On the consistency of multiclass classification methods

Binary classification is a well studied special case of the classification problem. Statistical properties of binary classifiers, such as consistency, have been investigated in a variety of settings. Binary classification methods can be generalized in many ways to handle multiple classes. It turns out that one can lose consistency in generalizing a binary classification method to deal with multiple classes. We study a rich family of multiclass methods and provide a necessary and sufficient condition for their consistency. We illustrate our approach by applying it to some multiclass methods proposed in the literature.

Implicit online learning

Online learning algorithms have recently risen to prominence due to their strong theoretical guarantees and an increasing number of practical applications for large-scale data analysis problems. In this paper, we analyze a class of online learning algorithms based on fixed potentials and nonlinearized losses, which yields algorithms with implicit update rules. We show how to efficiently compute these updates, and we prove regret bounds for the algorithms. We apply our formulation to several special cases where our approach has benefits over existing online learning methods. In particular, we provide improved algorithms and bounds for the online metric learning problem, and show improved robustness for online linear prediction problems. Results over a variety of data sets demonstrate the advantages of our framework.

Fault diagnostics and supervised testing : How fault diagnostic tools can be proactive?

The topic of fault detection and diagnostics (FDD) is studied from the perspective of proactive testing. Unlike most research focus in the diagnosis area in which system outputs are analyzed for diagnosis purposes, in this paper the focus is on the other side of the problem: manipulating system inputs for better diagnosis reasoning. In other words, the question of how diagnostic mechanisms can direct system inputs for better diagnosis analysis is addressed here. It is shown how the problem can be formulated as decision making problem coupled with a Bayesian Network based diagnostic mechanism. The developed mechanism is applied to the problem of supervised testing in HVAC systems.

Overcoming the complexity of diagnostic problems due to sensor network architecture

In fault detection and diagnostics, limitations coming from the sensor network architecture are one of the main challenges in evaluating a system’s health status. Usually the design of the sensor network architecture is not solely based on diagnostic purposes, other factors like controls, financial constraints, and practical limitations are also involved. As a result, it quite common to have one sensor (or one set of sensors) monitoring the behaviour of two or more components. This can significantly extend the complexity of diagnostic problems. In this paper a systematic approach is presented to deal with such complexities. It is shown how the problem can be formulated as a Bayesian network based diagnostic mechanism with latent variables. The developed approach is also applied to the problem of fault diagnosis in HVAC systems, an application area with considerable modeling and measurement constraints.

Optimal strategies and minimax lower bounds for online convex games

A number of learning problems can be cast as an Online Convex Game: on each round, a learner makes a prediction x from a convex set, the environment plays a loss function f, and the learner’s long-term goal is to minimize regret. Algorithms have been proposed by Zinkevich, when f is assumed to be convex, and Hazan et al., when f is assumed to be strongly convex, that have provably low regret. We consider these two settings and analyze such games from a minimax perspective, proving minimax strategies and lower bounds in each case. These results prove that the existing algorithms are essentially optimal.

Shifting : one-inclusion mistake bounds and sample compression

We present new expected risk bounds for binary and multiclass prediction, and resolve several recent conjectures on sample compressibility due to Kuzmin and Warmuth. By exploiting the combinatorial structure of concept class F, Haussler et al. achieved a VC(F)/n bound for the natural one-inclusion prediction strategy. The key step in their proof is a d=VC(F) bound on the graph density of a subgraph of the hypercube—one-inclusion graph. The first main result of this report is a density bound of n∙choose(n-1,≤d-1)/choose(n,≤d) < d, which positively resolves a conjecture of Kuzmin and Warmuth relating to their unlabeled Peeling compression scheme and also leads to an improved one-inclusion mistake bound. The proof uses a new form of VC-invariant shifting and a group-theoretic symmetrization. Our second main result is an algebraic topological property of maximum classes of VC-dimension d as being d-contractible simplicial complexes, extending the well-known characterization that d=1 maximum classes are trees. We negatively resolve a minimum degree conjecture of Kuzmin and Warmuth—the second part to a conjectured proof of correctness for Peeling—that every class has one-inclusion minimum degree at most its VC-dimension. Our final main result is a k-class analogue of the d/n mistake bound, replacing the VC-dimension by the Pollard pseudo-dimension and the one-inclusion strategy by its natural hypergraph generalization. This result improves on known PAC-based expected risk bounds by a factor of O(log n) and is shown to be optimal up to a O(log k) factor. The combinatorial technique of shifting takes a central role in understanding the one-inclusion (hyper)graph and is a running theme throughout

