We address the problem of quantification, a supervised learning task whose goal is, given a class, to estimate the relative frequency (or prevalence) of the class in a dataset of unlabeled items. Quantification has several applications in data and text mining, such as estimating the prevalence of positive reviews in a set of reviews of a given product or estimating the prevalence of a given support issue in a dataset of transcripts of phone calls to tech support. So far, quantification has been addressed by learning a general-purpose classifier, counting the unlabeled items that have been assigned the class, and tuning the obtained counts according to some heuristics. In this article, we depart from the tradition of using general-purpose classifiers and use instead a supervised learning model for structured prediction, capable of generating classifiers directly optimized for the (multivariate and nonlinear) function used for evaluating quantification accuracy. The experiments that we have run on 5,500 binary high-dimensional datasets (averaging more than 14,000 documents each) show that this method is more accurate, more stable, and more efficient than existing state-of-the-art quantification methods.
The idea of using a learner that directly optimizes a loss function specific to quantification was first proposed, although not implemented, in @cite_41 , which indeed proposes using SVM @math to directly optimize KLD; the present paper is thus the direct realization of that proposal. The first published work that implements and tests the idea of directly optimizing a quantification-specific loss function is @cite_6 , whose authors propose variants of decision trees and decision forests that directly optimize a loss combining classification accuracy and quantification accuracy. At the time of going to print we have become aware of a related paper @cite_32 whose authors, following @cite_41 , use @math to perform quantification; differently from the present paper, and similarly to @cite_6 , they use an evaluation function that combines classification accuracy and quantification accuracy.
@cite_41 @cite_32 @cite_6
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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. 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. The fixed slope lossy algorithm derived from the kth-order adaptive arithmetic codeword length function is extended to finite-state decoders or trellis-structured decoders. When this algorithm is used to encode a stationary, ergodic source with a continuous alphabet, the Lagrangian performance converges with probability one to a quantity computable as the infimum of an information-theoretic functional over a set of auxiliary random variables and reproduction levels, where spl lambda >0 and - spl lambda are designated to be the slope of the rate distortion function R(D) of the source at some D; the quantity is close to R(D)+ spl lambda D when the order k used in the arithmetic coding or the number of states in the decoders is large enough, An alternating minimization algorithm for computing the quantity is presented; this algorithm is based on a training sequence and in turn gives rise to a design algorithm for variable-rate trellis source codes. The resulting variable-rate trellis source codes are very efficient in low-rate regions. With k=8, the mean-squared error encoding performance at the rate 1 2 bits sample for memoryless Gaussian sources is comparable to that afforded by trellis-coded quantizers; with k=8 and the number of states in the decoder=32, the mean-squared error encoding performance at the rate 1 2 bits sample for memoryless Laplacian sources is about 1 dB better than that afforded by the trellis-coded quantizers with 256 states, with k=8 and the number of states in the decoder=256, the mean-squared error encoding performance at the rates of a fraction of 1 bit sample for highly dependent Gauss-Markov sources with correlation coefficient 0.9 is within about 0.6 dB of the distortion rate function.