Minimax convergence rates of a binary plug-in type classification procedure for time-homogeneous SDE paths under low-noise conditions

TL;DR

This paper investigates the minimax convergence rates of a binary classification method for time-homogeneous SDE paths under low-noise conditions, extending previous work to more complex diffusion models with space-dependent coefficients.

Contribution

It introduces a new analysis of convergence rates for a binary classifier in a complex diffusion setting, establishing faster rates under low-noise conditions and deriving lower bounds.

Findings

Faster convergence rates under low-noise conditions.

Established an exponential inequality crucial for analysis.

Derived lower bounds on excess risk.

Abstract

The study of minimax convergence rates for classification procedures adapted to SDE paths is rarely addressed in the literature. Only one paper established optimal convergence rates for a binary classifier for SDE paths constructed from the white noise model. In this paper, we consider a more complex diffusion model with space-dependent drift and diffusion coefficients where the drift depends on the class and the diffusion coefficient is common to all classes. We establish, under the low-noise condition, a faster convergence rate over a Holder space. This result will require the establishment of an exponential inequality, which is essential to obtain the expected rate. We then study the lower bound on the excess risk of the empirical classifier.