Most of real-world graphs are dynamic , i.e., they change over time. However, while the regression problem has been studied for static graphs, it has not been investigated for dynamic graphs, yet. In the current paper, first we present the notion of update-efficient matrix embedding that defines the conditions sufficient for a matrix embedding to be used for the dynamic graph regression problem, efficiently. We also show that some of the standard matrix embeddings, e.g., the (weighted) adjacency matrix, satisfy these conditions. Then, we prove that given an @math update-efficient matrix embedding, after an update operation in the graph, the optimal solution of the graph regression problem for the revised graph can be computed in @math time. In particular, using the (weighted) adjacency matrix as the matrix embedding of @math , it takes @math time to update the optimal solution, where @math is the number of nodes of the revised graph. To the best of our knowledge, this is the first result on updating the solution of the graph regression problem, in a time considerably less than the time of computing the solution from the scratch. Finally, we study a generalization of the dynamic graph regression problem and show that it can be solved in @math space.
In recent years, a number of algorithms have been proposed for different learning problems over nodes of an static graph. @cite_21 studied the classification problem for nodes of an static graph and showed the connection of their general formulation to Markov random fields. With @math the number of possible node labels, they also presented an @math -approximation algorithm for the classification problem, which is run in polynomial time. @cite_19 studied the problem of online label prediction of a graph with the perceptron. The key difference between online and dynamic settings is that online setting is used in the cases where it is computationally infeasible to solve the learning problem over the entire dataset. However, in dynamic setting the learning problem can be solved over the entite dataset and the challenge is to efficiently update the solution when the dataset changes. Some other examples of online learning for graphs include @cite_9 , @cite_18 , @cite_12 and @cite_2 .
@cite_18 @cite_9 @cite_21 @cite_19 @cite_2 @cite_12
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We consider the problem of using a large unlabeled sample to boost performance of a learning algorit,hrn when only a small set of labeled examples is available. In particular, we consider a problem setting motivated by the task of learning to classify web pages, in which the description of each example can be partitioned into two distinct views. For example, the description of a web page can be partitioned into the words occurring on that page, and the words occurring in hyperlinks t,hat point to that page. We assume that either view of the example would be sufficient for learning if we had enough labeled data, but our goal is to use both views together to allow inexpensive unlabeled data to augment, a much smaller set of labeled examples. Specifically, the presence of two distinct views of each example suggests strategies in which two learning algorithms are trained separately on each view, and then each algorithm’s predictions on new unlabeled examples are used to enlarge the training set of the other. Our goal in this paper is to provide a PAC-style analysis for this setting, and, more broadly, a PAC-style framework for the general problem of learning from both labeled and unlabeled data. We also provide empirical results on real web-page data indicating that this use of unlabeled examples can lead to significant improvement of hypotheses in practice. *This research was supported in part by the DARPA HPKB program under contract F30602-97-1-0215 and by NSF National Young investigator grant CCR-9357793. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. TO copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and or a fee. COLT 98 Madison WI USA Copyright ACM 1998 l-58113-057--0 98 7... 5.00 92 Tom Mitchell School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213-3891 mitchell+@cs.cmu.edu We consider the problem of active learning over the vertices in a graph, without feature representation. Our study is based on the common graph smoothness assumption, which is formulated in a Gaussian random field model. We analyze the probability distribution over the unlabeled vertices conditioned on the label information, which is a multivariate normal with the mean being the harmonic solution over the field. Then we select the nodes to label such that the total variance of the distribution on the unlabeled data, as well as the expected prediction error, is minimized. In this way, the classifier we obtain is theoretically more robust. Compared with existing methods, our algorithm has the advantage of selecting data in a batch offline mode with solid theoretical support. We show improved performance over existing label selection criteria on several real world data sets. We study the problem of online prediction of a noisy labeling of a graph with the perceptron. We address both label noise and concept noise. Graph learning is framed as an instance of prediction on a finite set. To treat label noise we show that the hinge loss bounds derived by Gentile [1] for online perceptron learning can be transformed to relative mistake bounds with an optimal leading constant when applied to prediction on a finite set. These bounds depend crucially on the norm of the learned concept. Often the norm of a concept can vary dramatically 