Support Vector Machines (SVMs).

Transcription

1 Support Vector Machines (SVMs). SemiSupervised Learning. SemiSupervised SVMs. MariaFlorina Balcan 3/25/215

2 Support Vector Machines (SVMs). One of the most theoretically well motivated and practically most effective classification algorithms in machine learning. Directly motivated by Margins and Kernels!

3 Geometric Margin WLOG homogeneous linear separators [w = ]. Definition: The margin of example x w.r.t. a linear sep. w is the distance from x to the plane w x =. Margin of example x 1 If w = 1, margin of x w.r.t. w is x w. x 1 w Margin of example x 2 x 2

4 Geometric Margin Definition: The margin of example x w.r.t. a linear sep. w is the distance from x to the plane w x =. Definition: The margin γ w of a set of examples S wrt a linear separator w is the smallest margin over points x S. Definition: The margin γ of a set of examples S is the maximum γ w over all linear separators w. γ γ w

5 Margin Important Theme in ML Both sample complexity and algorithmic implications. Sample/Mistake Bound complexity: If large margin, # mistakes Peceptron makes is small (independent on the dim of the space)! If large margin γ and if alg. produces a large margin classifier, then amount of data needed depends only on R/γ [Bartlett & ShaweTaylor 99]. Algorithmic Implications w γ γ Suggests searching for a large margin classifier SVMs

6 Support Vector Machines (SVMs) Directly optimize for the maximum margin separator: SVMs First, assume we know a lower bound on the margin γ Input: γ, S={(x 1, y 1 ),,(x m, y m )}; Find: some w where: w 2 = 1 For all i, y i w x i γ w γ γ Output: w, a separator of margin γ over S Realizable case, where the data is linearly separable by margin γ

7 Support Vector Machines (SVMs) Directly optimize for the maximum margin separator: SVMs E.g., search for the best possible γ Input: S={(x 1, y 1 ),,(x m, y m )}; Find: some w and maximum γ where: w 2 = 1 For all i, y i w x i γ w γ γ Output: maximum margin separator over S

8 Support Vector Machines (SVMs) Directly optimize for the maximum margin separator: SVMs Input: S={(x 1, y 1 ),,(x m, y m )}; Maximize γ under the constraint: w 2 = 1 For all i, y i w x i γ w γ γ

9 Support Vector Machines (SVMs) Directly optimize for the maximum margin separator: SVMs Input: S={(x 1, y 1 ),,(x m, y m )}; Maximize γ under the constraint: w 2 = 1 For all i, y i w x i γ This is a constrained optimization problem. objective function constraints Famous example of constrained optimization: linear programming, where objective fn is linear, constraints are linear (in)equalities

10 Support Vector Machines (SVMs) Directly optimize for the maximum margin separator: SVMs Input: S={(x 1, y 1 ),,(x m, y m )}; Maximize γ under the constraint: w 2 = 1 For all i, y i w x i γ w γ γ This constraint is nonlinear. In fact, it s even nonconvex w 1 w 2 2 w 1 w 2

11 Support Vector Machines (SVMs) Directly optimize for the maximum margin separator: SVMs Input: S={(x 1, y 1 ),,(x m, y m )}; Maximize γ under the constraint: w 2 = 1 For all i, y i w x i γ w = w/γ, then max γ is equiv. to minimizing w 2 (since w 2 = 1/γ 2 ). So, dividing both sides by γ and writing in terms of w we get: w γ γ Input: S={(x 1, y 1 ),,(x m, y m )}; 2 Minimize w under the constraint: For all i, y i w x i 1 w x = 1 w w x = 1

12 Support Vector Machines (SVMs) Directly optimize for the maximum margin separator: SVMs Input: S={(x 1, y 1 ),,(x m, y m )}; argmin w w 2 s.t.: For all i, y i w x i 1 This is a constrained optimization problem. The objective is convex (quadratic) All constraints are linear Can solve efficiently (in poly time) using standard quadratic programing (QP) software

13 Support Vector Machines (SVMs) Question: what if data isn t perfectly linearly separable? Issue 1: now have two objectives maximize margin minimize # of misclassifications. Ans 1: Let s optimize their sum: minimize w 2 C(# misclassifications) where C is some tradeoff constant. Issue 2: This is computationally hard (NPhard). [even if didn t care about margin and minimized # mistakes] NPhard [GuruswamiRaghavendra 6] w x = 1 w w x = 1

