Vladimir Pestov

Vladimir G. Pestov

We establish the last missing link allowing to describe those complete separable metric spaces $X$ in which the $k$ nearest neighbour classifier is universally consistent, both in combinatorial terms of dimension theory and via a fundamental property of real analysis. The following are equivalent: (1) The $k$-nearest neighbour classifier is universally consistent in $X$, (2) The strong Lebesgue--Besicovitch differentiation property holds in $X$ for every locally finite Borel measure, (3) $X$ is sigma-finite dimensional in the sense of Jun-Iti Nagata. The equivalence (2)$\iff$(3) was announced by Preiss (1983), while a detailed proof of the implication (3)$\Rightarrow$(2) has only appeared in Assouad and Quentin de Gromard (2006). The implication (2)$\Rightarrow$(1) was established by Cérou and Guyader (2006). We prove the implication (1)$\Rightarrow$(3). We further show that the weak (instead of strong) Lebesgue--Besicovitch property is insufficient for the consistency of the $k$-NN rule, as witnessed, for example, by the Heisenberg group (here we correct a wrong claim made in the previous article (Kumari and Pestov 2024)). A bit counter-intuitively, there is a metric on the real line uniformly equivalent to the usual distance but under which the $k$-NN classifier fails. Finally, another equivalent condition that can be added to the above is the Cover--Hart property: (4) the error of the $1$-nearest neighbour classifier is asymptotically at most twice as bad as the Bayes error.

Sushma Kumari, Vladimir G. Pestov

We continue to investigate the $k$ nearest neighbour ($k$-NN) learning rule in complete separable metric spaces. Thanks to the results of Cérou and Guyader (2006) and Preiss (1983), this rule is known to be universally consistent in every such metric space that is sigma-finite dimensional in the sense of Nagata. Here we show that the rule is strongly universally consistent in such spaces in the absence of ties. Under the tie-breaking strategy applied by Devroye, Györfi, Krzyżak, and Lugosi (1994) in the Euclidean setting, we manage to show the strong universal consistency in non-Archimedian metric spaces (that is, those of Nagata dimension zero). Combining the theorem of Cérou and Guyader with results of Assouad and Quentin de Gromard (2006), one deduces that the $k$-NN rule is universally consistent in metric spaces having finite dimension in the sense of de Groot. In particular, the $k$-NN rule is universally consistent in the Heisenberg group which is not sigma-finite dimensional in the sense of Nagata as follows from an example independently constructed by Korányi and Reimann (1995) and Sawyer and Wheeden (1992).

Vladimir Pestov

We present a universally consistent learning rule whose expected error is monotone non-increasing with the sample size under every data distribution. The question of existence of such rules was brought up in 1996 by Devroye, Györfi and Lugosi (who called them "smart"). Our rule is fully deterministic, a data-dependent partitioning rule constructed in an arbitrary domain (a standard Borel space) using a cyclic order. The central idea is to only partition at each step those cyclic intervals that exhibit a sufficient empirical diversity of labels, thus avoiding a region where the error function is convex.

Vladimir G. Pestov

We show that the $k$-nearest neighbour learning rule is universally consistent in a metric space $X$ if and only if it is universally consistent in every separable subspace of $X$ and the density of $X$ is less than every real-measurable cardinal. In particular, the $k$-NN classifier is universally consistent in every metric space whose separable subspaces are sigma-finite dimensional in the sense of Nagata and Preiss if and only if there are no real-valued measurable cardinals. The latter assumption is relatively consistent with ZFC, however the consistency of the existence of such cardinals cannot be proved within ZFC. Our results were inspired by an example sketched by Cérou and Guyader in 2006 at an intuitive level of rigour.

Benoît Collins, Sushma Kumari, Vladimir G. Pestov

The $k$ nearest neighbour learning rule (under the uniform distance tie breaking) is universally consistent in every metric space $X$ that is sigma-finite dimensional in the sense of Nagata. This was pointed out by Cérou and Guyader (2006) as a consequence of the main result by those authors, combined with a theorem in real analysis sketched by D. Preiss (1971) (and elaborated in detail by Assouad and Quentin de Gromard (2006)). We show that it is possible to give a direct proof along the same lines as the original theorem of Charles J. Stone (1977) about the universal consistency of the $k$-NN classifier in the finite dimensional Euclidean space. The generalization is non-trivial because of the distance ties being more prevalent in the non-euclidean setting, and on the way we investigate the relevant geometric properties of the metrics and the limitations of the Stone argument, by constructing various examples.

Vladimir G. Pestov

This is a set of lecture notes for an introductory course (advanced undergaduates or the 1st graduate course) on foundations of supervised machine learning (in Portuguese). The topics include: the geometry of the Hamming cube, concentration of measure, shattering and VC dimension, Glivenko-Cantelli classes, PAC learnability, universal consistency and the k-NN classifier in metric spaces, dimensionality reduction, universal approximation, sample compression. There are appendices on metric and normed spaces, measure theory, etc., making the notes self-contained. Este é um conjunto de notas de aula para um curso introdutório (curso de graduação avançado ou o 1o curso de pós) sobre fundamentos da aprendizagem de máquina supervisionada (em Português). Os tópicos incluem: a geometria do cubo de Hamming, concentração de medida, fragmentação e dimensão de Vapnik-Chervonenkis, classes de Glivenko-Cantelli, aprendizabilidade PAC, consistência universal e o classificador k-NN em espaços métricos, redução de dimensionalidade, aproximação universal, compressão amostral. Há apêndices sobre espaços métricos e normados, teoria de medida, etc., tornando as notas autosuficientes.

Hubert Haoyang Duan, Vladimir Pestov, Varun Singla

We present a supervised learning algorithm for text categorization which has brought the team of authors the 2nd place in the text categorization division of the 2012 Cybersecurity Data Mining Competition (CDMC'2012) and a 3rd prize overall. The algorithm is quite different from existing approaches in that it is based on similarity search in the metric space of measure distributions on the dictionary. At the preprocessing stage, given a labeled learning sample of texts, we associate to every class label (document category) a point in the space of question. Unlike it is usual in clustering, this point is not a centroid of the category but rather an outlier, a uniform measure distribution on a selection of domain-specific words. At the execution stage, an unlabeled text is assigned a text category as defined by the closest labeled neighbour to the point representing the frequency distribution of the words in the text. The algorithm is both effective and efficient, as further confirmed by experiments on the Reuters 21578 dataset.