to [CAL’ 92, BBL’ 06, H’ 07] Maria-Florina Balcan

Margin Based Active-Learning, Non-realizable Case Guarantee Assume PX is uniform over Sd. Assume that |P(Y=1|x)-P(Y=-1|x)| ¸ for all x. Assume w* is the Bayes classifier. Then The previous algorithm and proof extend naturally, and get again an exponential improvement. Maria-Florina Balcan

Margin Based Active-Learning, Non-realizable Case Guarantee Assume PX is uniform over Sd. Assume that |P(Y=1|x)-P(Y=-1|x)| ¸ for all x. Assume w* is the Bayes classifier. Then The previous algorithm and proof extend naturally, and get again an exponential improvement. Maria-Florina Balcan

Summary • Analyze a class of margin based active learning algorithms for learning linear separators. Open Problems • Analyze a wider class of distributions, e. g. log-concave. • Characterize the right sample complexity terms for the Active Learning setting. Maria-Florina Balcan

Maria-Florina Balcan

Also, special thanks to: Alina Beygelzimer, Sanjoy Dasgupta, and John Langford for useful discussions. Maria-Florina Balcan