Consistency in Models for Distributed Learning under Communication Constraints

Motivated by sensor networks and other distributed settings, several models for distributed learning are presented. The models differ from classical works in statistical pattern recognition by allocating observations of an independent and identically distributed (i.i.d.) sampling process amongst members of a network of simple learning agents. The agents are limited in their ability to communicate to a central fusion center and thus, the amount of information available for use in classification or regression is constrained. For several basic communication models in both the binary classification and regression frameworks, we question the existence of agent decision rules and fusion rules that result in a universally consistent ensemble. The answers to this question present new issues to consider with regard to universal consistency. Insofar as these models present a useful picture of distributed scenarios, this paper addresses the issue of whether or not the guarantees provided by Stone's Theorem in centralized environments hold in distributed settings.Comment: To appear in the IEEE Transactions on Information Theor

Learning with Symmetric Label Noise: The Importance of Being Unhinged

Convex potential minimisation is the de facto approach to binary classification. However, Long and Servedio [2010] proved that under symmetric label noise (SLN), minimisation of any convex potential over a linear function class can result in classification performance equivalent to random guessing. This ostensibly shows that convex losses are not SLN-robust. In this paper, we propose a convex, classification-calibrated loss and prove that it is SLN-robust. The loss avoids the Long and Servedio [2010] result by virtue of being negatively unbounded. The loss is a modification of the hinge loss, where one does not clamp at zero; hence, we call it the unhinged loss. We show that the optimal unhinged solution is equivalent to that of a strongly regularised SVM, and is the limiting solution for any convex potential; this implies that strong l2 regularisation makes most standard learners SLN-robust. Experiments confirm the SLN-robustness of the unhinged loss

Classification with the nearest neighbor rule in general finite dimensional spaces: necessary and sufficient conditions

Given an $n$-sample of random vectors $(X_i,Y_i)_{1 \leq i \leq n}$ whose joint law is unknown, the long-standing problem of supervised classification aims to \textit{optimally} predict the label $Y$ of a given a new observation $X$. In this context, the nearest neighbor rule is a popular flexible and intuitive method in non-parametric situations. Even if this algorithm is commonly used in the machine learning and statistics communities, less is known about its prediction ability in general finite dimensional spaces, especially when the support of the density of the observations is $\mathbb{R}^d$. This paper is devoted to the study of the statistical properties of the nearest neighbor rule in various situations. In particular, attention is paid to the marginal law of $X$, as well as the smoothness and margin properties of the \textit{regression function} $\eta(X) = \mathbb{E}[Y | X]$. We identify two necessary and sufficient conditions to obtain uniform consistency rates of classification and to derive sharp estimates in the case of the nearest neighbor rule. Some numerical experiments are proposed at the end of the paper to help illustrate the discussion.Comment: 53 Pages, 3 figure

Recursive Aggregation of Estimators by Mirror Descent Algorithm with Averaging

We consider a recursive algorithm to construct an aggregated estimator from a finite number of base decision rules in the classification problem. The estimator approximately minimizes a convex risk functional under the l1-constraint. It is defined by a stochastic version of the mirror descent algorithm (i.e., of the method which performs gradient descent in the dual space) with an additional averaging. The main result of the paper is an upper bound for the expected accuracy of the proposed estimator. This bound is of the order $\sqrt{(\log M)/t}$ with an explicit and small constant factor, where $M$ is the dimension of the problem and $t$ stands for the sample size. A similar bound is proved for a more general setting that covers, in particular, the regression model with squared loss.Comment: 29 pages; mai 200

