ACTIVE LEARNING ON WEIGHTED GRAPHS USING ADAPTIVE AND NON-ADAPTIVE APPROACHES. Eyal En Gad, Akshay Gadde, A. Salman Avestimehr and Antonio Ortega

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1 ACTIVE LEARNING ON WEIGHTED GRAPHS USING ADAPTIVE AND NON-ADAPTIVE APPROACHES Eya En Gad, Akshay Gadde, A. Saman Avestimehr and Antonio Ortega Department of Eectrica Engineering University of Southern Caifornia, Los Angees ABSTRACT This paper studies graph-based active earning, where the goa is to reconstruct a binary signa defined on the nodes of a weighted graph, by samping it on a sma subset of the nodes. A new samping agorithm is proposed, which sequentiay seects the graph nodes to be samped, based on an aggressive search for the boundary of the signa over the graph. The agorithm generaizes a recent method for samping nodes in unweighted graphs. The generaization improves the samping performance using the information gained from the avaiabe graph weights. An anaysis of the number of sampes required by the proposed agorithm is provided, and the gain over the unweighted method is further demonstrated in simuations. Additionay, the proposed method is compared with an aternative stateof-the-art method, which is based on the graph s spectra properties. It is shown that the proposed method significanty outperforms the spectra samping method, if the signa needs to be predicted with high accuracy. On the other hand, if a higher eve of inaccuracy is toerabe, then the spectra method outperforms the proposed aggressive search method. Consequenty, we propose a hybrid method, which is shown to combine the advantages of both approaches. Index Terms active earning on graphs, adaptive and nonadaptive samping of graph signas, samping compexity 1. INTRODUCTION This paper studies the probem of binary abe prediction on a graph. In this probem, we are given a graph G = (V, E), where the edges E (which can be weighted) capture the simiarity reationship between the objects represented by the nodes V. Each node has an initiay unknown abe associated with it, given by a signa f : V { 1, +1}. The goa is to reconstruct the entire signa by samping its vaues on a sma subset of the nodes. This is achievabe when the signa bears some degree of smoothness over the graph, which means that simiar objects are more ikey to have the same abe. Active earning aims to minimize the number of sampes needed by seecting the most informative nodes. This probem arises in many machine earning appications, where there is an abundance of unabeed data but abeed data is scarce and expensive to obtain, for exampe, requiring human expertise or eaborate experiments. Active earning is an effective way to minimize the cost of abeing in such scenarios [1]. A graph based approach to this probem starts by creating a graph where the nodes correspond to the data points X = {x 1,..., x n} and the edges capture the simiarity between them. Typicay, a sparse graph which connects each data point to few of its most simiar neighbors is used. This work is supported in part by NSF under grants CCF-1419 and CCF The unknown abes f i { 1, +1} associated with the data points define a binary function on the nodes. In datasets of interest, the signa is often notaby smooth on the graph. There are two approaches to active earning on graphs. The first approach focuses on identifying the nodes near the boundary region, where the signa changes from +1 to 1. The methods with this approach [2, 3] sampe nodes sequentiay, i.e., the nodes to be samped next are chosen based on the graph structure as we as previousy observed signa vaues. The second approach [4, 5, 6], in contrast, utiizes goba properties of the graph in order to identify the most informative nodes, and sampe them a at once. Such goba approaches usuay focus on providing a good approximation of the signa, rather than exact recovery. It is aso possibe to combine the two approaches, for exampe, as in [7]. The first contribution of this paper is a new samping agorithm, caed weighted S 2 which takes the boundary samping approach. The weighted S 2 agorithm is a generaization of a recenty proposed agorithm caed S 2 [2], which is defined ony for the case of unweighted edges. The purpose of the generaized agorithm is to take advantage of the additiona information avaiabe in the form of the edge weights, in order to reduce the samping compexity. We characterize the samping budget required for signa recovery by the weighted S 2 agorithm, as a function of the compexity of the signa with respect to the graph. We expain how this generaization can be usefu in reducing samping compexity, and demonstrate a significant reduction (neary 25% in one dataset), when neighboring nodes of opposite abes are consideraby ess simiar to each other than identicay abeed neighbors. We further compare the samping compexity of the weighted S 2 agorithm with an aternative state-of-the-art method caed the cutoff maximization method [4]. Unike the S 2 methods, which aim for a compete recovery of the signa by aggressivey searching for the boundary nodes, the cutoff maximization method is focused ony on providing a good approximation of the signa by ensuring that the unsamped nodes are we-connected to the samped nodes [8]. This method finds a samping set by optimizing a spectra function defined using the graph Lapacian. We perform the comparison on three reaistic data sets, and observe two interesting resuts: 1. The cutoff maximization method does not discover the entire boundary (i.e, nodes with oppositey abeed neighbors) uness the samping set contains amost a the nodes. In contrast, the number of nodes required by the S 2 methods to discover the boundary is not consideraby arger than the number of boundary nodes. 2. When the samping budget is quite imited, the cutoff maximization method provides a much better approximation of the signa. There exists a threshod in terms of the samping budget that determines which method offers better accuracy. Conversey, the toerabe degree of inaccuracy determines which of the methods offer ower samping compexity.

