Prediction on a Graph with a Perceptron

Transcription

1 Predition on a Graph with a Pereptron Mark Herbster, Massimiliano Pontil Department of Computer Siene University College London Gower Street, London WC1E 6BT, England, UK {m.herbster, m.pontil}@s.ul.a.uk Abstrat We study the problem of online predition of a noisy labeling of a graph with the pereptron. We address both label noise and onept noise. Graph learning is framed as an instane of predition on a finite set. To treat label noise we show that the hinge loss bounds derived by Gentile [1] for online pereptron learning an be transformed to relative mistake bounds with an optimal leading onstant when applied to predition on a finite set. These bounds depend ruially on the norm of the learned onept. Often the norm of a onept an vary dramatially with only small perturbations in a labeling. We analyze a simple transformation that stabilizes the norm under perturbations. We derive an upper bound that depends only on natural properties of the graph the graph diameter and the ut size of a partitioning of the graph whih are only indiretly dependent on the size of the graph. The impossibility of suh bounds for the graph geodesi nearest neighbors algorithm will be demonstrated. 1 Introdution We study the problem of robust online learning over a graph. Consider the following game for prediting the labeling of a graph. Nature presents a vertex v i1 ; the learner predits the label of the vertex ŷ 1 { 1, 1}; nature presents a label y 1 ; nature presents a vertex v i2 ; the learner predits ŷ 2 ; and so forth. The learner s goal is minimize the total number of mistakes ( {t : ŷ t y t } ). If nature is adversarial, the learner will always mispredit; but if nature is regular or simple, there is hope that a learner may make only a few mispreditions. Thus, a methodologial goal is to give learners whose total mispreditions an be bounded relative to the omplexity of nature s labeling. In this paper, we onsider the ut size as a measure of the omplexity of a graph s labeling, where the size of the ut is the number of edges between disagreeing labels. We will give bounds whih depend on the ut size and the diameter of the graph and thus do not diretly depend on the size of the graph. The problem of learning a labeling of a graph is a natural problem in the online learning setting, as well as a foundational tehnique for a variety of semisupervised learning methods [2, 3, 4, 5, 6]. For example, in the online setting, onsider a system whih serves advertisements on web pages. The web pages may be identified with the verties of a graph and the edges as links between pages. The online predition problem is then that, at a given time t the system may reeive a request to serve an advertisement on a partiular web page. For simpliity, we assume that there are two alternatives to be served: either advertisement A or advertisement B. The system then interprets the feedbak as the label and then may use this information in responding to the next request to predit an advertisement for a requested web page. 1.1 Related work There is a welldeveloped literature regarding learning on the graph. The early work of Blum and Chawla [2] presented an algorithm whih expliitly finds minuts of the label set. Bounds have been

2 Input: {(v it, y t)} l t=1 V M { 1,1}. Initialization: w 1 = 0; M A =. for t = 1,..., l do Predit: reeive v it ŷ t = sign(e i t w t) Update: reeive y t if ŷ t = y t then w t1 = w t else w t1 = w t y tv it M A = M A {t} end Figure 1: Pereptron on set V M. Figure 2: Barbell Figure 3:Barbell with onept noise Figure 4: Flower Figure 5: Otopus proven previously with smooth loss funtions [6, 7] in a bath setting. Kernels on graph labelings were introdued in [3, 5]. This urrent work builds upon our work in [8]. There it was shown that, given a fixed labeling of a graph, the number of mistakes made by an algorithm similar to the kernel pereptron [9] with a kernel that was the pseudoinverse of the graph Laplaian, ould be bounded by the quantity [8, Theorems 3.2, 4.1, and 4.2] 4Φ G (u)d G bal(u). (1) Here u { 1, 1} n is a binary vetor defining the labeling of the graph, Φ G (u) is the ut size 1 defined as Φ G (u) := {(i, j) E(G) : u i u j }, that is, the number of edges between positive and negative labels, D G is the diameter of the graph and bal(u) := ( 1 1 n u i ) 2 measures the label balane. This bound is interesting in that the mistakes of the algorithm ould be bounded in terms of simple