Downsampling or under-sampling is a technique that is utilized in the context of large and highly imbalanced classification models. We study optimal downsampling for imbalanced classification using generalized linear models (GLMs). We propose a pseudo maximum likelihood estimator and study its asymptotic normality in the context of increasingly imbalanced populations relative to an increasingly large sample size. We provide theoretical guarantees for the introduced estimator. Additionally, we compute the optimal downsampling rate using a criterion that balances statistical accuracy and computational efficiency. Our numerical experiments, conducted on both synthetic and empirical data, further validate our theoretical results, and demonstrate that the introduced estimator outperforms commonly available alternatives.

We consider the optimization of complex performance metrics in multi-label classification under the population utility framework. We mainly focus on metrics linearly decomposable into a sum of binary classification utilities applied separately to each label with an additional requirement of exactly $k$ labels predicted for each instance. These "macro-at-$k$" metrics possess desired properties for extreme classification problems with long tail labels. Unfortunately, the at-$k$ constraint couples the otherwise independent binary classification tasks, leading to a much more challenging optimization problem than standard macro-averages. We provide a statistical framework to study this problem, prove the existence and the form of the optimal classifier, and propose a statistically consistent and practical learning algorithm based on the Frank-Wolfe method. Interestingly, our main results concern even more general metrics being non-linear functions of label-wise confusion matrices. Empirical results provide evidence for the competitive performance of the proposed approach.

We consider sequential maximization of performance metrics that are general functions of a confusion matrix of a classifier (such as precision, F-measure, or G-mean). Such metrics are, in general, non-decomposable over individual instances, making their optimization very challenging. While they have been extensively studied under different frameworks in the batch setting, their analysis in the online learning regime is very limited, with only a few distinguished exceptions. In this paper, we introduce and analyze a general online algorithm that can be used in a straightforward way with a variety of complex performance metrics in binary, multi-class, and multi-label classification problems. The algorithm’s update and prediction rules are appealingly simple and computationally efficient without the need to store any past data. We show the algorithm attains $\mathcal{O}(\frac{\ln n}{n})$ regret for concave and smooth metrics and verify the efficiency of the proposed algorithm in empirical studies.

Extreme multi-label classification (XMLC) is the task of selecting a small subset of relevant labels from a very large set of possible labels. As such, it is characterized by long-tail labels, i.e., most labels have very few positive instances. With standard performance measures such as precision@k, a classifier can ignore tail labels and still report good performance. However, it is often argued that correct predictions in the tail are more "interesting" or "rewarding," but the community has not yet settled on a metric capturing this intuitive concept. The existing propensity-scored metrics fall short on this goal by confounding the problems of long-tail and missing labels. In this paper, we analyze generalized metrics budgeted "at k" as an alternative solution. To tackle the challenging problem of optimizing these metrics, we formulate it in the expected test utility (ETU) framework, which aims to optimize the expected performance on a given test set. We derive optimal prediction rules and construct their computationally efficient approximations with provable regret guarantees and being robust against model misspecification. Our algorithm, based on block coordinate descent, scales effortlessly to XMLC problems and obtains promising results in terms of long-tail performance.

The propensity model introduced by Jain et al has become a standard approach for dealing with missing and long-tail labels in extreme multi-label classification (XMLC). In this paper, we critically revise this approach showing that despite its theoretical soundness, its application in contemporary XMLC works is debatable. We exhaustively discuss the flaws of the propensity-based approach, and present several recipes, some of them related to solutions used in search engines and recommender systems, that we believe constitute promising alternatives to be followed in XMLC.

Multi-label classification (MLC) has wide practical importance, but the theoretical understanding of its statistical properties is still limited. As an attempt to fill this gap, we thoroughly study upper and lower regret bounds for two canonical MLC performance measures, Hamming loss and Precision@$\kappa$. We consider two different statistical and algorithmic settings, a non-parametric setting tackled by plug-in classifiers \`a la $k$-nearest neighbors, and a parametric one tackled by empirical risk minimization operating on surrogate loss functions. For both, we analyze the interplay between a natural MLC variant of the low noise assumption, widely studied in binary classification, and the label sparsity, the latter being a natural property of large-scale MLC problems. We show that those conditions are crucial in improving the bounds, but the way they are tangled is not obvious, and also different across the two settings.

