# Optimal rates for F-score binary classification

## TL;DR

This paper establishes optimal minimax rates for binary classification using F-score under smoothness and margin assumptions, proposing a semi-supervised method that efficiently estimates the classifier with proven optimality.

## Contribution

It introduces a semi-supervised classification procedure for F-score maximization that achieves minimax optimal rates under smoothness and margin conditions.

## Key findings

- Achieves the rate $O(n^{-(1+eta)eta/(2eta+d)})$ for excess F-score.
- Establishes the optimality of the proposed rates in a minimax sense.
- Shows that unlabeled data size does not affect convergence rates.

## Abstract

We study the minimax settings of binary classification with F-score under the $\beta$-smoothness assumptions on the regression function $\eta(x) = \mathbb{P}(Y = 1|X = x)$ for $x \in \mathbb{R}^d$. We propose a classification procedure which under the $\alpha$-margin assumption achieves the rate $O(n^{--(1+\alpha)\beta/(2\beta+d)})$ 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 $\eta$ on some level $\theta^*$ 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 $\theta^*$ 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 $\eta$ than the optimal threshold $\theta^*$. 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.