properties of a labeled graph. However, there are a variety of shortomings in this result. First, we observe that the bound above assumed a fixed labeling of the graph. In pratie, the online data sequene ould ontain multiple labels for a single vertex; this is the problem of label noise. Seond, for an unbalaned set of labels the bound is vauous, for example, if u = {1, 1,..., 1, 1} IR n then bal(u) = n 2. Third, onsider the prototypial easy instane for the algorithm of two dense lusters onneted by a few edges, for instane, two mliques onneted by a single edge (a barbell graph, see Figure 2). If eah lique is labeled with distint labels then we have that 4Φ G (u)d G bal(u) = = 12, whih is independent of m. Now suppose that, say, the first lique ontains one vertex whih is labeled as the seond lique (see Figure 3). Previously Φ G (u) = 1, but now Φ G (u) = m and the bound is vauous. This is the problem of onept noise; in this example, a Θ(1) perturbation of labeling inreases the bound multipliatively by Θ(m). 1.2 Overview A first aim of this paper is to improve upon the bounds in [8], partiularly, to address the three problems of label balane, label noise, and onept noise as disussed above. For this purpose, we apply the wellknown kernel pereptron [9] to the problem of online learning on the graph. We disuss the bakground material for this problem in setion 2, where we also show that the bounds of [1] an be speialized to relative mistake bounds when applied to, for example, predition on the graph. A seond important aim of this paper is to interpret the mistake bounds by an explanation in terms of high level graph properties. Hene, in setion 3, we refine a diameter based bound of [8, Theorem 4.2] to a sharper bound based on the resistane distane [10] on a weighted graph; whih we then losely math with a lower bound. In setion 4, we introdue a kernel whih is a simple augmentation of the pseudoinverse of the graph Laplaian and then prove a theorem on the performane of the pereptron with this kernel whih solves the three problems above. We onlude in setion 5, with a disussion omparing the mistake bounds for predition on the graph with the halving algorithm [11] and the knearest neighbors algorithm. 2 Preliminaries In this setion, we desribe our setup for Hilbert spaes on finite sets and its speifiation to the graph ase. We then reall a result of Gentile [1] on predition with the pereptron and disuss a speial ase in whih relative 0 1 loss (mistake) bounds are obtainable. 1 Later in the paper we extend the definition of ut size to weighted graphs.

3 2.1 Hilbert spaes of funtions defined on a finite set We denote matries by apital bold letters and vetors by small bold ase letters. So M denotes the n n matrix (M ij ) n i,j=1 and w the ndimensional vetor (w i) n i=1. The identity matrix is denoted by I. We also let 0 and 1 be the ndimensional vetors all of whose omponents equal to zero and one respetively, and e i the ith oordinate vetor of IR n. Let IN be the set of natural numbers and IN l := {1,..., l}. We denote a generi Hilbert spae with H. We identify V := IN n as the indies of a set of n objets, e.g. the verties of a graph. A vetor w IR n an alternatively be seen as a funtion f : V IR suh that f(i) = w i, i V. However, for simpliity we will use the notation w to denote both a vetor in IR n or the above funtion. A symmetri positive semidefinite matrix M indues a semiinner produt on IR n whih is defined as u, w M := u Mw, where denotes transposition. The reproduing kernel [12] assoiated with the above semiinner produt is K = M, where denotes pseudoinverse. We also define the oordinate spanning set V M := {v i := M e i : i = 1,..., n} (2) and let H(M) := span(v M ). The restrition of the semiinner produt, M to H(M) is an inner produt on H(M). The set V M ats as oordinates for H(M), that is, if w H(M) we have w i = e i M Mw = v i Mw = v i, w M, (3) although the vetors {v 1,..., v n } are not neessarily normalized and are linearly independent only if M is positive definite. We note that equation (3) is simply the reproduing kernel property [12] for kernel M. When V indexes the verties of an undireted graph G, a natural norm to use is that indued by the graph Laplaian. We explain this in detail now. Let A be the n n symmetri weight matrix of the graph suh that A ij 0, and define the edge set E(G) := {(i, j) : 0 < A ij, i < j}. The distane matrix assoiated with G is the perelement inverse of the weight matrix, that is, ij = 1 A ij ( may have as a matrix element). The graph Laplaian G is the n n matrix defined as G := D A where D = diag(d 1,..., d n ) and d i is the weighted degree of vertex i, d i = n j=1 A ij. The Laplaian is positive semidefinite and indues the seminorm w 2 G := w Gw = A ij (w i w j ) 2. (4) (i,j) E(G) When the graph is onneted, it follows from equation (4) that the null spae of G is spanned by the onstant vetor 1 only. In this paper, we always assume that the graph G is onneted. Where it is not ambiguous, we use the notation G to denote both the graph G and the graph Laplaian. 