2 Motivated by the second observation, we propose a hybrid approach (simiar in spirit to [7]) which sampes the first few nodes with the cutoff maximization method to approximate the boundary and then switches to the weighted S 2 method to refine the approximation. The experiments suggest that the hybrid approach combines the advantages of both methods. 2. S 2 ALGORITHM FOR WEIGHTED GRAPHS The S 2 agorithm was proposed in [2] for unweighted graphs. In this section we describe the principe of the agorithm, and then generaize the agorithm to weighted graphs. In the next section we anayze the query compexity of the generaized agorithm Origina S 2 Agorithm The goa of the agorithm is to find the signa f. To do this, the agorithm operates by finding the edges that connect oppositey abeed nodes. These edges are caed cut edges, and together the set of cut edges is caed the cut. The agorithm incrementay identifies the cut, with the rationae that once the entire cut is identified, the signa is competey recovered. To find the cut, the agorithm maintains a copy of the graph G, and each time it sampes a node that neighbors previousy samped nodes of opposite abe, it removes the newy discovered cut edges from the graph copy. This way, the remaining graph copy contains ony the undiscovered part of the cut, and the agorithm can more easiy focus on discovering these edges. The main idea of the agorithm is to ook for a pair of oppositey abeed nodes, and then to find the cut edge on the shortest path between the nodes, using a binary search procedure. The agorithm begins with a random samping phase. At this phase the agorithm queries a random node, according to the uniform distribution. After each sampe, the agorithm checks whether the samped node has any previousy samped neighbors of opposite abe. If such neighbors exist, then the connecting edges are newy discovered cut edges, and they are removed from the graph. After checking and potentiay removing newy discovered cut edges, the agorithm checks whether the remaining graph contains any pair of connected nodes of opposite abes. If no such pair exists, the agorithm proceeds with a random sampe. If pairs of connected, oppositey abeed nodes do exist, the agorithm ooks for the shortest path among a the paths connecting such pairs, and sampe the node in the midde of that path (breaking ties arbitrariy). After each samping operation, either a random sampe or a bisecting sampe, the agorithm again removes a newy discovered cut edges. The S 2 agorithm is described more formay in Agorithm 1. The agorithm is given a budget, which determines the number of queries to perform. Once the budget is exhausted, the agorithm cas a abe competion function, which predicts the abes of the unsamped nodes. Severa such abe competion agorithms are known, such as the POCS method in [9]. The S 2 agorithm uses a function caed Midde Shortest Path (MSP), which returns the node in the midde of the shortest path among a the paths connecting a pair of oppositey abes nodes in the remaining graph Generaization for Weighted Graphs The S 2 agorithm in [2] is defined ony for unweighted graphs. Since many earning scenarios provide a weighted graph, we extend the agorithm to expoit the additiona avaiabe information by modifying the MSP function in the agorithm. Our modification is based on the assumption that the signa is smooth, which means that high-weight edges connect mosty nodes of the same abe. Therefore, the weight of cut edges is generay ow. Agorithm 1 S 2 Agorithm Inputs: Graph G, BUDGET n. 1: L 2: whie 1 do 3: x Randomy chosen unabeed node. 4: do 5: Add (x, f(x)) to L. 6: Remove newy discovered cut edges from G. 7: if L = BUDGET then 8: return LabeCompetion(G, L) 9: end if 1: whie x MSP(G,L) exists 11: end whie Fig. 1: An iustration of advantage of weighted S 2 over unweighted S 2. In the unweighted S 2 agorithm, each MSP query reduces the number of nodes in the shortest of the paths between any two oppositey abeed nodes by approximatey one haf. The main idea in our generaization is to take advantage of the ow weights of cut edges in order to reduce this number by more than a haf with each query. To do this, we first switch our perspective, for convenience, from the edge weights to the distances associated with the edges, which are inversey proportiona to the weights. The distance between nonneighboring nodes is defined as the sum of the engths of the edges in the shortest path connecting the nodes. Since the weights are a decreasing function of