Maria-Florina Balcan A Theoretical Model for Learning from Labeled and Unlabeled Data Maria-Florina Balcan & Avrim Blum Carnegie Mellon University, Computer Science Department

Maria-Florina Balcan What is Machine Learning? Design of programs that adapt from experience, identify patterns in data. Used to: –recognize speech, faces, … –categorize documents, info retrieval,... Goals of ML theory: develop models, analyze algorithmic and statistical issues involved.

Maria-Florina Balcan Outline of the talk Brief Overview of Supervised Learning –PAC Model Semi-Supervised Learning –An Augmented PAC Style Model

Maria-Florina Balcan Usual Supervised Learning Problem Decide which messages are spam and which are important. Might represent each message by n features. (e.g., keywords, spelling, etc.). Take a sample S of data, labeled according to whether they were/weren't spam. Goal of algorithm is to use data seen so far to produce good prediction rule h (a "hypothesis") for future data.

Maria-Florina Balcan The Concept Learning Setting E.g., Given data, some reasonable rules might be: Predict SPAM if unknown AND (money OR pills) Predict SPAM if money + pills – known > 0... examplelabel

Maria-Florina Balcan Supervised Learning, Big Questions Algorithm Design. How to optimize? –How might we automatically generate rules that do well on observed data? Sample Complexity/Confidence Bound –Real goal is to do well on new data. –What kind of confidence do we have that rules that do well on sample will do well in the future? for a given learning alg, how much data do we need...

Maria-Florina Balcan Supervised Learning: Formalization (PAC) PAC model – standard model for learning from labeled data. Have sample S = {(x,l)} drawn from some distrib D over examples x 2 X, labeled by some target function c *. Alg does optimization over S to produce some hypothesis h 2 C (e.g., C = linear separators). Goal is for h to be close to c * over D. –err(h)=Pr x 2 D (h(x)  c * (x)) Allow failure with small probability  (to allow for chance that S is not representative).

Maria-Florina Balcan The Issue of Sample-Complexity We want to do well on D, but all we have is S. –Are we in trouble? –How big does S have to be so that low error on S implies low error on D? Luckily, sample-complexity bounds. Algorithm: Pick a concept that agrees with S. Sample Complexity Statement: –If |S| ¸ (1/  )[log|C| + log 1/  ], then with probability at least (1-  ), all h 2 C that agree with sample S have true error · .

Maria-Florina Balcan Outline of the talk Brief Overview of Supervised Learning –PAC Model Semi-Supervised Learning –An Augmented PAC Style Model

Maria-Florina Balcan Combining Labeled and Unlabeled Data Hot topic in recent years in Machine Learning. Many applications have lots of unlabeled data, but labeled data is rare or expensive: Web page, document classification OCR, Image classification Several methods have been developed to try to use unlabeled data to improve performance, e.g.: Transductive SVM Co-training Graph-based methods

Maria-Florina Balcan An Augmented PAC style Model for Semi-Supervised Learning Extends PAC naturally to the case of learning from both labeled and unlabeled data. Unlabeled data is useful if we have beliefs not only about the form of the target, but also about its relationship with the underlying distribution. –Different algorithms are based on different assumptions about how data should behave. –Question – how to capture many of the assumptions typically used?

Maria-Florina Balcan Example of “typical” assumption The separator goes through low density regions of the space/large margin. –assume we are looking for linear separator –belief: there should exist one with large separation + + _ _ Labeled data only + + _ _ + + _ _ Transductive SVM SVM

Maria-Florina Balcan Another Example Agreement between two parts : co-training. –examples contain two sufficient sets of features i.e. an example is x= h x 1, x 2 i –belief: the two parts of the example are consistent 9 c 1, c 2 such that c 1 (x 1 )=c 2 (x 2 )=c * (x) –for example, if we want to classify web pages:

Maria-Florina Balcan Another Example, cont Agreement between two parts : co-training. –examples contain two sufficient sets of features i.e. an example is x= h x 1, x 2 i –belief: the two parts of the example are consistent 9 c 1, c 2 such that c 1 (x 1 )=c 2 (x 2 )=c * (x) –for example, if we want to classify web pages: My AdvisorProf. Avrim BlumMy AdvisorProf. Avrim Blum x 2 - Text info x 1 - Link info x - Link info & Text info x = h x 1, x 2 i

Maria-Florina Balcan Co-Training Works by using unlabeled data to propagate learned information. Text info Link info X1X1 X2X2 My Advisor +

Maria-Florina Balcan Semi-Supervised Learning Formalization. Main Idea Augment the notion of a concept class C with a notion of compatibility  between a concept and the data distribution (  (h,D) 2 [0,1]). “Learn C” becomes “learn (C,  )” (i.e. learn class C under compatibility notion  ). Express relationships that one hopes the target function and underlying distribution will possess.

Maria-Florina Balcan Semi-Supervised Learning Formalization. Main Idea Augment the notion of a concept class C with a notion of compatibility  between a concept and the data distribution (  (h,D) 2 [0,1]). “Learn C” becomes “learn (C,  )” (i.e. learn class C under compatibility notion  ). Express relationships that one hopes the target function and underlying distribution will possess. Use unlabeled data & the belief that the target is compatible to reduce C down to just {the highly compatible functions in C}.

Maria-Florina Balcan Semi-Supervised Learning Formalization. Main Idea, cont Use unlabeled data & our belief to reduce size(C) down to size(highly compatible functions in C) in our sample complexity bounds. Need to analyze how much unlabeled data is needed to uniformly estimate compatibilities well. Require that the degree of compatibility be something that can be estimated from a finite sample.

Maria-Florina Balcan Margins, Compatibility Margins: belief is that there should exist a separator with margin .  ( h, D) =1-(the probability mass within distance  of h). can be can be estimated from a finite sample. + + _ _ Highly compatible

Maria-Florina Balcan Types of Results in Our Model As in the usual PAC model, can discuss algorithmic and sample complexity issues. Can analyze how much unlabeled data we need to see: –depends both on the complexity of C and the complexity of our notion of compatibility. Can analyze the ability of a finite unlabeled sample to reduce our dependence on labeled examples: –as a function of compatibility of the target function and various measures of the helpfulness of the distribution.

Maria-Florina Balcan Examples of Results in Our Model Algorithm: pick a compatible concept that agrees with the labeled sample. Sample Complexity Statement:

Maria-Florina Balcan Examples of Results in Our Model, cont. Algorithm: pick a compatible concept that agrees with the labeled sample. Sample Complexity Statement: Highly compatible + + _ _

Maria-Florina Balcan Summary Provided a PAC style model for semi-supervised learning. Captures many of the ways in which unlabeled data is typically used. Unified framework for analyzing when and why unlabeled data can help. Can get much better bounds in terms of labeled examples.

Maria-Florina Balcan