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 cut size of a partitioning of the graph – which are only indirectly dependent on the size of the graph. The impossibility of such bounds for the graph geodesic nearest neighbors algorithm will be demonstrated. We consider dynamic evaluation of algebraic functions such as computing determinant, matrix adjoint, matrix inverse and solving linear system of equations. We show that in the dynamic setup the above problems can be solved faster than evaluating everything from scratch. In the case when rows and columns of the matrix can change we show an algorithm that achieves O(n sup 2 ) arithmetic operations per update and O(1) arithmetic operations per query. Next, we describe two algorithms, with different tradeoffs, for updating the inverse and determinant when single entries of the matrix are changed. The fastest update for the first tradeoff is O(n sup 1.575 ) arithmetic operations per update and O(n sup 0.575 ) arithmetic operations per query. The second tradeoff gives O(n sup 1.495 ) arithmetic operations per update and O(n sup 1.495 ) arithmetic operations per query. We also consider the case when some number of columns or rows can change. We use dynamic determinant computations to solve the following problems in the dynamic setup: computing the number of spanning trees in a graph and testing if an edge in a graph is contained in some perfect matching. These are the first dynamic algorithms for these problems. Next, with the use of dynamic matrix inverse, we solve fully dynamic transitive closure in general directed graphs. The bounds on arithmetic operations for dynamic matrix inverse translate directly to time bounds for dynamic transitive closure. Thus we obtain the first known algorithm with O(n sup 2 ) worst-case update time and constant query time and two algorithms for transitive closure in general digraphs with subquadratic update and query times. Our algorithms for transitive closure are randomized with one-sided error. We also consider for the first time the case when the edges incident with a part of vertices of the graph can be changed. In this paper we present a Bayesian framework for parsing images into their constituent visual patterns. The parsing algorithm optimizes the posterior probability and outputs a scene representation as a "parsing graph", in a spirit similar to parsing sentences in speech and natural language. The algorithm constructs the parsing graph and re-configures it dynamically using a set of moves, which are mostly reversible Markov chain jumps. This computational framework integrates two popular inference approaches--generative (top-down) methods and discriminative (bottom-up) methods. The former formulates the posterior probability in terms of generative models for images defined by likelihood functions and priors. The latter computes discriminative probabilities based on a sequence (cascade) of bottom-up tests filters. In our Markov chain algorithm design, the posterior probability, defined by the generative models, is the invariant (target) probability for the Markov chain, and the discriminative probabilities are used to construct proposal probabilities to drive the Markov chain. Intuitively, the bottom-up discriminative probabilities activate top-down generative models. In this paper, we focus on two types of visual patterns--generic visual patterns, such as texture and shading, and object patterns including human faces and text. These types of patterns compete and cooperate to explain the image and so image parsing unifies image segmentation, object detection, and recognition (if we use generic visual patterns only then image parsing will correspond to image segmentation (Tu and Zhu, 2002. IEEE Trans. PAMI, 24(5):657--673). We illustrate our algorithm on natural images of complex city scenes and show examples where image segmentation can be improved by allowing object specific knowledge to disambiguate low-level segmentation cues, and conversely where object detection can be improved by using generic visual patterns to explain away shadows and occlusions. Graph-based approaches have been most successful in semisupervised learning. In this paper, we focus on label propagation in graph-based semisupervised learning. One essential point of label propagation is that the performance is heavily affected by incorporating underlying manifold of given data into the input graph. The other more important point is that in many recent real-world applications, the same instances are represented by multiple heterogeneous data sources. A key challenge under this setting is to integrate different data representations automatically to achieve better predictive performance. In this paper, we address the issue of obtaining the optimal linear combination of multiple different graphs under the label propagation setting. For this problem, we propose a new formulation with the sparsity (in coefficients of graph combination) property which cannot be rightly achieved by any other existing methods. This unique feature provides two important advantages: 1) the improvement of prediction performance by eliminating irrelevant or noisy graphs and 2) the interpretability of results, i.e., easily identifying informative graphs on classification. We propose efficient optimization algorithms for the proposed approach, by which clear interpretations of the mechanism for sparsity is provided. Through various synthetic and two real-world data sets, we empirically demonstrate the advantages of our proposed approach not only in prediction performance but also in graph selection ability. Over the last