2.2 Online predition of funtions on a finite set with the pereptron Gentile [1] bounded the performane of the pereptron algorithm on nonseparable data with the linear hinge loss. Here, we apply his result to study the problem of predition on a finite set with the pereptron (see Figure 1). In this ase, the inputs are the oordinates in the set V M H(M) defined above. We additionally assume that matrix M is positive definite (not just positive semidefinite as in the previous subsetion). This assumption, along with the fat that the inputs are oordinates, enables us to upper bound the hinge loss and hene we may give a relative mistake bound in terms of the omplete set of base lassifiers { 1, 1} n. Theorem 2.1. Let M be a symmetri positive definite matrix. If {(v it, y t )} l t=1 V M { 1, 1} is a sequene of examples, M A denotes the set of trials in whih the pereptron algorithm predited inorretly and X = max t MA v it M, then the umulative number of mistakes M A of the algorithm is bounded by M A 2 M A M u u 2 M X2 2 2 M A M u u 2 M X2 u 4 M X4 4 for all u { 1, 1} n, where M u = {t IN l : u it y t }. In partiular, if M u = 0 then M A u 2 M X2. (5)

4 Proof. This bound follows diretly from [1, Theorem 8] with p = 2, γ = 1, and w 1 = 0. Sine M is assumed to be symmetri positive definite, it follows that { 1, 1} n H(M). Thus, the hinge loss L u,t := max(0, 1 y t u, v it M ) of any lassifier u { 1, 1} n with any example (v it, y t ) is either 0 or 2, sine u, v it M = 1 by equation (3). This allows us to bound the hinge loss term of [1, Theorem 8] diretly with mistakes. We emphasize that our hypothesis on M does not imply linear separability sine multiple instanes of an input vetor in the training sequene may have distint target labels. Moreover, we note that, for deterministi predition the onstant 2 in the first term of the right hand side of equation (5) is optimal for an online algorithm as a mistake may be fored on every trial. 3 Interpretation of the spae H(G) The bound for predition on a finite set in equation (5) involves two quantities, namely the squared norm of a lassifier u { 1, 1} n and the maximum of the squared norms of the oordinates v V M. In the ase of predition on the graph, reall from equation (4) that u 2 G := Gu = u (i,j) E(G) A ij(u i u j ) 2. Therefore, we may identify this seminorm with the weighted ut size Φ G (u) := 1 4 u 2 G (6) of the labeling indued when u { 1, 1} n. In partiular, with boolean weighted edges (A ij {0, 1}) the ut simply ounts the number of edges spanning disagreeing labels. The norm v w G is a metri distane for v, w span(v G ) however, surprisingly, the square of the norm v p v q 2 G when restrited to graph oordinates v p, v q V G is also a metri known as the resistane distane [10], r G (p, q) := (e p e q ) G (e p e q ) = v p v q 2 G. (7) It is interesting to note that the resistane distane between vertex p and vertex q is the effetive resistane between verties p and q, where the graph is the iruit and edge (i, j) is a resistor with the resistane ij = A 1 ij. As we shall see, our bounds in setion 4 depend on v p 2 G = v p 0 2 G. Formally, this is not an effetive resistane between vertex p and another vertex 0. The vetor 0, informally however, is P v V v G the enter of the graph as 0 = V G, sine 1 is in the null spae of H(G). In the following, we further haraterize v p 2 G. First, we observe qualitatively that the more interonneted the graph the smaller the term v p 2 G (Corollary 3.1). Seond, in Theorem 3.2 we quantitatively upper bound v p 2 G by the average (over q) of the effetive resistane between vertex p and eah vertex q in the graph (inluding q = p), whih in turn may be upper bounded by the eentriity of p. We proeed with the following useful lemma and theorem, as a basis for our later results. Lemma 3.1. Let x H then x 2 = min w H { w 2 : w, x = 1 }. The proof is straightforward and we do not elaborate on the details. Theorem 3.1. If M and M are symmetri positive semidefinite matries with span(v M ) = span(v M ) and, for every w span(v M ), w 2 M w 2 M then n 2 a i v i n a i v i 2, M M i=1 where v i V M, v i V M and a IRn. Proof. Let x = n i=1 a iv i and x = n i=1 a iv i then 2 x 2 M = x x 2 x 2 M M x 2 x 2 M M x 2 = x 2 M, M M x where the first inequality follows sine x, = 1, hene x 2 M x M is a feasible solution to x 2 M the minimization problem in the right hand side of Lemma 3.1. While the seond one follows immediately from the assumption that w 2 M w 2 M. i=1

