We study consistency properties of surrogate loss functions for general multiclass learning problems, defined by a general multiclass loss matrix. We extend the notion of classification calibration, which has been studied for binary and multiclass 0-1 classification problems (and for certain other specific learning problems), to the general multiclass setting, and derive necessary and sufficient conditions for a surrogate loss to be calibrated with respect to a loss matrix in this setting. We then introduce the notion of convex calibration dimension of a multiclass loss matrix, which measures the smallest size' of a prediction space in which it is possible to design a convex surrogate that is calibrated with respect to the loss matrix. We derive both upper and lower bounds on this quantity, and use these results to analyze various loss matrices. In particular, we apply our framework to study various subset ranking losses, and use the convex calibration dimension as a tool to show both the existence and non-existence of various types of convex calibrated surrogates for these losses. Our results strengthen recent results of (2010) and (2012) on the non-existence of certain types of convex calibrated surrogates in subset ranking. We anticipate the convex calibration dimension may prove to be a useful tool in the study and design of surrogate losses for general multiclass learning problems.
We study losses for binary classification and class probability estimation and extend the understanding of them from margin losses to general composite losses which are the composition of a proper loss with a link function. We characterise when margin losses can be proper composite losses, explicitly show how to determine a symmetric loss in full from half of one of its partial losses, introduce an intrinsic parametrisation of composite binary losses and give a complete characterisation of the relationship between proper losses and "classification calibrated" losses. We also consider the question of the "best" surrogate binary loss. We introduce a precise notion of "best" and show there exist situations where two convex surrogate losses are incommensurable. We provide a complete explicit characterisation of the convexity of composite binary losses in terms of the link function and the weight function associated with the proper loss which make up the composite loss. This characterisation suggests new ways of "surrogate tuning" as well as providing an explicit characterisation of when Bregman divergences on the unit interval are convex in their second argument. Finally, in an appendix we present some new algorithm-independent results on the relationship between properness, convexity and robustness to misclassification noise for binary losses and show that all convex proper losses are non-robust to misclassification noise.  The probability of error of classification methods based on convex combinations of simple base classifiers by boosting algorithms is investigated. The main result of the paper is that certain regularized boosting algorithms provide Bayes-risk consistent classifiers under the sole assumption that the Bayes classifier may be approximated by a convex combination of the base classifiers. Nonasymptotic distribution-free bounds are also developed which offer interesting new insight into how boosting works and help explain its success in practical classification problems. This paper proposes evaluation methods based on the use of non-dichotomous relevance judgements in IR experiments. It is argued that evaluation methods should credit IR methods for their ability to retrieve highly relevant documents. This is desirable from the user point of view in modem large IR environments. The proposed methods are (1) a novel application of P-R curves and average precision computations based on separate recall bases for documents of different degrees of relevance, and (2) two novel measures computing the cumulative gain the user obtains by examining the retrieval result up to a given ranked position. We then demonstrate the use of these evaluation methods in a case study on the effectiveness of query types, based on combinations of query structures and expansion, in retrieving documents of various degrees of relevance. The test was run with a best match retrieval system (In- Query I) in a text database consisting of newspaper articles. The results indicate that the tested strong query structures are most effective in retrieving highly relevant documents. The differences between the query types are practically essential and statistically significant. More generally, the novel evaluation methods and the case demonstrate that non-dichotomous relevance assessments are applicable in IR experiments, may reveal interesting phenomena, and allow harder testing of IR methods. We consider the problem of @math -class classification ( @math ), where the classifier can choose to abstain from making predictions at a given cost, say, a factor @math of the cost of misclassification. Designing consistent algorithms for such @math -class classification problems with a reject option' is the main goal of this paper, thereby extending and generalizing previously known results for @math . We show that the Crammer-Singer surrogate and the one vs all hinge loss, albeit with a different predictor than the standard argmax, yield consistent algorithms for this problem when @math . More interestingly, we design a new convex surrogate that is also consistent for this problem when @math and operates on a much lower dimensional space ( @math as opposed to @math ). We also generalize all three surrogates to be consistent for any @math . It is shown that various classifiers that are based on minimization of a regularized risk are universally consistent, i.e., they can asymptotically learn in every classification task. The role of the loss functions used in these algorithms is considered in detail. As an application of our general framework, several types of support vector machines (SVMs) as well as regularization networks are treated. Our methods combine techniques from stochastics, approximation theory, and functional analysis Recent experiments and theoretical studies show that AdaBoost can overfit in the limit of large time. If running the algorithm forever is suboptimal, a natural question is how low can the prediction error be during the process of AdaBoost? We show under general regularity conditions that during the process of AdaBoost a consistent prediction is generated, which has the prediction error approximating the optimal Bayes error as the sample size increases. This result suggests that, while running the algorithm forever can be suboptimal, it is reasonable to expect that some regularization method via truncation of the process may lead to a near-optimal performance for sufficiently large sample size. Surrogate losses underlie numerous state-of-the-art binary classification algorithms, such as support vector machines and boosting. The impact of a surrogate loss on the statistical performance of an algorithm is well-understood in symmetric classification settings, where the misclassification costs are equal and the loss is a margin loss. In particular, classification-calibrated losses are known to imply desirable properties such as consistency. While numerous efforts have been made to extend surrogate loss-based algorithms to asymmetric settings, to deal with unequal misclassification costs or training data imbalance, considerably less attention has been paid to whether the modified loss is still calibrated in some sense. This article extends the theory of classification-calibrated losses to asymmetric problems. As in the symmetric case, it is shown that calibrated asymmetric surrogate losses give rise to excess risk bounds, which control the expected misclassification cost in terms of the excess surrogate risk. This theory is illustrated on the class of uneven margin losses, and the uneven hinge, squared error, exponential, and sigmoid losses are treated in detail.