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
2912269121 2101557761 2022465039 2150477941 2134829400 2793731154 1969623397 2768012150 2803642127 2023163512 2188209982 2032708784 1964840769 2113290770 1982032418 2151532749 2134686390 2963113424 2362855512 2963185791
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. While convex losses for binary classification are attractive due to the existence of numerous (provably) efficient methods for finding their global optima, they are sensitive to outliers. On the other hand, while the nonconvex 0-1 loss is robust to outliers, it is NP-hard to optimize and thus rarely directly optimized in practice. In this paper, however, we do just that: we explore a variety of practical methods for direct (approximate) optimization of the 0-1 loss based on branch and bound search, combinatorial search, and coordinate descent on smooth, differentiable relaxations of 0-1 loss. Empirically, we compare our proposed algorithms to logistic regression, SVM, and the Bayes point machine showing that the proposed 0-1 loss optimization algorithms perform at least as well and offer a clear advantage in the presence of outliers. To this end, we believe this work reiterates the importance of 0-1 loss and its robustness properties while challenging the notion that it is difficult to directly optimize. We present a formulation of deep learning that aims at producing a large margin classifier. The notion of margin, minimum distance to a decision boundary, has served as the foundation of several theoretically profound and empirically successful results for both classification and regression tasks. However, most large margin algorithms are applicable only to shallow models with a preset feature representation; and conventional margin methods for neural networks only enforce margin at the output layer. Such methods are therefore not well suited for deep networks. In this work, we propose a novel loss function to impose a margin on any chosen set of layers of a deep network (including input and hidden layers). Our formulation allows choosing any norm on the metric measuring the margin. We demonstrate that the decision boundary obtained by our loss has nice properties compared to standard classification loss functions. Specifically, we show improved empirical results on the MNIST, CIFAR-10 and ImageNet datasets on multiple tasks: generalization from small training sets, corrupted labels, and robustness against adversarial perturbations. The resulting loss is general and complementary to existing data augmentation (such as random adversarial input transform) and regularization techniques (such as weight decay, dropout, and batch norm). 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. In knowledge graph embedding models, the margin-based ranking loss as the common loss function is usually used to encourage discrimination between golden triplets and incorrect triplets, which has proved effective in many translation-based models for knowledge graph embedding. However, we find that the loss function cannot ensure the fact that the scoring of correct triplets must be low enough to fulfill the translation. In this paper, we present a limit-based scoring loss to provide lower scoring of a golden triplet, and then to extend two basic translation models TransE and TransH, separately to TransE-RS and TransH-RS by combining limit-based scoring loss with margin-based ranking loss. Both the presented models have low complexities of parameters benefiting for application on large scale graphs. In experiments, we evaluate our models on two typical tasks including triplet classification and link prediction, and also analyze the scoring distributions of positive and negative triplets by different models. Experimental results show that the introduced limit-based scoring loss is effective to improve the capacities of knowledge graph embedding. We study binary classification in the presence of class-conditional random noise, where the learner gets to see labels that are flipped independently with some probability, and where the flip probability depends on the class. Our goal is to devise learning algorithms that are efficient and statistically consistent with respect to commonly used utility measures. In particular, we look at a family of measures motivated by their application in domains where cost-sensitive learning is necessary (for example, when there is class imbalance). In contrast to most of the existing literature on consistent classification that are limited to the classical 0-1 loss, our analysis includes more general utility measures such as the AM measure (arithmetic mean of True Positive Rate and True Negative Rate). For this problem of cost-sensitive learning under class-conditional random noise, we develop two approaches that are based on suitably modifying surrogate losses. First, we provide a simple unbiased estimator of any loss, and obtain performance bounds for empirical utility maximization in the presence of i.i.d. data with noisy labels. If the loss function satis_es a simple symmetry condition, we show that using unbiased estimator leads to an efficient algorithm for empirical maximization. Second, by leveraging a reduction of risk minimization under noisy labels to classification with weighted 0-1 loss, we suggest the use of a simple weighted surrogate loss, for which we are able to obtain strong utility bounds. This approach implies that methods already used in practice, such as biased SVM and weighted logistic regression, are provably noise-tolerant. For two practically important measures in our family, we show that the proposed methods are competitive with respect to recently proposed methods for dealing with label noise in several benchmark data sets. We study how closely the optimal Bayes error rate can be approximately reached using a classification algorithm that computes a classifier by minimizing a convex upper bound of the classification error function. The measurement of closeness is characterized by the loss function used in the estimation. We show that such a classification scheme can be generally regarded as a (nonmaximum-likelihood) conditional in-class probability estimate, and we use this analysis to compare various convex loss functions that have appeared in the literature. Furthermore, the theoretical insight allows us to design good loss functions with desirable properties. Another aspect of our analysis is to demonstrate the consistency of certain classification methods using convex risk minimization. This study sheds light on the good performance of some recently proposed linear classification methods including boosting and support vector machines. It also shows their limitations and suggests possible improvements. 