5 As a orollary to the above theorem we have the following when M is a graph Laplaian. Corollary 3.1. Given onneted graphs G and G with distane matries and suh that ij ij then for all p, q V, we have that v p 2 G v p 2 G and r G (p, q) r G (p, q). The first inequality in the above orollary demonstrates that v p 2 G is noninreasing in a graph that is stritly more onneted. The seond inequality is the wellknown Rayleigh s monotoniity law whih states that if any resistane in a iruit is dereased then the effetive resistane between any two points annot inrease. We define the geodesi distane between verties p, q V to be d G (p, q) := min P (p, q) where the minimum is taken with respet to all paths P(p, q) from p to q, with the path length defined as P(p, q) := (i,j) E(P(p,q)) ij. The eentriity d G (p) of a vertex p V is the geodesi distane on the graph between p and the furthest vertex on the graph to p, that is, d G (p) = max q V d G (p, q) D G, and D G is the (geodesi) diameter of the graph, D G := max p V d G (p). A graph G is onneted when D G <. A tree is an nvertex onneted graph with n 1 edges. The following lemma, a well known result (see e.g. [10]), establishes that the resistane distane an be be equated with the geodesi distane when the graph is a tree. Lemma 3.2. If the graph T is a tree with graph Laplaian T then r T (p, q) = d T (p, q). The next theorem provides a quantitative relationship between v p 2 G and two measures of the onnetivity of vertex p, namely its eentriity and the mean of the effetive resistanes between vertex p and eah vertex on the graph. Theorem 3.2. If G is a onneted graph then v p 2 G 1 n r G (p, q) d G (p). (8) n q=1 Proof. Reall that r G (p, q) = v p v q 2 G (see equation (7)) and use n q=1 v q = 0 to obtain that 1 n n q=1 v p v q 2 G = v p Gv p 1 n n q=1 v q Gv q whih implies the left inequality in (8). Next, by Corollary 3.1, if T is the Laplaian of a tree T G then r G (p, q) r T (p, q) for p, q V. Therefore, from Lemma 3.2 we onlude that r G (p, q) d T (p, q). Moreover, sine T G an be any tree, we have that r G (p, q) min T d T (p, q) where the minimum is over all trees T G. Sine the geodesi path from p to q is neessarily ontained in some tree T G it follows that min T d T (p, q) = d G (p, q) and, so, r G (p, q) d G (p, q). Now the theorem follows by maximizing d G (p, q) over q and the definition of d G (p). We identify the resistane diameter of a graph G as R G := max p,q V r G (p, q); thus, from the previous theorem, we may also onlude that max v p 2 p V G R G D G. (9) We omplete this setion by showing that there exists a family of graphs for whih the above inequality is nearly tight. Speifially, we onsider the flower graph (see Figure 4) obtained by onneting the first vertex of a hain with p 1 verties to the root vertex of an mary tree of depth one. We index the verties of this graph so that verties 1 to p orrespond to stem verties and verties p 1 to p m to petals. Clearly, this graph has diameter equal to p, hene our upper bound above establishes that v 1 2 G p. We now argue that as m grows this bound is almost tight. From Lemma 3.1 we have that v 1 2 G = min { w H(G) w 2 G : w, v 1 = 1 }. We note that by symmetry, the solution ŵ = (ŵ i : i IN pm ) to the problem above satisfies ŵ i = z if i p 1 sine{ ŵ must take the same value on the petal verties. Consequently, it follows that v 1 2 G = min m(z w p ) 2 p 1 i=1 (w i w i1 ) 2 p : w 1 = 1, i=1 w i mz = 0}. We upper bound this minimum by hoosing w i = p i p 1 for 1 i p. Thus, w 1 = 1 as it is required, w p = 0 and we ompute z by the onstraint set of the above minimization problem as z = p 2m. A diret omputation gives v 1 2 G 1 (p 1) p2 4m from whih using a first order Taylor expansion it follows that v 1 2 G (p 1) (p 1)2 p 2 4m. Therefore, as m the upper bound on v 1 2 G (equation (8)) for the flower graph is mathed by a lower bound with a gap of 1.