years, deep convolutional neural networks (ConvNets) have transformed the field of computer vision thanks to their unparalleled capacity to learn high level semantic image features. However, in order to successfully learn those features, they usually require massive amounts of manually labeled data, which is both expensive and impractical to scale. Therefore, unsupervised semantic feature learning, i.e., learning without requiring manual annotation effort, is of crucial importance in order to successfully harvest the vast amount of visual data that are available today. In our work we propose to learn image features by training ConvNets to recognize the 2d rotation that is applied to the image that it gets as input. We demonstrate both qualitatively and quantitatively that this apparently simple task actually provides a very powerful supervisory signal for semantic feature learning. We exhaustively evaluate our method in various unsupervised feature learning benchmarks and we exhibit in all of them state-of-the-art performance. Specifically, our results on those benchmarks demonstrate dramatic improvements w.r.t. prior state-of-the-art approaches in unsupervised representation learning and thus significantly close the gap with supervised feature learning. For instance, in PASCAL VOC 2007 detection task our unsupervised pre-trained AlexNet model achieves the state-of-the-art (among unsupervised methods) mAP of 54.4 that is only 2.4 points lower from the supervised case. We get similarly striking results when we transfer our unsupervised learned features on various other tasks, such as ImageNet classification, PASCAL classification, PASCAL segmentation, and CIFAR-10 classification. The code and models of our paper will be published on: this https URL . This work addresses graph-based semi-supervised classification and betweenness computation in large, sparse, networks (several millions of nodes). The objective of semi-supervised classification is to assign a label to unlabeled nodes using the whole topology of the graph and the labeling at our disposal. Two approaches are developed to avoid explicit computation of pairwise proximity between the nodes of the graph, which would be impractical for graphs containing millions of nodes. The first approach directly computes, for each class, the sum of the similarities between the nodes to classify and the labeled nodes of the class, as suggested initially in [1,2]. Along this approach, two algorithms exploiting different state-of-the-art kernels on a graph are developed. The same strategy can also be used in order to compute a betweenness measure. The second approach works on a trellis structure built from biased random walks on the graph, extending an idea introduced in [3]. These random walks allow to define a biased bounded betweenness for the nodes of interest, defined separately for each class. All the proposed algorithms have a linear computing time in the number of edges while providing good results, and hence are applicable to large sparse networks. They are empirically validated on medium-size standard data sets and are shown to be competitive with state-of-the-art techniques. Finally, we processed a novel data set, which is made available for benchmarking, for multi-class classification in a large network: the U.S. patents citation network containing 3M nodes (of six different classes) and 38M edges. The three proposed algorithms achieve competitive results (around 85 classification rate) on this large network-they classify the unlabeled nodes within a few minutes on a standard workstation. We consider how to assign labels to any undirected graph with n nodes such that, given the labels of two nodes and no other information regarding the graph, it is possible to determine the distance between the two nodes. The challenge in such a distance labeling scheme is primarily to minimize the maximum label lenght and secondarily to minimize the time needed to answer distance queries (decoding). Previous schemes have offered different trade-offs between label lengths and query time. This paper presents a simple algorithm with shorter labels and shorter query time than any previous solution, thereby improving the state-of-the-art with respect to both label length and query time in one single algorithm. Our solution addresses several open problems concerning label length and decoding time and is the first improvement of label length for more than three decades. More specifically, we present a distance labeling scheme with label size (log 3) 2 + o(n) (logarithms are in base 2) and O(1) decoding time. This outperforms all existing results with respect to both size and decoding time, including Winkler's (Combinatorica 1983) decade-old result, which uses labels of size (log 3)n and O(n log n) decoding time, and (SODA'01), which uses labels of size 11n + o(n) and O(loglog n) decoding time. In addition, our algorithm is simpler than the previous ones. In the case of integral edge weights of size at most W, we present almost matching upper and lower bounds for label sizes. For r-additive approximation schemes, where distances can be off by an additive constant r, we give both upper and lower bounds. In particular, we present an upper bound for 1-additive approximation schemes which, in the unweighted case, has the same size (ignoring second order terms) as an adjacency scheme: n 2. We also give results for bipartite graphs and for exact and 1-additive distance oracles. In this paper, we propose a general framework for graph matching which is suitable for different problems of pattern recognition. The pattern