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. In the interest of deriving classifiers that are robust to outlier observations, we present integer programming formulations of Vapnik's support vector machine (SVM) with the ramp loss and hard margin loss. The ramp loss allows a maximum error of 2 for each training observation, while the hard margin loss calculates error by counting the number of training observations that are in the margin or misclassified outside of the margin. SVM with these loss functions is shown to be a consistent estimator when used with certain kernel functions. In computational studies with simulated and real-world data, SVM with the robust loss functions ignores outlier observations effectively, providing an advantage over SVM with the traditional hinge loss when using the linear kernel. Despite the fact that training SVM with the robust loss functions requires the solution of a quadratic mixed-integer program (QMIP) and is NP-hard, while traditional SVM requires only the solution of a continuous quadratic program (QP), we are able to find good solutions and prove optimality for instances with up to 500 observations. Solution methods are presented for the new formulations that improve computational performance over industry-standard integer programming solvers alone. The goal of binary classification is to estimate a discriminant function y from observations of covariate vectors and corresponding binary labels. We consider an elaboration of this problem in which the covariates are not available directly but are transformed by a dimensionality-reducing quantizer Q. We present conditions on loss functions such that empirical risk minimization yields Bayes consistency when both the discriminant function and the quantizer are estimated. These conditions are stated in terms of a general correspondence between loss functions and a class of functionals known as Ali-Silvey or -divergence functionals. Whereas this correspondence was established by Blackwell [Proc. 2nd Berkeley Symp. Probab. Statist. 1 (1951) 93-102. Univ. California Press, Berkeley] for the 0-1 loss, we extend the correspondence to the broader class of surrogate loss functions that play a key role in the general theory of Bayes consistency for binary classification. Our result makes it possible to pick out the (strict) subset of surrogate loss functions that yield Bayes consistency for joint estimation of the discriminant function and the quantizer. In this paper, we theoretically study the problem of binary classification in the presence of random classification noise—the learner, instead of seeing the true labels, sees labels that have independently been flipped with some small probability. Moreover, random label noise is class-conditional— the flip probability depends on the class. We provide two approaches to suitably modify any given surrogate loss function. First, we provide a simple unbiased estimator of any loss, and obtain performance bounds for empirical risk minimization in the presence of iid data with noisy labels. If the loss function satisfies a simple symmetry condition, we show that the method leads to an efficient algorithm for empirical minimization. Second, by leveraging a reduction of risk minimization under noisy labels to classification with weighted 0-1 loss, we suggest the use of a simple weighted surrogate loss, for which we are able to obtain strong empirical risk bounds. This approach has a very remarkable consequence — methods used in practice such as biased SVM and weighted logistic regression are provably noise-tolerant. On a synthetic non-separable dataset, our methods achieve over 88 accuracy even when 40 of the labels are corrupted, and are competitive with respect to recently proposed methods for dealing with label noise in several benchmark datasets. Many of the classification algorithms developed in the machine learning literature, including the support vector machine and boosting, can be viewed as minimum contrast methods that minimize a convex surrogate of the 0–1 loss function. The convexity makes these algorithms computationally efficient. The use of a surrogate, however, has statistical consequences that must be balanced against the computational virtues of convexity. To study these issues, we provide a general quantitative relationship between the risk as assessed using the 0–1 loss and the risk as assessed using any nonnegative surrogate loss function. We show that this relationship gives nontrivial upper bounds on excess risk under the weakest possible condition on the loss function—that it satisfies a pointwise form of Fisher consistency for classification. The relationship is based on a simple variational transformation of the loss function that is easy to compute in many applications. We also present a refined version of this result in the... Problems of data classification can be studied in the framework of regularization theory as ill-posed problems. In this framework, loss functions play an important role in the application of regularization theory to classification. In this paper, we review some important convex loss functions, including hinge loss, square loss, modified square loss, exponential loss, logistic regression loss, as well as some non-convex loss functions, such as sigmoid loss, @math -loss, ramp loss, normalized