the distances, it foows that cut edges are typicay onger than other edges. We take advantage of this fact by modifying the MSP function to sampe the node cosest to the midpoint of the path, where the midpoint is computed in terms of ength, rather than the number of edges in the path. With each query, the proposed samping rue can potentiay reduce the number of nodes aong the shortest of the paths between any two oppositey abeed nodes by more than haf if the cut edges contribute significanty more than the non-cut edges to the ength of the path. Thus, utimatey it requires ess sampes to discover a cut edge. This intuition is demonstrated in Figure 1. In this exampe, the nodes abeed +1 are connected with an edge of ength /2, the nodes abeed 1 are connected with an edge of ength and the cut edge is of ength 3. Given the abes of the end nodes of this path, the binary search phase of the unweighted S 2 agorithm needs to sampe abes of 3 extra nodes to discover the cut edge. The weighted S 2 agorithm, on the other hand, finds the cut edge with ony 2 sampes. This type of situation arises more prominenty in an unbaanced data set, where the number of nodes in one cass is much arger than the other. The advantage of weighted S 2 agorithm in such a case is experimentay verified in Section ANALYSIS OF THE WEIGHTED S 2 ALGORITHM 3.1. Notation Note that f partitions the vertices of G into a coection of connected components with identicay abeed vertices. Let V 1, V 2,..., V k be these k connected components. Notice that the first node that S 2 queries in each coection V i is often queried randomy, and not by a bisection query. Define V i β min 1 i k n.

3 Fig. 2: The interva of interest is at east haved after two queries. If β is sma, more random queries are required by S 2. Let C be the set of cut edges in G. The ength of the shortest cut edge in G is denoted by cut. Let C be the set of nodes which share an edge with at east one oppositey abeed node. The nodes in C are caed boundary nodes. For 1 i < j k, et C i,j be the subset of C for which each edge {x, y} C i,j satisfies x V i and y V j. If C i,j is not empty for some 1 i < j k, then it is caed a cut component. The number of cut components is denoted by m. For a pair of nodes v 1 and v 2, et d(v 1, v 2) denote the ength of the shortest path between v 1 and v 2. Define n = max v1,v 2 V d(v 1, v 2) Cut Custering For nodes x, y V, et d G (x, y) be the ength of the shortest path connecting x and y in G. Let e 1 = {x 1, y 1} and e 2 = {x 2, y 2} be a pair of cut edges in G such that f(x 1) = f(x 2) and f(y 1) = f(y 2). Define δ(e 1, e 2) = d G C (x 1, x 2) + d G C (y 1, y 2) + max{ e1, e2 }, where G C is the graph G with a the cut edges removed. Let H r = (C, E) be the meta graph whose nodes are the cut edges of G, and {e, e } E iff δ(e, e ) r. Let κ be the smaest number for which H κ has m connected components. The motivation for the definition of κ is demonstrated in the foowing emma. Lemma 1. Consider a case in which after the remova of an edge e by the weighted S 2 agorithm, there exist an undiscovered cut edge e in the same cut component of e. Then the ength of the shortest path between two oppositey abeed nodes in the remaining graph is at most κ. Proof. Consider the connected component of the meta graph H κ that contains the removed cut edge (as a meta node). By the assumptions of the emma, there must be at east one meta edge in this connected component that connects a discovered cut edge to an undiscovered cut edge. This meta edge corresponds to a path ength at most κ in the remaining graph, proving the emma Query Compexity Theorem 1. Suppose that a graph G = (V, E) and a signa f are such that the induced cut set C has m components with cut custering κ. Then for any ɛ >, the weighted S 2 wi recover C with probabiity at east 1 ɛ if the BUDGET is at east m 2 og 2 ( n cut +( C m) 2 og 2 ( κ cut + og(1/(βɛ)) og(1/(1 β)). The proof of Theorem 1 uses the fact that after a pair of connected and oppositey abeed nodes is found, the number of queries unti a boundary node is samped is at most ogarithmic in (/ cut), where is the ength of the path between the nodes. We show this fact in the foowing emma. Lemma 2. The ( cut-edge of ength cut is found after no more than r = 2 og 2 aggressive steps. cut Proof. In Figure 2, et C and D denote the queried nodes, such that C is samped first. Let E denote the midpoint of the interva between A and B, where there may not be a node. By considering a the cases based on abes of C and D