arXiv subjects

Evgenii Chzhen

Publications and source records attributed to Evgenii Chzhen.

23 records · Page 2Linked to original sources

arXiv2019

Optimal rates for F-score binary classification

We study the minimax settings of binary classification with F-score under the $β$-smoothness assumptions on the regression function $η(x) = \mathbb{P}(Y = 1|X = x)$ for $x \in \mathbb{R}^d$. We propose a classification procedure which under the $α$-margin assumption achieves the rate $O(n^{--(1+α)β/(2β+d)})$ for the excess F-score. In this context, the Bayes optimal classifier for the F-score can be obtained by thresholding the aforementioned regression function $η$ on some level $θ^*$ to be estimated. The proposed procedure is performed in a semi-supervised manner, that is, for the estimation of the regression function we use a labeled dataset of size $n \in \mathbb{N}$ and for the estimation of the optimal threshold $θ^*$ we use an unlabeled dataset of size $N \in \mathbb{N}$. Interestingly, the value of $N \in \mathbb{N}$ does not affect the rate of convergence, which indicates that it is "harder" to estimate the regression function $η$ than the optimal threshold $θ^*$. This further implies that the binary classification with F-score behaves similarly to the standard settings of binary classification. Finally, we show that the rates achieved by the proposed procedure are optimal in the minimax sense up to a constant factor.

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arXiv2019

Minimax semi-supervised confidence sets for multi-class classification

In this work we study the semi-supervised framework of confidence set classification with controlled expected size in minimax settings. We obtain semi-supervised minimax rates of convergence under the margin assumption and a H{ö}lder condition on the regression function. Besides, we show that if no further assumptions are made, there is no supervised method that outperforms the semi-supervised estimator proposed in this work. We establish that the best achievable rate for any supervised method is n^{--1/2} , even if the margin assumption is extremely favorable. On the contrary, semi-supervised estimators can achieve faster rates of convergence provided that sufficiently many unlabeled samples are available. We additionally perform numerical evaluation of the proposed algorithms empirically confirming our theoretical findings.

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arXiv2019

Classification of sparse binary vectors

In this work we consider a problem of multi-label classification, where each instance is associated with some binary vector. Our focus is to find a classifier which minimizes false negative discoveries under constraints. Depending on the considered set of constraints we propose plug-in methods and provide non-asymptotic analysis under margin type assumptions. Specifically, we analyze two particular examples of constraints that promote sparse predictions: in the first one, we focus on classifiers with $\ell_0$-type constraints and in the second one, we address classifiers with bounded false positive discoveries. Both formulations lead to different Bayes rules and, thus, different plug-in approaches. The first considered scenario is the popular multi-label top-$K$ procedure: a label is predicted to be relevant if its score is among the $K$ largest ones. For this case, we provide an excess risk bound that achieves so called `fast' rates of convergence under a generalization of the margin assumption to this settings. The second scenario differs significantly from the top-$K$ settings, as the constraints are distribution dependent. We demonstrate that in this scenario the almost sure control of false positive discoveries is impossible without extra assumptions. To alleviate this issue we propose a sufficient condition for the consistent estimation and provide non-asymptotic upper-bound.

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arXiv2018

On Lasso refitting strategies

A well-know drawback of l_1-penalized estimators is the systematic shrinkage of the large coefficients towards zero. A simple remedy is to treat Lasso as a model-selection procedure and to perform a second refitting step on the selected support. In this work we formalize the notion of refitting and provide oracle bounds for arbitrary refitting procedures of the Lasso solution. One of the most widely used refitting techniques which is based on Least-Squares may bring a problem of interpretability, since the signs of the refitted estimator might be flipped with respect to the original estimator. This problem arises from the fact that the Least-Squares refitting considers only the support of the Lasso solution, avoiding any information about signs or amplitudes. To this end we define a sign consistent refitting as an arbitrary refitting procedure, preserving the signs of the first step Lasso solution and provide Oracle inequalities for such estimators. Finally, we consider special refitting strategies: Bregman Lasso and Boosted Lasso. Bregman Lasso has a fruitful property to converge to the Sign-Least-Squares refitting (Least-Squares with sign constraints), which provides with greater interpretability. We additionally study the Bregman Lasso refitting in the case of orthogonal design, providing with simple intuition behind the proposed method. Boosted Lasso, in contrast, considers information about magnitudes of the first Lasso step and allows to develop better oracle rates for prediction. Finally, we conduct an extensive numerical study to show advantages of one approach over others in different synthetic and semi-real scenarios.

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arXiv2017

On the benefits of output sparsity for multi-label classification

The multi-label classification framework, where each observation can be associated with a set of labels, has generated a tremendous amount of attention over recent years. The modern multi-label problems are typically large-scale in terms of number of observations, features and labels, and the amount of labels can even be comparable with the amount of observations. In this context, different remedies have been proposed to overcome the curse of dimensionality. In this work, we aim at exploiting the output sparsity by introducing a new loss, called the sparse weighted Hamming loss. This proposed loss can be seen as a weighted version of classical ones, where active and inactive labels are weighted separately. Leveraging the influence of sparsity in the loss function, we provide improved generalization bounds for the empirical risk minimizer, a suitable property for large-scale problems. For this new loss, we derive rates of convergence linear in the underlying output-sparsity rather than linear in the number of labels. In practice, minimizing the associated risk can be performed efficiently by using convex surrogates and modern convex optimization algorithms. We provide experiments on various real-world datasets demonstrating the pertinence of our approach when compared to non-weighted techniques.

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