A Multiclass Classification Approach to Label Ranking
This work addresses a more ambitious task in multiclass classification for scenarios where ranking all labels is needed, but it is incremental as it builds on existing ranking median regression methods.
The paper tackles the problem of label ranking, which involves sorting all possible labels by their posterior probabilities rather than just predicting the top label, by analyzing it as a variant of ranking median regression and proving that the One-Versus-One multiclassification approach can yield an optimal ranking under certain noise conditions, with experimental results supporting its relevance.
In multiclass classification, the goal is to learn how to predict a random label $Y$, valued in $\mathcal{Y}=\{1,\; \ldots,\; K \}$ with $K\geq 3$, based upon observing a r.v. $X$, taking its values in $\mathbb{R}^q$ with $q\geq 1$ say, by means of a classification rule $g:\mathbb{R}^q\to \mathcal{Y}$ with minimum probability of error $\mathbb{P}\{Y\neq g(X) \}$. However, in a wide variety of situations, the task targeted may be more ambitious, consisting in sorting all the possible label values $y$ that may be assigned to $X$ by decreasing order of the posterior probability $η_y(X)=\mathbb{P}\{Y=y \mid X \}$. This article is devoted to the analysis of this statistical learning problem, halfway between multiclass classification and posterior probability estimation (regression) and referred to as label ranking here. We highlight the fact that it can be viewed as a specific variant of ranking median regression (RMR), where, rather than observing a random permutation $Σ$ assigned to the input vector $X$ and drawn from a Bradley-Terry-Luce-Plackett model with conditional preference vector $(η_1(X),\; \ldots,\; η_K(X))$, the sole information available for training a label ranking rule is the label $Y$ ranked on top, namely $Σ^{-1}(1)$. Inspired by recent results in RMR, we prove that under appropriate noise conditions, the One-Versus-One (OVO) approach to multiclassification yields, as a by-product, an optimal ranking of the labels with overwhelming probability. Beyond theoretical guarantees, the relevance of the approach to label ranking promoted in this article is supported by experimental results.