and their positions reative to E, it can be shown that after two queries, the ength of the interva of interest (i.e., the interva containing the cut-edge) is at east haved. The detais are omitted in the interest of space. After i (where i is even) queries, the ength of the interva of interest is at most. Note that the cut-edge is found when the 2 ength of the interva i/2 of interest is ess than or equa to cut. If r is the maximum number of queries required to ocate the cut-edge, then = cut r = 2r/2 2 og 2 ( cut Proof of Theorem 1. The random samping phase foows the same argument as in [2], which gives the term og(1/(βɛ)). og(1/(1 β)) A sequence of bisection queries that commences after a random query or an edge remova, and terminates with an edge remova, is ca a run. In each run, at east one boundary node is being queried. Therefore, the number of runs is no greater than C. In each cut component, after the first cut edge and boundary node are discovered, ( the rest of the boundary nodes are discovered in 2 og κ 2 cut queries each, according to Lemmas 1 and 2. For discovering the first cut edge in each cut ( components, we triviay bound the number of queries by 2 og n 2 cut. Since there are m cut components, we bound the tota number of bisection queries by ( ( n κ m 2 og 2 + ( C m) 2 og 2. cut cut 4. EXPERIMENTS We consider the foowing graph based cassification probems: 1. USPS handwritten digit recognition [1]: For our experiments, we consider two binary cassification sub-probems, namey, 7 vs. 9 and 2 vs. 4, consisting of 2 randomy seected images of each cass represented as vectors of dimension 256. The distance between two data points is d(i, j) = x i x j. An unweighted graph G is constructed by connecting a pair of nodes (i, j) if j is a k-nearest neighbor (k-nn with k = 4) 1 of i or vice versa. A weighted dissimiarity graph G d is defined to have the same topoogy as G but the weight associated with edge (i, j) is set to d(i, j). A weighted simiarity graph G w is defined to have the same topoogy as G but the weight associated with edge (i, j) is set to w(i, j) = exp ( d(i, j) 2 /2σ 2). The parameter σ is set to be 1/3-rd of the average distance to the k-th nearest neighbor for a datapoints. 2. Newsgroups text cassification [11]: For our experiments, we consider a binary cassification sub-probem Baseba vs. Hockey, where each cass contains 2 randomy seected documents. Each document i is represented by a 3 dimensiona vector x i whose eements are the tf-idf statistics of the 3 most frequent words in the dataset [8]. The cosine simiarity between a pair data points (i, j) is given by w(i, j) = x i,x j x i x j. The distance between them is defined as d(i, j) = 1 w(i, j) 2. The k-nn unweighted graph G (with k = 4), the dissimiarity graph G d and the simiarity graph G w are constructed using these distance and simiarity measures as in the previous exampe. 1 Due to ack of space, we do not study the effect of k in detai. (1)

4 Tabe 1: Number of sampes needed to discover a the cut edges. S 2 methods invove random samping. Average over 3 trias is reported. mean( Data n C C cut) mean( non-cut) n switch Two circes v v Baseba v Hockey Cassification error Cassification error Cassification error Number of sampes (a) 7 v Number of sampes (b) 2 v. 4 Fig. 3: Cassification error against the number of samped nodes Number of sampes (c) Baseba v. Hockey Fig. 4: A synthetic two circes dataset In addition to the above datasets, we generate a synthetic two circes dataset, shown in Figure 4, in order to demonstrate the advantage of weighted S 2 over unweighted S 2. It contains 9 points in one cass (marked red) eveny distributed on the inner circe of mean radius 1 and variance.5 and 1 points in the second cass (marked bue) on the outer circe of mean radius 1.1 and variance.45. A 4-nn graph is constructed using the Eucidean distance between the coordinates of the points. We compare the performance of the foowing active earning methods: (1) unweighted S 2 method [2] with graph G, (2) weighted S 2 method with dissimiarity graph G d, (3) cutoff maximization method [8] with simiarity graph G w and (4) a hybrid approach combining cutoff maximization and weighted S 2 method. After the nodes seected by each method have been samped, we reconstruct the unknown abe signa using the approximate POCS based bandimited reconstruction scheme [8] to get the soft abes. We threshod these soft abes to get the fina abe predictions. The hybrid approach uses the non-adaptive cutoff maximization