Set-valued prediction is a well-known concept in multi-class classification. When a classifier is uncertain about the class label for a test instance, it can predict a set of classes instead of a single class. In this paper, we focus on hierarchical multi-class classification problems, where valid sets (typically) correspond to internal nodes of the hierarchy. We argue that this is a very strong restriction, and we propose a relaxation by introducing the notion of representation complexity for a predicted set. In combination with probabilistic classifiers, this leads to a challenging inference problem for which specific combinatorial optimization algorithms are needed. We propose three methods and evaluate them on benchmark datasets: a naïve approach that is based on matrix-vector multiplication, a reformulation as a knapsack problem with conflict graph, and a recursive tree search method. Experimental results demonstrate that the last method is computationally more efficient than the other two approaches, due to a hierarchical factorization of the conditional class distribution.

Extreme multi-label classification (XMLC) refers to the task of tagging instances with small subsets of relevant labels coming from an extremely large set of all possible labels. Recently, XMLC has been widely applied to diverse web applications such as automatic content labeling, online advertising, or recommendation systems. In such environments, label distribution is often highly imbalanced, consisting mostly of very rare tail labels, and relevant labels can be missing. As a remedy to these problems, the propensity model has been introduced and applied within several XMLC algorithms. In this work, we focus on the problem of optimal predictions under this model for probabilistic label trees, a popular approach for XMLC problems. We introduce an inference procedure, based on the A*-search algorithm, that efficiently finds the optimal solution, assuming that all probabilities and propensities are known. We demonstrate the attractiveness of this approach in a wide empirical study on popular XMLC benchmark datasets.

Extreme multi-label classification (XMLC) is a problem of tagging an instance with a small subset of relevant labels chosen from an extremely large pool of possible labels. Large label spaces can be efficiently handled by organizing labels as a tree, like in the hierarchical softmax (HSM) approach commonly used for multi-class problems. In this paper, we investigate probabilistic label trees (PLTs) that have been recently devised for tackling XMLC problems. We show that PLTs are a no-regret multi-label generalization of HSM when precision@$k$ is used as a model evaluation metric. Critically, we prove that pick-one-label heuristic---a reduction technique from multi-label to multi-class that is routinely used along with HSM---is not consistent in general. We also show that our implementation of PLTs, referred to as extremeText (XT), obtains significantly better results than HSM with the pick-one-label heuristic and XML-CNN, a deep network specifically designed for XMLC problems. Moreover, XT is competitive to many state-of-the-art approaches in terms of statistical performance, model size and prediction time which makes it amenable to deploy in an online system.

In this paper, we present a novel approach to estimate the relative depth of regions in monocular images. There are several contributions. First, the task of monocular depth estimation is considered as a learning-to-rank problem which offers several advantages compared to regression approaches. Second, monocular depth clues of human perception are modeled in a systematic manner. Third, we show that these depth clues can be modeled and integrated appropriately in a Rankboost framework. For this purpose, a space-efficient version of Rankboost is derived that makes it applicable to rank a large number of objects, as posed by the given problem. Finally, the monocular depth clues are combined with results from a deep learning approach. Experimental results show that the error rate is reduced by adding the monocular features while outperforming state-of-the-art systems.

Statistical learning theory is at an inflection point enabled by recent advances in understanding and optimizing a wide range of metrics. Of particular interest are non-decomposable metrics such as the F-measure and the Jaccard measure which cannot be represented as a simple average over examples. Non-decomposability is the primary source of difficulty in theoretical analysis, and interestingly has led to two distinct settings and notions of consistency. In this manuscript we analyze both settings, from statistical and algorithmic points of view, to explore the connections and to highlight differences between them for a wide range of metrics. The analysis complements previous results on this topic, clarifies common confusions around both settings, and provides guidance to the theory and practice of binary classification with complex metrics.

We consider optimization of generalized performance metrics for binary classification by means of surrogate losses. We focus on a class of metrics, which are linear-fractional functions of the false positive and false negative rates (examples of which include $$F_{\beta }$$ -measure, Jaccard similarity coefficient, AM measure, and many others). Our analysis concerns the following two-step procedure. First, a real-valued function f is learned by minimizing a surrogate loss for binary classification on the training sample. It is assumed that the surrogate loss is a strongly proper composite loss function (examples of which include logistic loss, squared-error loss, exponential loss, etc.). Then, given f, a threshold $$\widehat{\theta }$$ is tuned on a separate validation sample, by direct optimization of the target performance metric. We show that the regret of the resulting classifier (obtained from thresholding f on $$\widehat{\theta }$$ ) measured with respect to the target metric is upperbounded by the regret of f measured with respect to the surrogate loss. We also extend our results to cover multilabel classification and provide regret bounds for micro- and macro-averaging measures. Our findings are further analyzed in a computational study on both synthetic and real data sets.

