Recently, a parametrized class of loss functions called @math -loss, @math , has been introduced for classification. This family, which includes the log-loss and the 0-1 loss as special cases, comes with compelling properties including an equivalent margin-based form which is classification-calibrated for all @math . We introduce a generalization of this family to the entire range of @math and establish how the parameter @math enables the practitioner to choose among a host of operating conditions that are important in modern machine learning tasks. We prove that smaller @math values are more conducive to faster optimization; in fact, @math -loss is convex for @math and quasi-convex for @math . Moreover, we establish bounds to quantify the degradation of the local-quasi-convexity of the optimization landscape as @math increases; we show that this directly translates to a computational slow down. On the other hand, our theoretical results also suggest that larger @math values lead to better generalization performance. This is a consequence of the ability of the @math -loss to limit the effect of less likely data as @math increases from 1, thereby facilitating robustness to outliers and noise in the training data. We provide strong evidence supporting this assertion with several experiments on benchmark datasets that establish the efficacy of @math -loss for @math in robustness to errors in the training data. Of equal interest is the fact that, for @math , our experiments show that the decreased robustness seems to counteract class imbalances in training data.
While the previous result establishes an equivalent margin-based form for @math -loss, our next result establishes some of its basic optimization characteristics. Finally, we conclude this section with another basic property of @math -loss that highlights its suitability for classification. For binary classification and margin-based losses, Bartlett in @cite_18 introduce as a means to compare the performance of a loss function relative to the 0-1 loss. A margin-based loss function @math is classification-calibrated if its minimum conditional risk given @math is attained by a @math such that @math , where @math is the true posterior. Building upon such a result in @cite_9 for @math -loss, the following proposition shows that @math is classification-calibrated for all @math . The proof of Proposition is given in Appendix .
@cite_9 @cite_18
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We present @math -loss, @math , a tunable loss function for binary classification that bridges log-loss ( @math ) and @math - @math loss ( @math ). We prove that @math -loss has an equivalent margin-based form and is classification-calibrated, two desirable properties for a good surrogate loss function for the ideal yet intractable @math - @math loss. For logistic regression-based classification, we provide an upper bound on the difference between the empirical and expected risk for @math -loss by exploiting its Lipschitzianity along with recent results on the landscape features of empirical risk functions. Finally, we show that @math -loss with @math performs better than log-loss on MNIST for logistic regression. 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. In many classification procedures, the classification function is obtained by minimizing a certain empirical risk on the training sample. The classification is then based on the sign of the classification function. In recent years, there have been a host of classification methods proposed that use different margin-based loss functions. The margin-based loss functions are often motivated as upper bounds of the misclassification loss, but this cannot explain the statistical properties of the classification procedures. We show that a large family of margin-based loss functions are Fisher consistent for classification. That is, the population minimizer of the loss function leads to the Bayes optimal rule of classification. Our result covers almost all margin-based loss functions that have been proposed in the literature. We give an inequality that links the Fisher consistency of margin-based loss functions with the consistency of methods based on these loss functions. We use this inequality to obtain the rate of convergence for the method of sieves based on a class of margin-based loss functions. We present a margin-based loss function for classification, inspired by the recently proposed similarity measure called correntropy. We show that correntropy induces a nonconvex loss function that is a closer approximation to the misclassification loss (ideal 0–1 loss). We show that the discriminant function obtained by optimizing the proposed loss function using a neural network is insensitive to outliers and has better generalization performance as compared to using the squared loss function which is common in neural network classifiers. The proposed method of training classifiers is a practical way of obtaining better results on real world classification problems, that uses a simple gradient based online training procedure for minimizing the empirical risk.