14 Support Vector Machines (SVMs) Question: what if data isn t perfectly linearly separable? Issue 1: now have two objectives maximize margin minimize # of misclassifications. Ans 1: Let s optimize their sum: minimize w 2 C(# misclassifications) where C is some tradeoff constant. Issue 2: This is computationally hard (NPhard). [even if didn t care about margin and minimized # mistakes] NPhard [GuruswamiRaghavendra 6] w x = 1 w w x = 1

15 Support Vector Machines (SVMs) Question: what if data isn t perfectly linearly separable? Replace # mistakes with upper bound called hinge loss Input: S={(x 1, y 1 ),,(x m, y m )}; Minimize w Input: S={(x 1, y 1 ),,(x m, y m )}; Find 2 under the constraint: For all i, y i w x i 1 argmin w,ξ1,,ξ m w 2 C For all i, y i w x i 1 ξ i ξ i ξ i are slack variables i ξ i s.t.: w x = 1 w w x = 1 w w x = 1 w x = 1

16 Support Vector Machines (SVMs) Question: what if data isn t perfectly linearly separable? Replace # mistakes with upper bound called hinge loss Input: S={(x 1, y 1 ),,(x m, y m )}; Find argmin w,ξ1,,ξ m w 2 C For all i, y i w x i 1 ξ i ξ i ξ i are slack variables i ξ i s.t.: C controls the relative weighting between the twin goals of making the w 2 small (margin is large) and ensuring that most examples have functional margin 1. w x = 1 l w, x, y w w x = 1 = max (,1 y w x)

17 Support Vector Machines (SVMs) Question: what if data isn t perfectly linearly separable? Replace # mistakes with upper bound called hinge loss Input: S={(x 1, y 1 ),,(x m, y m )}; Find argmin w,ξ1,,ξ m w 2 C For all i, y i w x i 1 ξ i ξ i i ξ i s.t.: Replace the number of mistakes with the hinge loss w x = 1 w w x = 1 w 2 C(# misclassifications) l w, x, y = max (,1 y w x)

18 Support Vector Machines (SVMs) Question: what if data isn t perfectly linearly separable? Replace # mistakes with upper bound called hinge loss Input: S={(x 1, y 1 ),,(x m, y m )}; Find argmin w,ξ1,,ξ m w 2 C For all i, y i w x i 1 ξ i ξ i i ξ i s.t.: w x = 1 Total amount have to move the points to get them on the correct side of the lines w x = 1/ 1, where the distance between the lines w x = and w x = 1 counts as 1 unit. w w x = 1 l w, x, y = max (,1 y w x)

19 What if the data is far from being linearly separable? Example: vs No good linear separator in pixel representation. SVM philosophy: Use a Kernel

20 Support Vector Machines (SVMs) Input: S={(x 1, y 1 ),,(x m, y m )}; Find argmin w,ξ1,,ξ m w 2 C i ξ i s.t.: Primal form For all i, y i w x i 1 ξ i ξ i Which is equivalent to: Input: S={(x 1, y 1 ),,(x m, y m )}; Find argmin α 1 2 For all i, i j y i y j α i α j x i x j i α i s.t.: α i C i Lagrangian Dual i y i α i =

21 SVMs (Lagrangian Dual) Input: S={(x 1, y 1 ),,(x m, y m )}; Find argmin α 1 2 i j y i y j α i α j x i x j i α i s.t.: For all i, α i C i y i α i = Final classifier is: w = α i y i x i i The points x i for which α i are called the support vectors i w x = 1 w w x = 1

22 Kernelizing the Dual SVMs Input: S={(x 1, y 1 ),,(x m, y m )}; Find argmin α 1 2 For all i, i j y i y j α i α j x i x j i α i s.t.: α i C i Replace x i x j with K x i, x j. i y i α i = Final classifier is: w = α i y i x i i The points x i for which α i are called the support vectors With a kernel, classify x using α i y i K(x, x i ) i

23 Support Vector Machines (SVMs). One of the most theoretically well motivated and practically most effective classification algorithms in machine learning. Directly motivated by Margins and Kernels!

24 What you should know The importance of margins in machine learning. The primal form of the SVM optimization problem The dual form of the SVM optimization problem. Kernelizing SVM. Think about how it s related to Regularized Logistic Regression.

