CS1420_Lecture_13.pdf - CS142 Machine Learning Spring 2017 Lecture 13 Instructor Pedro Felzenszwalb Scribes Dan Xiang Tyler Dae Devlin The

CS1420_Lecture_13.pdf - CS142 Machine Learning Spring 2017...

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CS142: Machine Learning Spring 2017 Lecture 13 Instructor: Pedro Felzenszwalb Scribes: Dan Xiang, Tyler Dae Devlin The generalization problem In this lecture we address what is perhaps the most important theoretical question in all of machine learning: How do we know that low training error will translate to low test error? PAC learning “PAC” is an acronym that stands for probably approximately correct . Before we can understand where this phrase comes from, we first need to recall the setup for binary classification. The goal of binary classification is to approximate some unknown target func- tion f : X → {- 1 , +1 } . Suppose D is an unknown distribution over the input space X . We observe labeled examples ( x 1 , y 1 ) , . . . , ( x n , y n ), where the x i ’s are sampled i.i.d. according to the distribution D and the corresponding y i ’s are given by the hidden target function f , i.e. y i = f ( x i ). Given these labeled examples, our task is to select a function h from a hypothesis set H such that h f . For the linear classifier, the hypothesis space H is the set of all functions of the form h ( x ) = ( +1 w T x > 0 - 1 w T x 0 , and a particular hypothesis h is completely specified by its weight vector w . Two types of errors 1. A finite set of training data may not be sufficient to identify the target function f exactly. As a result, the hypothesis h that we end up choosing may be close — but not exactly equal — to f . We can quantify how different h
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