approach in the beginning and switches to the weighted S 2 method after samping a certain number of nodes n switch. In order to determine n switch, after samping the i-th node with the cutoff method, we compute 1 ˆf i,ˆf i 1, where ˆf i denotes the vector of predicted soft abes. Once this vaue fas beow beow.1, indicating that ˆf i ˆf i 1 the e newy added abe ony marginay changed the predictions, hybrid approach switches to weighted S 2. Tabe 1 ists the number of sampes required by each of the samping methods to discover a the cut edges using the observed abes. It shows that weighted S 2 can reduce the sampe compexity significanty (by 25%) compared to unweighted S 2 if the ratio of mean ength of cut edges and mean ength of non-cut edges is high as is the case in the unbaanced two circes dataset. In rest of the datasets, the gain offered by weighted S 2 is negigibe since the cut edges are ony sighty onger than non-cut edges and as a resut, taking engths into account in the bisection phase does not offer much advantage. We aso observe that the number of sampes required by the weighted S 2 method is cose to the size of the cut C in most of the datasets. Tabe 1 aso shows that the adaptive methods S 2 and weighted S 2 are very efficient at recovering the entire cut exacty, compared to the non-adaptive cutoff maximization method. In practice, it is not necessary to reconstruct the signa exacty and some reconstruction error is aowed. Figure 3 pots the cassification error against the number of samped nodes. It shows that the cassification error of the cutoff maximization method decreases rapidy in the eary stages of samping when very few sampes are observed. However, the decrease is sow in ater stages of samping. S 2 methods, on the other hand, are good at reducing the error in the ater stages, but performs poory with ony a few sampes. The figure aso shows that the hybrid method performs as we as the better method in each region. 5. CONCLUSIONS The paper generaizes the S 2 agorithm for the case of weighted graphs. The samping compexity of the generaized agorithm is anayzed, and the gain over the unweighted version is demonstrated by simuation. Additiona experiments identify the region of toerabe reconstruction error in which the S 2 agorithms outperforms a graph frequency based goba approach. A hybrid approach is proposed with the advantages of both methods. It remains open to anayticay characterize of the gain of the weighted S 2 method over the unweighted version. Another interesting avenue for future work is to provide a performance anaysis for the spectra samping method which can suggest an optima switching criterion for the hybrid method.

5 6. REFERENCES [1] Burr Settes, Active earning iterature survey, Computer Sciences Technica Report 1648, University of Wisconsin Madison, 21. [2] Gautam Dasarathy, Robert Nowak, and Xiaojin Zhu, S 2 : An efficient graph based active earning agorithm with appication to nonparametric cassification, Journa of Machine Learning Research, vo. 4, pp. 1 2, 215. [3] Xiaojin Zhu, John Lafferty, and Zoubin Ghahramani, Combining active earning and semi-supervised earning using gaussian fieds and harmonic functions, in ICML 23 workshop on the continuum from abeed to unabeed data in machine earning and data mining, 23, pp [4] A. Anis, A. Gadde, and A. Ortega, Towards a samping theorem for signas on arbitrary graphs, in Acoustics, Speech and Signa Processing (ICASSP), IEEE Internationa Conference on, 214. [5] M. Ji and J. Han, A variance minimization criterion to active earning on graphs, in Internationa Conference on Artificia Inteigence and Statistics (AISTATS), 212, vo. 22, pp [6] Andrew Guiory and Jeff Bimes, Active semi-supervised earning using submoduar functions, in Proceedings of 27th Conference on Uncertainty in Artificia Inteigence, 211, pp [7] Thomas Osugi, Deng Kim, and Stephen Scott, Baancing exporation and expoitation: A new agorithm for active machine earning, in Data Mining, Fifth IEEE Internationa Conference on. IEEE, 25, pp. 8 pp. [8] A. Gadde, A. Anis, and A. Ortega, Active semi-supervised earning using samping theory for graph signas, in ACM SIGKDD Internationa Conference on Knowedge Discovery and Data Mining, 214, pp [9] S. K. Narang, A. Gadde, E. Sanou, and A. Ortega, Locaized iterative methods for interpoation in graph structured data, in Signa and Information Processing (GobaSIP), IEEE Goba Conference on, 213. [1] USPS handwritten digits data, edu/ roweis/data.htm. [11] 2 newsgroups data, jason/ 2Newsgroups/.