Pairwise Classification and Pairwise Support Vector Machines

Several modifications have been suggested to extend binary classifiers to multiclass classification, for instance the One Against All technique, the One Against One technique, or Directed Acyclic Graphs. A recent approach for multiclass classification is the pairwise classification, which relies on two input examples instead of one and predicts whether the two input examples belong to the same class or to different classes. A Support Vector Machine (SVM), which is able to handle pairwise classification tasks, is called pairwise SVM. A common pairwise classification task is face recognition. In this area, a set of images is given for training and another set of images is given for testing. Often, one is interested in the interclass setting. The latter means that any person which is represented by an image in the training set is not represented by any image in the test set. From the mentioned multiclass classification techniques only the pairwise classification technique provides meaningful results in the interclass setting. For a pairwise classifier the order of the two examples should not influence the classification result. A common approach to enforce this symmetry is the use of selected kernels. Relations between such kernels and certain projections are provided. It is shown, that those projections can lead to an information loss. For pairwise SVMs another approach for enforcing symmetry is the symmetrization of the training sets. In other words, if the pair (a,b) of examples is a training pair then (b,a) is a training pair, too. It is proven that both approaches do lead to the same decision function for selected parameters. Empirical tests show that the approach using selected kernels is three to four times faster. For a good interclass generalization of pairwise SVMs training sets with several million training pairs are needed. A technique is presented which further speeds up the training time of pairwise SVMs by a factor of up to 130 and thus enables the learning of training sets with several million pairs. Another element affecting time is the need to select several parameters. Even with the applied speed up techniques a grid search over the set of parameters would be very expensive. Therefore, a model selection technique is introduced that is much less computationally expensive. In machine learning, the training set and the test set are created by using some data generating process. Several pairwise data generating processes are derived from a given non pairwise data generating process. Advantages and disadvantages of the different pairwise data generating processes are evaluated. Pairwise Bayes' Classifiers are introduced and their properties are discussed. It is shown that pairwise Bayes' Classifiers for interclass generalization tasks can differ from pairwise Bayes' Classifiers for interexample generalization tasks. In face recognition the interexample task implies that each person which is represented by an image in the test set is also represented by at least one image in the training set. Moreover, the set of images of the training set and the set of images of the test set are disjoint. Pairwise SVMs are applied to four synthetic and to two real world datasets. One of the real world datasets is the Labeled Faces in the Wild (LFW) database while the other one is provided by Cognitec Systems GmbH. Empirical evidence for the presented model selection heuristic, the discussion about the loss of information and the provided speed up techniques is given by the synthetic databases and it is shown that classifiers of pairwise SVMs lead to a similar quality as pairwise Bayes' classifiers. Additionally, a pairwise classifier is identified for the LFW database which leads to an average equal error rate (EER) of 0.0947 with a standard error of the mean (SEM) of 0.0057. This result is better than the result of the current state of the art classifier, namely the combined probabilistic linear discriminant analysis classifier, which leads to an average EER of 0.0993 and a SEM of 0.0051.Es gibt verschiedene Ansätze, um binäre Klassifikatoren zur Mehrklassenklassifikation zu nutzen, zum Beispiel die One Against All Technik, die One Against One Technik oder Directed Acyclic Graphs. Paarweise Klassifikation ist ein neuerer Ansatz zur Mehrklassenklassifikation. Dieser Ansatz basiert auf der Verwendung von zwei Input Examples anstelle von einem und bestimmt, ob diese beiden Examples zur gleichen Klasse oder zu unterschiedlichen Klassen gehören. Eine Support Vector Machine (SVM), die für paarweise Klassifikationsaufgaben genutzt wird, heißt paarweise SVM. Beispielsweise werden Probleme der Gesichtserkennung als paarweise Klassifikationsaufgabe gestellt. Dazu nutzt man eine Menge von Bildern zum Training und ein andere Menge von Bildern zum Testen. Häufig ist man dabei an der Interclass Generalization interessiert. Das bedeutet, dass jede Person, die auf wenigstens einem Bild der Trainingsmenge dargestellt ist, auf keinem Bild der Testmenge vorkommt. Von allen erwähnten Mehrklassenklassifikationstechniken liefert nur die paarweise Klassifikationstechnik sinnvolle Ergebnisse für die Interclass Generalization. Die Entscheidung eines paarweisen Klassifikators sollte nicht von der Reihenfolge der zwei Input Examples abhängen. Diese Symmetrie wird häufig durch die Verwendung spezieller Kerne gesichert. Es werden Beziehungen zwischen solchen Kernen und bestimmten Projektionen hergeleitet. Zudem wird gezeigt, dass diese Projektionen zu einem Informationsverlust führen können. Für paarweise SVMs ist die Symmetrisierung der Trainingsmengen ein weiter Ansatz zur Sicherung der Symmetrie. Das bedeutet, wenn das Paar (a,b) von Input Examples zur Trainingsmenge gehört, dann muss das Paar (b,a) ebenfalls zur Trainingsmenge gehören. Es wird bewiesen, dass für bestimmte Parameter beide Ansätze zur gleichen Entscheidungsfunktion führen. Empirische Messungen zeigen, dass der Ansatz mittels spezieller Kerne drei bis viermal schneller ist. Um eine gute Interclass Generalization zu erreichen, werden bei paarweisen SVMs Trainingsmengen mit mehreren Millionen Paaren benötigt. Es wird eine Technik eingeführt, die die Trainingszeit von paarweisen SVMs um bis zum 130-fachen beschleunigt und es somit ermöglicht, Trainingsmengen mit mehreren Millionen Paaren zu verwenden. Auch die Auswahl guter Parameter für paarweise SVMs ist im Allgemeinen sehr zeitaufwendig. Selbst mit den beschriebenen Beschleunigungen ist eine Gittersuche in der Menge der Parameter sehr teuer. Daher wird eine Model Selection Technik eingeführt, die deutlich geringeren Aufwand erfordert. Im maschinellen Lernen werden die Trainingsmenge und die Testmenge von einem Datengenerierungsprozess erzeugt. Ausgehend von einem nicht paarweisen Datengenerierungsprozess werden unterschiedliche paarweise Datengenerierungsprozesse abgeleitet und ihre Vor- und Nachteile bewertet. Es werden paarweise Bayes-Klassifikatoren eingeführt und ihre Eigenschaften diskutiert. Es wird gezeigt, dass sich diese Bayes-Klassifikatoren für Interclass Generalization Aufgaben und für Interexample Generalization Aufgaben im Allgemeinen unterscheiden. Bei der Gesichtserkennung bedeutet die Interexample Generalization, dass jede Person, die auf einem Bild der Testmenge dargestellt ist, auch auf mindestens einem Bild der Trainingsmenge vorkommt. Außerdem ist der Durchschnitt der Menge der Bilder der Trainingsmenge mit der Menge der Bilder der Testmenge leer. Paarweise SVMs werden an vier synthetischen und an zwei Real World Datenbanken getestet. Eine der verwendeten Real World Datenbanken ist die Labeled Faces in the Wild (LFW) Datenbank. Die andere wurde von Cognitec Systems GmbH bereitgestellt. Die Annahmen der Model Selection Technik, die Diskussion über den Informationsverlust, sowie die präsentierten Beschleunigungstechniken werden durch empirische Messungen mit den synthetischen Datenbanken belegt. Zudem wird mittels dieser Datenbanken gezeigt, dass Klassifikatoren von paarweisen SVMs zu ähnlich guten Ergebnissen wie paarweise Bayes-Klassifikatoren führen. Für die LFW Datenbank wird ein paarweiser Klassifikator bestimmt, der zu einer durchschnittlichen Equal Error Rate (EER) von 0.0947 und einem Standard Error of The Mean (SEM) von 0.0057 führt. Dieses Ergebnis ist besser als das des aktuellen State of the Art Klassifikators, dem Combined Probabilistic Linear Discriminant Analysis Klassifikator. Dieser führt zu einer durchschnittlichen EER von 0.0993 und einem SEM von 0.0051