6 4 Predition on the graph We define the following symmetri positive definite graph kernel, K b := G b11 I, (0 < b, 0 ), (10) where G b = (K b ) 1 is the matrix of the assoiated Hilbert spae H(G b ). In Lemma 4.1 below we prove the needed properties of H(G b ) as a neessary step for the bound in Theorem 4.2. As we shall see, these properties moderate the onsequenes of label imbalane and onept noise. To prove Lemma 4.1, we use the following theorem whih is a speial ase of [12, Thm I, I.6]. Theorem 4.1. If M 1 and M 2 are n n symmetri positive semidefinite matries, and we set M := (M 1 M 2 ) then w 2 M = inf{ w 1 2 M 1 w 2 2 M 2 : w i H(M i ), w 1 w 2 = w} for every w H(M). Next, we define β u [0, 1] as a measure of the balane of a labeling u { 1, 1} n as β u := ( 1 n n i=1 u i) 2. Note that for a perfetly balaned labeling β u = 0, while β u = 1 for a perfetly unbalaned one. Lemma 4.1. Given a vertex p with assoiated oordinates v p V G and v p V G b we have that v p 2 = v G b p 2 G b. (11) Moreover, if u, u { 1, 1} n and where k := {i : u i u i} we have that u 2 G b u 2 G β u b 4k. (12) Proof. To prove equation (11) we reall equation (3) and note that v p 2 G b = v p, v p b1e p G b = v p, v p G b v p, b1e p G b = v p 2 G b. To prove inequality (12) we proeed in two steps. First, we show that u 2 G b 0 = u 2 G β u b. (13) Indeed, we an uniquely deompose u as the sum of a vetor in H(G) and one in H( 11 n 2 b ) as u = (u 1 1 n n i=1 u i) 1 1 n n i=1 u i. Therefore, by Theorem 4.1 we onlude that u 2 = G b 0 u β u 1 2 G β u 1 2 = u 2 11 G βu b, where u β u 1 2 G = u 2 G sine 1 H (G). n 2 b Seond, we show, for any symmetri positive definite matrix M, u, u { 1, 1} n and > 0, that u 2 M u 2 M 4k, (14) where M := (M 1 I) 1 and k := {i : u i u i}. To this end, we deompose u as a sum of two elements of H(M) and H( 1 I) as u = u (u u) and observe that u u 2 1 I = 4k. By Theorem 4.1 it then follows that u 2 M u 2 M u u 2 1 I = u 2 M 4k. Now inequality (12) follows by ombining equations (13) and (14) with M = G b 0. We an now state our relative mistake bound for online predition on the graph. Theorem 4.2. Let G be a onneted graph. If {(v it, y t )} l t=1 V G b { 1, 1} is a sequene of examples and M A denotes the set of trials in whih the pereptron algorithm predited inorretly, then the umulative number of mistakes M A of the algorithm is bounded by M A 2 M A M u Z 2 2 M A M u Z Z2 4, (15) for all u,u { 1, 1} n, where k = {i:u i u i}, β u =( 1 n n i=1 u i )2, M u ={t IN l :u it y t }, and ( 4Φ G (u ) β u b Z = 4k In partiular, if b = 1, = 0, k = 0 and M u = 0 then )( R G b ). M A (4Φ G (u) β u )(R G 1). (16)

7 Proof. The proof follows by Theorem 2.1 with M = G b, then bounding u 2 G b and v t 2 G b via Lemma 4.1, and then using max t MA v it 2 G R G by equation (9). The upper bound of the theorem is more resilient to label imbalane, onept noise, and label noise than the bound in [8, Theorems 3.2, 4.1, and 4.2] (see equation (1)). For example, given the noisy barbell graph in Figure 3 but with k n noisy verties the bound (1) is O(kn) while the bound (15) with b = 1, = 1, and M u = 0 is O(k). A similar argument may be given for label imbalane. In the bound above, for easy interpretability, one may upper bound the resistane diameter R G by the geodesi diameter D G. However, the resistane diameter makes for a sharper bound in a number of natural situations. For example now onsider (a thik barbell) two mliques (one labeled 1, one 1 ) with l edges (l < m) between the liques. We observe between any two verties there are at least l edgedisjoint paths of length no more than five, therefore the resistane diameter is at most 5/l by the resistorsinparallel rule while the geodesi diameter is 3. Thus, for thik barbells if