representation we assume is at the same time highly structured, like for classic syntactic and structural approaches, and of subsymbolic nature with real-valued features, like for connectionist and statistic approaches. We show that random walk based models, inspired by Google's PageRank, give rise to a spectral theory that nicely enhances the graph topological features at node level. As a straightforward consequence, we derive a polynomial algorithm for the classic graph isomorphism problem, under the restriction of dealing with Markovian spectrally distinguishable graphs (MSD), a class of graphs that does not seem to be easily reducible to others proposed in the literature. The experimental results that we found on different test-beds of the TC-15 graph database show that the defined MSD class "almost always" covers the database, and that the proposed algorithm is significantly more efficient than top scoring VF algorithm on the same data. Most interestingly, the proposed approach is very well-suited for dealing with partial and approximate graph matching problems, derived for instance from image retrieval tasks. We consider the objects of the COIL-100 visual collection and provide a graph-based representation, whose node's labels contain appropriate visual features. We show that the adoption of classic bipartite graph matching algorithms offers a straightforward generalization of the algorithm given for graph isomorphism and, finally, we report very promising experimental results on the COIL-100 visual collection. Over the last years, deep convolutional neural networks (ConvNets) have transformed the field of computer vision thanks to their unparalleled capacity to learn high level semantic image features. However, in order to successfully learn those features, they usually require massive amounts of manually labeled data, which is both expensive and impractical to scale. Therefore, unsupervised semantic feature learning, i.e., learning without requiring manual annotation effort, is of crucial importance in order to successfully harvest the vast amount of visual data that are available today. In our work we propose to learn image features by training ConvNets to recognize the 2d rotation that is applied to the image that it gets as input. We demonstrate both qualitatively and quantitatively that this apparently simple task actually provides a very powerful supervisory signal for semantic feature learning. We exhaustively evaluate our method in various unsupervised feature learning benchmarks and we exhibit in all of them state-of-the-art performance. Specifically, our results on those benchmarks demonstrate dramatic improvements w.r.t. prior state-of-the-art approaches in unsupervised representation learning and thus significantly close the gap with supervised feature learning. For instance, in PASCAL VOC 2007 detection task our unsupervised pre-trained AlexNet model achieves the state-of-the-art (among unsupervised methods) mAP of 54.4 that is only 2.4 points lower from the supervised case. We get similar striking results when we transfer our unsupervised learned features on various other tasks, such as ImageNet classification, PASCAL classification, PASCAL segmentation, and CIFAR-10 classification. We explore scalable and accurate dynamic pattern detection methods in graph-based data sets. We apply our proposed Dynamic Subset Scan method to the task of detecting, tracking, and source-tracing contaminant plumes spreading through a water distribution system equipped with noisy, binary sensors. While static patterns affect the same subset of data over a period of time, dynamic patterns may affect different subsets of the data at each time step. These dynamic patterns require a new approach to define and optimize penalized likelihood ratio statistics in the subset scan framework, as well as new computational techniques that scale to large, real-world networks. To address the first concern, we develop new subset scan methods that allow the detected subset of nodes to change over time, while incorporating temporal consistency constraints to reward patterns that do not dramatically change between adjacent time steps. Second, our Additive Graph Scan algorithm allows our novel scan statistic to process small graphs (500 nodes) in 4.1 seconds on average while maintaining an approximation ratio over 99 compared to an exact optimization method, and to scale to large graphs with over 12,000 nodes in 30 minutes on average. Evaluation results across multiple detection, tracking, and source-tracing tasks demonstrate substantial performance gains achieved by the Dynamic Subset Scan approach. The authors consider the problem of learning a linear threshold function (a halfspace in n dimensions, also called a "perceptron"). Methods for solving this problem generally fall into two categories. In the absence of noise, this problem can be formulated as a linear program and solved in polynomial time with the ellipsoid algorithm (or interior point methods). On the other hand, simple greedy algorithms such as the perceptron algorithm seem to work well in practice and can be made noise tolerant; but, their running time depends on a separation parameter (which quantifies the amount of "wiggle room" available) and can be exponential in the description length of the input. They show how simple greedy methods can be used to find weak hypotheses (hypotheses that classify noticeably more than half of the examples) in polynomial time, without dependence on any separation parameter. This results in a polynomial-time algorithm for learning linear threshold functions in the PAC model