25 Modern (Partially) Supervised Machine Learning Using Unlabeled Data and Interaction for Learning

26 Classic Paradigm Insufficient Nowadays Modern applications: massive amounts of raw data. Only a tiny fraction can be annotated by human experts. Protein sequences Billions of webpages Images

27 SemiSupervised Learning raw data face not face Labeled data Expert Labeler Classifier

28 Active Learning raw data face not face Expert Labeler O O O Classifier

29 SemiSupervised Learning Prominent paradigm in past 15 years in Machine Learning. Most applications have lots of unlabeled data, but labeled data is rare or expensive: Web page, document classification Computer Vision Computational Biology,.

30 SemiSupervised Learning Data Source Learning Algorithm Unlabeled examples Unlabeled examples Expert / Oracle Labeled Examples Algorithm outputs a classifier S l ={(x 1, y 1 ),,(x ml, y ml )} x i drawn i.i.d from D, y i = c (x i ) S u ={x 1,,x mu } drawn i.i.d from D Goal: h has small error over D. err D h = Pr x~ D (h x c (x))

31 Semisupervised learning: no querying. Just have lots of additional unlabeled data. A bit puzzling since unclear what unlabeled data can do for you. Key Insight Unlabeled data useful if we have beliefs not only about the form of the target, but also about its relationship with the underlying distribution.

32 Combining Labeled and Unlabeled Data Several methods have been developed to try to use unlabeled data to improve performance, e.g.: Transductive SVM [Joachims 99] Cotraining [Blum & Mitchell 98] Graphbased methods [B&C1], [ZGL3] Workshops [ICML 3, ICML 5, ] Books: SemiSupervised Learning, MIT 26 O. Chapelle, B. Scholkopf and A. Zien (eds) Test of time awards at ICML! Introduction to SemiSupervised Learning, Morgan & Claypool, 29 Zhu & Goldberg

33 Example of typical assumption: Margins The separator goes through low density regions of the space/large margin. assume we are looking for linear separator belief: should exist one with large separation SVM Labeled data only Transductive SVM

34 Transductive Support Vector Machines Optimize for the separator with large margin wrt labeled and unlabeled data. [Joachims 99] argmin w w 2 s.t.: y i w x i 1, for all i {1,, m l } S u ={x 1,,x mu } y u w x u 1, for all u {1,, m u } y u { 1, 1} for all u {1,, m u } w w x = 1 w x = 1 Input: S l ={(x 1, y 1 ),,(x ml, y ml )} Find a labeling of the unlabeled sample and w s.t. w separates both labeled and unlabeled data with maximum margin.

35 Transductive Support Vector Machines argmin w w 2 C ξ i i C ξ u u y i w x i 1ξ i, for all i {1,, m l } S u ={x 1,,x mu } y i w x u 1 ξ u, for all u {1,, m u } y u { 1, 1} for all u {1,, m u } w w x = 1 w x = 1 Input: S l ={(x 1, y 1 ),,(x ml, y ml )} Find a labeling of the unlabeled sample and w s.t. w separates both labeled and unlabeled data with maximum margin. Optimize for the separator with large margin wrt labeled and unlabeled data. [Joachims 99]

36 Transductive Support Vector Machines Optimize for the separator with large margin wrt labeled and unlabeled data. Input: S l ={(x 1, y 1 ),,(x ml, y ml )} S u ={x 1,,x mu } argmin w w 2 C i ξ i C u ξ u y i w x i 1ξ i, for all i {1,, m l } y i w x u 1 ξ u, for all u {1,, m u } y u { 1, 1} for all u {1,, m u } NPhard.. Convex only after you guessed the labels too many possible guesses

37 Transductive Support Vector Machines Optimize for the separator with large margin wrt labeled and unlabeled data. Heuristic (Joachims) high level idea: First maximize margin over the labeled points Use this to give initial labels to unlabeled points based on this separator. Try flipping labels of unlabeled points to see if doing so can increase margin Keep going until no more improvements. Finds a locallyoptimal solution.

38 Experiments [Joachims99]

39 Transductive Support Vector Machines Helpful distribution Nonhelpful distributions Highly compatible 1/ 2 clusters, all partitions separable by large margin

40 SemiSupervised Learning Prominent paradigm in past 15 years in Machine Learning. Key Insight Unlabeled data useful if we have beliefs not only about the form of the target, but also about its relationship with the underlying distribution. Prominent techniques Transductive SVM [Joachims 99] Cotraining [Blum & Mitchell 98] Graphbased methods [B&C1], [ZGL3]