we use the geodesi diameter we have a mistake bound of 16l (substituting β u = 0, and R G 3 into (16)) while surprisingly with the resistane diameter the bound (substituting b = 1 4n, = 0, M u = 0, β u = 0, and R G 5/l into (15)) is independent of l and is Disussion In this paper, we have provided a bound on the performane of the pereptron on the graph in terms of strutural properties of the graph and its labeling whih are only indiretly dependent on the number of verties in the graph, in partiular, they depend on the ut size and the diameter. In the following, we ompare the pereptron with two other approahes. First, we ompare the pereptron with the graph kernel K 1 0 to the oneptually simpler knearest neighbors algorithm with either the graph geodesi distane or the resistane distane. In partiular, we prove the impossibility of bounding performane of knearest neighbors only in terms of the diameter and the ut size. Speifially, we give a parameterized family of graphs for whih the number of mistakes of the pereptron is upper bounded by a fixed onstant independent of the graph size while knearest neighbors provably inurs mistakes linearly in the graph size. Seond, we ompare the pereptron with the graph kernel K 1 0 with a simple appliation of the lassial halving algorithm [11]. Here, we onlude that the upper bound for the pereptron is better for graphs with a small diameter while the halving algorithm s upper bound is better for graphs with a large diameter. In the following, for simpliity we limit our disussion to binaryweighted graphs, noisefree data (see equation (16)) and upper bound the resistane diameter R G with the geodesi diameter D G (see equation (9)). 5.1 Knearest neighbors on the graph We onsider the knearest neighbors algorithms on the graph with both the resistane distane (see equation (7)) and the graph geodesi distane. The geodesi distane between two verties is the length of the shortest path between the two verties (reall the disussion in setion 3). In the following, we use the emphasis distane to refer simultaneously to both distanes. Now, onsider the family of O l,m,p of otopus graphs. An otopus graph (see Figure 5) onsists of a head whih is an llique (C (l) ) with verties denoted by 1,..., l, and a set of m tentales ({T i } m i=1 ), where eah tentale is a line graph of length p. The verties of tentale i are denoted by {t i,0, t i,1,..., t i,p }; the t i,0 are all identified as one vertex r whih ats as the root of the m tentales. There is an edge (the body) onneting root r to the vertex 1 on the head. Thus, this graph has diameter D = max(p 2, 2p) and there are l mp 1 verties in total; an otopus O m,p is balaned if l = mp 1. Note that the distane of every vertex in the head to every other vertex in the graph is no more than p 2, and every tentale tip t i,p is distane 2p to other tips t j,p : j i. We now argue that knearest neighbors may inur mistakes linear in the number of tentales. To this end, hoose p 3 and suppose we have the following online data sequene {( 1, 1), (t 1,p, 1), ( 2, 1), (t 2,p, 1),..., ( m, 1), (t m,p, 1)}. Note that knearest neighbors will make a mistake on every instane (t i,p, 1) and so, even assuming that it predits orretly on ( 1, 1) it will always make m mistakes. We now ontrast this result with the performane of the pereptron with the graph kernel K 1 0 (see equation (10)). By equation (16), the number of mistakes will be upper bounded by 10p 5 beause there is a ut of size 1 and the