in the presence of random classification noise. The algorithm is based on a new method for removing outliers in data. Specifically, for any set S of points in R sup n , each given to b bits of precision, they show that one can remove only a small fraction of S so that in the remaining set T, for every vector v, max sub x spl epsiv T (v spl middot x) sup 2 spl les poly(n,b)|T| sup -1 spl Sigma sub x spl epsiv T (v spl middot x) sup 2 . After removing these outliers, they are able to show that a modified version of the perceptron learning algorithm works in polynomial time, even in the presence of random classification noise. A common classifier for unlabeled nodes on undirected graphs uses label propagation from the labeled nodes, equivalent to the harmonic predictor on Gaussian random fields (GRFs). For active learning on GRFs, the commonly used V-optimality criterion queries nodes that reduce the L2 (regression) loss. V-optimality satisfies a submodularity property showing that greedy reduction produces a (1 - 1 e) globally optimal solution. However, L2 loss may not characterise the true nature of 0 1 loss in classification problems and thus may not be the best choice for active learning. We consider a new criterion we call Σ-optimality, which queries the node that minimizes the sum of the elements in the predictive covariance. Σ-optimality directly optimizes the risk of the surveying problem, which is to determine the proportion of nodes belonging to one class. In this paper we extend submodularity guarantees from V-optimality to Σ-optimality using properties specific to GRFs. We further show that GRFs satisfy the suppressor-free condition in addition to the conditional independence inherited from Markov random fields. We test Σ-optimality on real-world graphs with both synthetic and real data and show that it outperforms V-optimality and other related methods on classification. Machine learning research has, to a great extent, ignored an important aspect of many real world applications: time. Existing concept learners predominantly operate on a static set of attributes; for example, classifying flowers described by leaf size, pet al colour and pet al count. However, many real datasets are not static; they cannot sensibly be represented as a fixed set of attributes. Rather, the examples are expressed as features that vary temporally, and it is the temporal variation itself that is used for classification. Consider a simple gesture recognition domain, in which the temporal features are the position of the hands, finger bends, and so on. Looking at the position of the hand at one point in time is not likely to lead to a successful classification; it is only by analysing changes in position that recognition is possible. This thesis presents a new technique for temporal classification. By extracting sub-events from the training instances and parameterising them to allow feature construction for a subsequent learning process, it is able to employ background knowledge and express learnt concepts in terms of the background knowledge. The novel results of the thesis are: a temporal learner capable of producing comprehensible and accurate classifiers for multivariate time series that can learn from a small number of instances and can integrate non-temporal features; a feature construction technique that parameterises sub-events of the training set and clusters them to construct features for a propositional learner; and a technique for post-processing classification rules produced by the learner to give a comprehensible description expressed in the same form as the original background knowledge. The thesis discusses the implementation of TClass, a temporal learner. Results show rules that are comprehensible in many cases and accuracy results close to or better than existing technique—over 98 per cent for sign language and 72 per cent for ECGs (equivalent to the accuracy of a human cardiologist). One further surprising result is that a small set of very primitive sub-events proves to be functional, avoiding the need for labour-intensive background knowledge if it is not available. We present two algorithms for dynamically maintaining a spanning forest of a graph undergoing edge insertions and deletions. Our algorithms guarantee worst-case update time and work against an adaptive adversary, meaning that an edge update can depend on previous outputs of the algorithms. We provide the first polynomial improvement over the long-standing O(√n) bound of [Frederickson STOC'84, Eppstein, Galil, Italiano and Nissenzweig FOCS'92] for such type of algorithms. The previously best improvement was O(√n (loglogn)2 logn) [Kejlberg-Rasmussen, Kopelowitz, Pettie and Thorup ESA'16]. We note however that these bounds were obtained by deterministic algorithms while our algorithms are randomized. Our first algorithm is Monte Carlo and guarantees an O(n0.4+o(1)) worst-case update time, where the o(1) term hides the O(√loglogn logn) factor. Our second algorithm is Las Vegas and guarantee an O(n0.49306) worst-case update time with high probability. Algorithms with better update time either needed to assume that the adversary is oblivious (e.g. [Kapron, King and Mountjoy SODA'13]) or can only guarantee an amortized update time. Our second result answers an open problem by To the best of our knowledge, our algorithms are among a few non-trivial randomized dynamic algorithms that work against adaptive adversaries. The key to our results is a decomposition of graphs into subgraphs that either have high expansion or sparse. This decomposition serves as an interface between recent