Graph-based Estimation of Information Divergence Functions

abstract: Information divergence functions, such as the Kullback-Leibler divergence or the Hellinger distance, play a critical role in statistical signal processing and information theory; however estimating them can be challenge. Most often, parametric assumptions are made about the two distributions to estimate the divergence of interest. In cases where no parametric model fits the data, non-parametric density estimation is used. In statistical signal processing applications, Gaussianity is usually assumed since closed-form expressions for common divergence measures have been derived for this family of distributions. Parametric assumptions are preferred when it is known that the data follows the model, however this is rarely the case in real-word scenarios. Non-parametric density estimators are characterized by a very large number of parameters that have to be tuned with costly cross-validation. In this dissertation we focus on a specific family of non-parametric estimators, called direct estimators, that bypass density estimation completely and directly estimate the quantity of interest from the data. We introduce a new divergence measure, the $D_p$-divergence, that can be estimated directly from samples without parametric assumptions on the distribution. We show that the $D_p$-divergence bounds the binary, cross-domain, and multi-class Bayes error rates and, in certain cases, provides provably tighter bounds than the Hellinger divergence. In addition, we also propose a new methodology that allows the experimenter to construct direct estimators for existing divergence measures or to construct new divergence measures with custom properties that are tailored to the application. To examine the practical efficacy of these new methods, we evaluate them in a statistical learning framework on a series of real-world data science problems involving speech-based monitoring of neuro-motor disorders.Dissertation/ThesisDoctoral Dissertation Electrical Engineering 201

Some Problems in Statistical Pattern Recognition

1 online resource (PDF, 25 pages

Change-point Problem and Regression: An Annotated Bibliography

The problems of identifying changes at unknown times and of estimating the location of changes in stochastic processes are referred to as the change-point problem or, in the Eastern literature, as disorder . The change-point problem, first introduced in the quality control context, has since developed into a fundamental problem in the areas of statistical control theory, stationarity of a stochastic process, estimation of the current position of a time series, testing and estimation of change in the patterns of a regression model, and most recently in the comparison and matching of DNA sequences in microarray data analysis. Numerous methodological approaches have been implemented in examining change-point models. Maximum-likelihood estimation, Bayesian estimation, isotonic regression, piecewise regression, quasi-likelihood and non-parametric regression are among the methods which have been applied to resolving challenges in change-point problems. Grid-searching approaches have also been used to examine the change-point problem. Statistical analysis of change-point problems depends on the method of data collection. If the data collection is ongoing until some random time, then the appropriate statistical procedure is called sequential. If, however, a large finite set of data is collected with the purpose of determining if at least one change-point occurred, then this may be referred to as non-sequential. Not surprisingly, both the former and the latter have a rich literature with much of the earlier work focusing on sequential methods inspired by applications in quality control for industrial processes. In the regression literature, the change-point model is also referred to as two- or multiple-phase regression, switching regression, segmented regression, two-stage least squares (Shaban, 1980), or broken-line regression. The area of the change-point problem has been the subject of intensive research in the past half-century. The subject has evolved considerably and found applications in many different areas. It seems rather impossible to summarize all of the research carried out over the past 50 years on the change-point problem. We have therefore confined ourselves to those articles on change-point problems which pertain to regression. The important branch of sequential procedures in change-point problems has been left out entirely. We refer the readers to the seminal review papers by Lai (1995, 2001). The so called structural change models, which occupy a considerable portion of the research in the area of change-point, particularly among econometricians, have not been fully considered. We refer the reader to Perron (2005) for an updated review in this area. Articles on change-point in time series are considered only if the methodologies presented in the paper pertain to regression analysis