sigmoid loss, and the loss function of 2 layer neural network. Based on the analysis of these loss functions, we propose a new differentiable non-convex loss function, called smoothed 0-1 loss function, which is a natural approximation of the 0-1 loss function. To compare the performance of different loss functions, we propose two binary classification algorithms for binary classification, one for convex loss functions, the other for non-convex loss functions. A set of experiments are launched on several binary data sets from the UCI repository. The results show that the proposed smoothed 0-1 loss function is robust, especially for those noisy data sets with many outliers. Existing cellular network analyses, and even simulations, typically use the standard path loss model where received power decays like @math over a distance @math . This standard path loss model is quite idealized, and in most scenarios the path loss exponent @math is itself a function of @math , typically an increasing one. Enforcing a single path loss exponent can lead to orders of magnitude differences in average received and interference powers versus the true values. In this paper, we study multi-slope path loss models, where different distance ranges are subject to different path loss exponents. We focus on the dual-slope path loss function, which is a piece-wise power law and continuous and accurately approximates many practical scenarios. We derive the distributions of SIR, SNR, and finally SINR before finding the potential throughput scaling, which provides insight on the observed cell-splitting rate gain. The exact mathematical results show that the SIR monotonically decreases with network density, while the converse is true for SNR, and thus the network coverage probability in terms of SINR is maximized at some finite density. With ultra-densification (network density goes to infinity), there exists a phase transition in the near-field path loss exponent @math : if @math unbounded potential throughput can be achieved asymptotically; if $ 0 , ultra-densification leads in the extreme case to zero throughput. We prove that the empirical risk of most well-known loss functions factors into a linear term aggregating all labels with a term that is label free, and can further be expressed by sums of the same loss. This holds true even for non-smooth, non-convex losses and in any RKHS. The first term is a (kernel) mean operator -- the focal quantity of this work -- which we characterize as the sufficient statistic for the labels. The result tightens known generalization bounds and sheds new light on their interpretation. Factorization has a direct application on weakly supervised learning. In particular, we demonstrate that algorithms like SGD and proximal methods can be adapted with minimal effort to handle weak supervision, once the mean operator has been estimated. We apply this idea to learning with asymmetric noisy labels, connecting and extending prior work. Furthermore, we show that most losses enjoy a data-dependent (by the mean operator) form of noise robustness, in contrast with known negative results. The choice of the loss function is critical in extreme multi-label learning where the objective is to annotate each data point with the most relevant subset of labels from an extremely large label set. Unfortunately, existing loss functions, such as the Hamming loss, are unsuitable for learning, model selection, hyperparameter tuning and performance evaluation. This paper addresses the issue by developing propensity scored losses which: (a) prioritize predicting the few relevant labels over the large number of irrelevant ones; (b) do not erroneously treat missing labels as irrelevant but instead provide unbiased estimates of the true loss function even when ground truth labels go missing under arbitrary probabilistic label noise models; and (c) promote the accurate prediction of infrequently occurring, hard to predict, but rewarding tail labels. Another contribution is the development of algorithms which efficiently scale to extremely large datasets with up to 9 million labels, 70 million points and 2 million dimensions and which give significant improvements over the state-of-the-art. This paper's results also apply to tagging, recommendation and ranking which are the motivating applications for extreme multi-label learning. They generalize previous attempts at deriving unbiased losses under the restrictive assumption that labels go missing uniformly at random from the ground truth. Furthermore, they provide a sound theoretical justification for popular label weighting heuristics used to recommend rare items. Finally, they demonstrate that the proposed contributions align with real world applications by achieving superior clickthrough rates on sponsored search advertising in Bing. Multiclass classification problems such as image annotation can involve a large number of classes. In this context, confusion between classes can occur, and single label classification may be misleading. We provide in the present paper a general device that, given an unlabeled dataset and a score function defined as the minimizer of some empirical and convex risk, outputs a set of class labels, instead of a single one. Interestingly, this procedure does not require that the unlabeled dataset explores the whole classes. Even more, the method is calibrated to control the expected size of the output set while minimizing the classification risk. We show the statistical optimality of the procedure and establish rates of convergence under the Tsybakov margin condition. It turns out that these rates are linear on the number of labels. We apply our methodology to convex aggregation of confidence sets based on the V-fold cross validation principle also known as the superlearning principle. We illustrate the numerical performance of the procedure on real data and demonstrate in particular that with moderate expected size, w.r.t. the number of labels, the procedure provides significant improvement of the classification risk.