8 diameter is 2p. Thus, for balaned otopi O m,p with p 3, as m grows the number of mistakes of the kernel pereptron will be bounded by a fixed onstant. Whereas distane knearest neighbors will inur mistakes linearly in m. 5.2 Halving algorithm We now ompare the performane of our algorithm to the lassial halving algorithm [11]. The halving algorithm operates by prediting on eah trial as the majority of the lassifiers in the onept lass whih have been onsistent over the trial sequene. Hene, the number of mistakes of the halving algorithm is upper bounded by the logarithm of the ardinality of the onept lass. Let KG k = {u { 1, 1}n : Φ G (u) = k} be the set all of all lassifiers with a ut size equal to k on a fixed graph G. The ardinality of KG k is upper bounded by ( ) n(n 1) k sine any lassifier (ut) in K k G an be uniquely identified by a hoie of k edges and 1 bit whih determines the sign of the verties in the same of partition (however we overount as not every set of edges determines a lassifier). The number of mistakes of the halving algorithm is upper bounded by O(k log n k ). For example, on a line graph with a ut size of 1 the halving algorithm has an upper bound of log n while the upper bound for the number of mistakes of the pereptron as given in equation (16) is 5n 5. Although the halving algorithm has a sharper bound on suh large diameter graphs as the line graph, it unfortunately has a logarithmi dependene on n. This ontrasts to the bound of the pereptron whih is essentially independent of n. Thus, the bound for the halving algorithm is roughly sharper on graphs with a diameter ω(log n k ), while the pereptron bound is roughly sharper on graphs with a diameter o(log n k ). We emphasize that this analysis of upper bounds is quite rough and sharper bounds for both algorithms ould be obtained for example, by inluding a term representing the minimal possible ut, that is, the minimum number of edges neessary to disonnet a graph. For the halving algorithm this would enable a better bound on the ardinality of KG k (see [13]). While, for the pereptron the larger the onnetivity of the graph, the weaker the diameter upper bound in Theorem 3.2 (see for example the disussion of thik barbells at the end of setion 4). Aknowledgments We wish to thank the anonymous reviewers for their useful omments. This work was supported by EPSRC Grant GR/T18707/01 and by the IST Programme of the European Community, under the PASCAL Network of Exellene IST Referenes [1] C. Gentile. The robustness of the pnorm algorithms. Mahine Learning, 53(3): , [2] A. Blum and S. Chawla. Learning from labeled and unlabeled data using graph minuts. In ICML 2002, pages Morgan Kaufmann, San Franiso, CA, [3] R. I. Kondor and J. Lafferty. Diffusion kernels on graphs and other disrete input spaes. In ICML 2002, pages Morgan Kaufmann, San Franiso, CA, [4] X. Zhu, Z. Ghahramani, and J. Lafferty. Semisupervised learning using gaussian fields and harmoni funtions. In ICML 2003, pages , [5] A. Smola and R.I. Kondor. Kernels and regularization on graphs. In COLT 2003, pages , [6] M. Belkin, I. Matveeva, and P. Niyogi. Regularization and semisupervised learning on large graphs. In COLT 2004, pages , Banff, Alberta, Springer. [7] T. Zhang and R. Ando. Analysis of spetral kernel design based semisupervised learning. In Y. Weiss, B. Shölkopf, and J. Platt, editors, NIPS 18, pages MIT Press, Cambridge, MA, [8] M. Herbster, M. Pontil, and L. Wainer. Online learning over graphs. In ICML 2005, pages , New York, NY, USA, ACM Press. [9] Y. Freund and R. E. Shapire. Large margin lassifiation using the pereptron algorithm. Mahine Learning, 37(3): , [10] D. Klein and M. Randić. Resistane distane. Journal of Mathematial Chemistry, 12(1):81 95, [11] J. M. Barzdin and R. V. Frievald. On the predition of general reursive funtions. Soviet Math. Doklady, 13: , [12] N. Aronszajn. Theory of reproduing kernels. Trans. Amer. Math. So., 68: , [13] D. Karger and C. Stein. A new approah to the minimum ut problem. JACM, 43(4): , 1996.