Learning from Labeled and Unlabeled Data: Semi-supervised Learning and Ranking p. 1/31

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1 Learning from Labeled and Unlabeled Data: Semi-supervised Learning and Ranking Dengyong Zhou Dept. Schölkopf, Max Planck Institute for Biological Cybernetics, Germany Learning from Labeled and Unlabeled Data: Semi-supervised Learning and Ranking p. 1/31

2 Learning from Examples Input space X, and output space Y = {1, 1}. Training set S = {z 1 = (x 1, y 1 ),..., z l = (x l, y l )} in Z = X Y drawn i.i.d. from some unknown distribution. Classifier f : X Y. Learning from Labeled and Unlabeled Data: Semi-supervised Learning and Ranking p. 2/31

3 Transductive Setting Input space X = {x 1,..., x n }, and output space Y = {1, 1}. Training set S = {z 1 = (x 1, y 1 ),..., z l = (x l, y l )}. Classifier f : X Y. Learning from Labeled and Unlabeled Data: Semi-supervised Learning and Ranking p. 3/31

4 Intuition about classification: Manifold Local consistency. Nearby points are likely to have the same label. Global consistency. Points on the same structure (typically referred to as a cluster or manifold) are likely to have the same label. Learning from Labeled and Unlabeled Data: Semi-supervised Learning and Ranking p. 4/31

5 A Toy Dataset (Two Moons) (a) Toy Data (Two Moons) unlabeled point labeled point 1 labeled point (b) SVM (RBF Kernel) (c) k NN 1.5 (d) Ideal Classification Learning from Labeled and Unlabeled Data: Semi-supervised Learning and Ranking p. 5/31

6 Algorithm 1. Form the affinity matrix W defined by W ij = exp( x i x j 2 /2σ 2 ) if i j and W ii = Construct the matrix S = D 1/2 W D 1/2 in which D is a diagonal matrix with its (i, i)-element equal to the sum of the i-th row of W. 3. Iterate f(t + 1) = αsf(t) + (1 α)y until convergence, where α is a parameter in (0, 1). 4. Let f denote the limit of the sequence {f(t)}. Label each point x i as y i = sgn(f i ). Learning from Labeled and Unlabeled Data: Semi-supervised Learning and Ranking p. 6/31

7 Convergence Theorem. The sequence {f(t)} converges to f = β(i αs) 1 y, where β = 1 α. Proof. Suppose F (0) = Y. By the iteration equation, we have t 1 f(t) = (αs) t 1 Y + (1 α) (αs) i Y. (1) i=0 Since 0 < α < 1 and the eigenvalues of S in [ 1, 1], lim t (αs)t 1 = 0, and lim t t 1 i=0 (αs) i = (I αs) 1. (2) Learning from Labeled and Unlabeled Data: Semi-supervised Learning and Ranking p. 7/31

8 Regularization Framework Cost function Q(f) = 1 [ n 2 W ij ( 1 Dii f i 1 Djj f j ) 2 + µ n ( fi y i ) 2 ] i,j=1 i=1 Smoothness term. Measure the changes between nearby points. Fitting term. Measure the changes from the initial label assignments. Learning from Labeled and Unlabeled Data: Semi-supervised Learning and Ranking p. 8/31

9 Regularization Framework Theorem. f = arg min f F Q(f). Proof. Differentiating Q(f) with respect to f, we have Q f = f Sf + µ(f y) = 0, (1) f=f which can be transformed into f µ Sf µ y = 0. (2) 1 + µ Let α = 1/(1 + µ) and β = µ/(1 + µ). Then (I αs)f = βy. (3) Learning from Labeled and Unlabeled Data: Semi-supervised Learning and Ranking p. 9/31

10 Two Variants Substitute P = D 1 W for S in the iteration equation. Then f = (I αp ) 1 y. Replace S with P T, the transpose of P. Then f = (I αp T ) 1 y, which is equivalent to f = (D αw ) 1 y. Learning from Labeled and Unlabeled Data: Semi-supervised Learning and Ranking p. 10/31

11 Toy Problem 1.5 (a) t = (b) t = (c) t = (d) t = Learning from Labeled and Unlabeled Data: Semi-supervised Learning and Ranking p. 11/31

12 Toy Problem Learning from Labeled and Unlabeled Data: Semi-supervised Learning and Ranking p. 12/31

13 Handwritten Digit Recognition (USPS) k NN (k = 1) SVM (RBF kernel) consistency variant (1) variant (2) 0.3 test error # labeled points Dimension: 16x16. Size: (α = 0.95) Learning from Labeled and Unlabeled Data: Semi-supervised Learning and Ranking p. 13/31

14 Handwritten Digit Recognition (USPS) consistency variant (1) variant (2) test error values of parameter α Size of labeled data: l = 50. Learning from Labeled and Unlabeled Data: Semi-supervised Learning and Ranking p. 14/31

15 Text Classification (20-newsgroups) k NN (k = 1) SVM (RBF kernel) consistency variant (1) variant (2) 0.6 test error # labeled points Dimension: Size: (α = 0.95) Learning from Labeled and Unlabeled Data: Semi-supervised Learning and Ranking p. 15/31