Exponentiated gradient algorithms for large-margin structured classification

We consider the problem of structured classification, where the task is to predict a label y from an input x, and y has meaningful internal structure. Our framework includes supervised training of Markov random fields and weighted context-free grammars as special cases. We describe an algorithm that solves the large-margin optimization problem defined in [12], using an exponential-family (Gibbs distribution) representation of structured objects. The algorithm is efficient—even in cases where the number of labels y is exponential in size—provided that certain expectations under Gibbs distributions can be calculated efficiently. The method for structured labels relies on a more general result, specifically the application of exponentiated gradient updates [7, 8] to quadratic programs.

Adaboost is consistent

The risk, or probability of error, of the classifier produced by the AdaBoost algorithm is investigated. In particular, we consider the stopping strategy to be used in AdaBoost to achieve universal consistency. We show that provided AdaBoost is stopped after n1-ε iterations---for sample size n and ε ∈ (0,1)---the sequence of risks of the classifiers it produces approaches the Bayes risk.

A unifying view of multiple kernel learning

Recent research on multiple kernel learning has lead to a number of approaches for combining kernels in regularized risk minimization. The proposed approaches include different formulations of objectives and varying regularization strategies. In this paper we present a unifying general optimization criterion for multiple kernel learning and show how existing formulations are subsumed as special cases. We also derive the criterion's dual representation, which is suitable for general smooth optimization algorithms. Finally, we evaluate multiple kernel learning in this framework analytically using a Rademacher complexity bound on the generalization error and empirically in a set of experiments.

Matrix regularization techniques for online multitask learning

In this paper we examine the problem of prediction with expert advice in a setup where the learner is presented with a sequence of examples coming from different tasks. In order for the learner to be able to benefit from performing multiple tasks simultaneously, we make assumptions of task relatedness by constraining the comparator to use a lesser number of best experts than the number of tasks. We show how this corresponds naturally to learning under spectral or structural matrix constraints, and propose regularization techniques to enforce the constraints. The regularization techniques proposed here are interesting in their own right and multitask learning is just one application for the ideas. A theoretical analysis of one such regularizer is performed, and a regret bound that shows benefits of this setup is reported.

Closing the gap between bandit and full-information online optimization : high-probability regret bound

We demonstrate a modification of the algorithm of Dani et al for the online linear optimization problem in the bandit setting, which allows us to achieve an O( \sqrt{T ln T} ) regret bound in high probability against an adaptive adversary, as opposed to the in expectation result against an oblivious adversary of Dani et al. We obtain the same dependence on the dimension as that exhibited by Dani et al. The results of this paper rest firmly on those of Dani et al and the remarkable technique of Auer et al for obtaining high-probability bounds via optimistic estimates. This paper answers an open question: it eliminates the gap between the high-probability bounds obtained in the full-information vs bandit settings.

Adaptive online gradient descent

We study the rates of growth of the regret in online convex optimization. First, we show that a simple extension of the algorithm of Hazan et al eliminates the need for a priori knowledge of the lower bound on the second derivatives of the observed functions. We then provide an algorithm, Adaptive Online Gradient Descent, which interpolates between the results of Zinkevich for linear functions and of Hazan et al for strongly convex functions, achieving intermediate rates between [square root T] and [log T]. Furthermore, we show strong optimality of the algorithm. Finally, we provide an extension of our results to general norms.