We consider the problem of (macro) F-measure maximization in the context of extreme multi-label classification (XMLC), i.e., multi-label classification with extremely large label spaces. We investigate several approaches based on recent results on the maximization of complex performance measures in binary classification. According to these results, the F-measure can be maximized by properly thresholding conditional class probability estimates. We show that a naive adaptation of this approach can be very costly for XMLC and propose to solve the problem by classifiers that efficiently deliver sparse probability estimates (SPEs), that is, probability estimates restricted to the most probable labels. Empirical results provide evidence for the strong practical performance of this approach.

The F-measure is an important and commonly used performance metric for binary prediction tasks. By combining precision and recall into a single score, it avoids disadvantages of simple metrics like the error rate, especially in cases of imbalanced class distributions. The problem of optimizing the F-measure, that is, of developing learning algorithms that perform optimally in the sense of this measure, has recently been tackled by several authors. In this paper, we study the problem of F-measure maximization in the setting of online learning. We propose an efficient online algorithm and provide a formal analysis of its convergence properties. Moreover, first experimental results are presented, showing that our method performs well in practice.

We compare the plug-in rule approach for optimizing the F-measure in multi-label classification with an approach based on structured loss minimization, such as the structured support vector machine (SSVM). Whereas the former derives an optimal prediction from a probabilistic model in a separate inference step, the latter seeks to optimize the F-measure directly during the training phase. We introduce a novel plug-in rule algorithm that estimates all parameters required for a Bayes-optimal prediction via a set of multinomial regression models, and we compare this algorithm with SSVMs in terms of computational complexity and statistical consistency. As a main theoretical result, we show that our plug-in rule algorithm is consistent, whereas the SSVM approaches are not. Finally, we present results of a large experimental study showing the benefits of the introduced algorithm.

Most of the multi-label classification (MLC) methods proposed in recent years intended to exploit, in one way or the other, dependencies between the class labels. Comparing to simple binary relevance learning as a baseline, any gain in performance is normally explained by the fact that this method is ignoring such dependencies. Without questioning the correctness of such studies, one has to admit that a blanket explanation of that kind is hiding many subtle details, and indeed, the underlying mechanisms and true reasons for the improvements reported in experimental studies are rarely laid bare. Rather than proposing yet another MLC algorithm, the aim of this paper is to elaborate more closely on the idea of exploiting label dependence, thereby contributing to a better understanding of MLC. Adopting a statistical perspective, we claim that two types of label dependence should be distinguished, namely conditional and marginal dependence. Subsequently, we present three scenarios in which the exploitation of one of these types of dependence may boost the predictive performance of a classifier. In this regard, a close connection with loss minimization is established, showing that the benefit of exploiting label dependence does also depend on the type of loss to be minimized. Concrete theoretical results are presented for two representative loss functions, namely the Hamming loss and the subset 0/1 loss. In addition, we give an overview of state-of-the-art decomposition algorithms for MLC and we try to reveal the reasons for their effectiveness. Our conclusions are supported by carefully designed experiments on synthetic and benchmark data.

The F-measure, originally introduced in information retrieval, is nowadays routinely used as a performance metric for problems such as binary classification, multi-label classification, and structured output prediction. Optimizing this measure remains a statistically and computationally challenging problem, since no closed-form maximizer exists. Current algorithms are approximate and typically rely on additional assumptions regarding the statistical distribution of the binary response variables. In this paper, we present an algorithm which is not only computationally efficient but also exact, regardless of the underlying distribution. The algorithm requires only a quadratic number of parameters of the joint distribution (with respect to the number of binary responses). We illustrate its practical performance by means of experimental results for multi-label classification.

Ordinal classification problems with monotonicity constraints (also referred to as multicriteria classification problems) often appear in real-life applications, however, they are considered relatively less frequently in theoretical studies than regular classification problems. We introduce a rule induction algorithm based on the statistical learning approach that is tailored for this type of problems. The algorithm first monotonizes the dataset (excludes strongly inconsistent objects), using Stochastic Dominance-based Rough Set Approach, and then uses forward stagewise additive modeling framework for generating a monotone rule ensemble. Experimental results indicate that taking into account knowledge about order andmonotonicity constraints in the classifier can improve the prediction accuracy.

We propose a new rule induction algorithm for solving classification problems via probability estimation. The main advantage of decision rules is their simplicity and good interpretability. While the early approaches to rule induction were based on sequential covering, we follow an approach in which a single decision rule is treated as a base classifier in an ensemble. The ensemble is built by greedily minimizing the negative loglikelihood which results in estimating the class conditional probability distribution. The introduced approach is compared with other decision rule induction algorithms such as SLIPPER, LRI and RuleFit.