16 Text Classification (20-newsgroups) consistency variant (1) variant (2) 0.4 test error values of parameter α Size of labeled data: l = 50. Learning from Labeled and Unlabeled Data: Semi-supervised Learning and Ranking p. 16/31

17 Spectral Graph Theory Normalized graph Laplacian = D 1/2 (D W )D 1/2. Linear operator on the space of functions defined on the Graph. Theorem. i,j W ij ( ) 2 1 Dii f i 1 f j = f, f. Djj Discrete analogy of Laplace-Beltrami operator on Riemannian Manifold which satisfies f 2 = (f)f. M M Discrete Laplace equation f = y. Green s function G =. Learning from Labeled and Unlabeled Data: Semi-supervised Learning and Ranking p. 17/31

18 Reversible Markov Chains Lazy random walk defined by the transition probability matrix P = (1 α)i + αd 1 W, α (0, 1). Hitting time H ij = E{ number of steps required for a random walk to reach a position x j with an initial position x i }. Commute time C ij = H ij + H ji. Theorem. Let L = (D αw ) 1. Then C ij L ii + L jj L ij L ji Learning from Labeled and Unlabeled Data: Semi-supervised Learning and Ranking p. 18/31

19 Ranking Problem Problem setting. Given a set of point X = {x 1,..., x q, x q+1,..., x n } R m, the first q points are the queries. The task is to rank the remaining points according to their relevances to the queries. Examples. Image, document, movie, book, protein ("killer application"),... Learning from Labeled and Unlabeled Data: Semi-supervised Learning and Ranking p. 19/31

20 Intuition of Ranking: Manifold (a) Two moons ranking problem (b) Ideal ranking query The relevant degrees of points in the upper moon to the query should decrease along the moon shape. All points in the upper moon should be more relevant to the query than the points in the lower moon. Learning from Labeled and Unlabeled Data: Semi-supervised Learning and Ranking p. 20/31

21 Toy Ranking (a) Connected graph Simply ranking the data according to the shortest paths on the graph does not work well. Robust solution is to assemble all paths between two points: f = i αi S i y. Learning from Labeled and Unlabeled Data: Semi-supervised Learning and Ranking p. 21/31

22 Toy Ranking Learning from Labeled and Unlabeled Data: Semi-supervised Learning and Ranking p. 22/31

23 Connection to Google Theorem. For the task of ranking data represented by a connected and undirected graph without queries, f and PageRank yield the same ranking list. Learning from Labeled and Unlabeled Data: Semi-supervised Learning and Ranking p. 23/31

24 Personalized Google: a variant The ranking scores given by PageRank: π(t + 1) = αp T π(t). (4) Add a query term on the right-hand side for the query-based ranking, π(t + 1) = αp T π(t) + (1 α)y. (5) This can be viewed as the personalized version of PageRank. Learning from Labeled and Unlabeled Data: Semi-supervised Learning and Ranking p. 24/31

25 Image Ranking The top-left digit in each panel is the query. The left panel shows the top 99 by our method; and the right panel shows the top 99 by the Euclidean distance. Learning from Labeled and Unlabeled Data: Semi-supervised Learning and Ranking p. 25/31

26 Document Ranking 0.9 (a) autos 0.9 (b) motorcycles inner product inner product inner product manifold ranking (c) baseball manifold ranking inner product manifold ranking (d) hockey manifold ranking Learning from Labeled and Unlabeled Data: Semi-supervised Learning and Ranking p. 26/31

27 Related Work Graph/disffusion/cluster kernel (Kondor et al 2002; Chapelle et al. 2002; Smola et al. 2003). Spectral clustering (Shi et al. 1997; Ng et al. 2001). Manifold learning (nonlinear data reduction)(tenenbaum et al. 2000; Roweis et al. 2000) Learning from Labeled and Unlabeled Data: Semi-supervised Learning and Ranking p. 27/31

28 Related Work Random walks (Szummer et al. 2001). Graph min-cuts (Blum et al. 2001) Learning on manifolds (Belkin et al. 2001). Gaussian random fields (Zhu et al. 2003). Learning from Labeled and Unlabeled Data: Semi-supervised Learning and Ranking p. 28/31

29 Conclusion Proposed a general semi-supervised learning algorithm. Proposed a general example-based ranking algorithm. Learning from Labeled and Unlabeled Data: Semi-supervised Learning and Ranking p. 29/31

30 Next Work Model selection. Active learning. Generalization theory of learning from labeled and unlabeled data. Specifical problems & large-scale problems. Learning from Labeled and Unlabeled Data: Semi-supervised Learning and Ranking p. 30/31

31 References 1. Zhou, D., Bousquet, O., Lal, T.N., Weston, J. and Schölkopf, B.: Learning with Local and Global Consistency. NIPS, Zhou, D., Weston, J., Gretton, A., Bousquet, O. and Schölkopf, B.: Ranking on Data Manifolds. NIPS, Weston, J., C. Leslie, D. Zhou, A. Elisseeff and W. S. Noble: Semi-Supervised Protein Classification using Cluster Kernels. NIPS, Learning from Labeled and Unlabeled Data: Semi-supervised Learning and Ranking p. 31/31