developments on (static) flow computation and many old ideas in dynamic graph algorithms: On the one hand, we can combine previous dynamic graph techniques to get faster dynamic spanning forest algorithms if such decomposition is given. On the other hand, we can adapt flow-related techniques (e.g. those from [Khandekar, Rao and Vazirani STOC'06], [Peng SODA'16], and [Orecchia and Zhu SODA'14]) to maintain such decomposition. To the best of our knowledge, this is the first time these flow techniques are used in fully dynamic graph algorithms. Multi-label learning refers to the problems where an instance can be assigned to more than one category. In this paper, we present a novel Semi-supervised algorithm for Multi-label learning by solving a Sylvester Equation (SMSE). Two graphs are first constructed on instance level and category level respectively. For instance level, a graph is defined based on both labeled and unlabeled instances, where each node represents one instance and each edge weight reflects the similarity between corresponding pairwise instances. Similarly, for category level, a graph is also built based on all the categories, where each node represents one category and each edge weight reflects the similarity between corresponding pairwise categories. A regularization framework combining two regularization terms for the two graphs is suggested. The regularization term for instance graph measures the smoothness of the labels of instances, and the regularization term for category graph measures the smoothness of the labels of categories. We show that the labels of unlabeled data finally can be obtained by solving a Sylvester Equation. Experiments on RCV1 data set show that SMSE can make full use of the unlabeled data information as well as the correlations among categories and achieve good performance. In addition, we give a SMSE’s extended application on collaborative filtering. We propose a new graph-based label propagation algorithm for transductive learning. Each example is associated with a vertex in an undirected graph and a weighted edge between two vertices represents similarity between the two corresponding example. We build on Adsorption, a recently proposed algorithm and analyze its properties. We then state our learning algorithm as a convex optimization problem over multi-label assignments and derive an efficient algorithm to solve this problem. We state the conditions under which our algorithm is guaranteed to converge. We provide experimental evidence on various real-world datasets demonstrating the effectiveness of our algorithm over other algorithms for such problems. We also show that our algorithm can be extended to incorporate additional prior information, and demonstrate it with classifying data where the labels are not mutually exclusive. We tackle the problem of object detection and pose estimation in a shared space downtown environment. For perception multiple laser scanners with 360 coverage were fused in a dynamic occupancy grid map (DOGMa). A single-stage deep convolutional neural network is trained to provide object hypotheses comprising of shape, position, orientation and an existence score from a single input DOGMa. Furthermore, an algorithm for offline object extraction was developed to automatically label several hours of training data. The algorithm is based on a two-pass trajectory extraction, forward and backward in time. Typical for engineered algorithms, the automatic label generation suffers from misdetections, which makes hard negative mining impractical. Therefore, we propose a loss function counteracting the high imbalance between mostly static background and extremely rare dynamic grid cells. Experiments indicate, that the trained network has good generalization capabilities since it detects objects occasionally lost by the label algorithm. Evaluation reaches an average precision (AP) of 75.9 We study resource-limited online learning, motivated by the problem of conditional-branch outcome prediction in computer architecture. In particular, we consider (parallel) time and space-efficient ensemble learners for online settings, empirically demonstrating benefits similar to those shown previously for offline ensembles. Our learning algorithms are inspired by the previously published “boosting by filtering” framework as well as the offline Arc-x4 boosting-style algorithm. We train ensembles of online decision trees using a novel variant of the ID4 online decision-tree algorithm as the base learner, and show empirical results for both boosting and bagging-style online ensemble methods. Our results evaluate these methods on both our branch prediction domain and online variants of three familiar machine-learning benchmarks. Our data justifies three key claims. First, we show empirically that our extensions to ID4 significantly improve performance for single trees and additionally are critical to achieving performance gains in tree ensembles. Second, our results indicate significant improvements in predictive accuracy with ensemble size for the boosting-style algorithm. The bagging algorithms we tried showed poor performance relative to the boosting-style algorithm (but still improve upon individual base learners). Third, we show that ensembles of small trees are often able to outperform large single trees with the same number of nodes (and similarly outperform smaller ensembles of larger trees that use the same total number of nodes). This makes online boosting particularly useful in domains such as branch prediction with tight space restrictions (i.e., the available real-estate on a microprocessor chip).