Minimax convergence rates of a binary classification procedure for time-homogeneous SDE paths

TL;DR

This paper investigates the optimal rates at which a classifier can learn to distinguish between two classes of trajectories generated by time-homogeneous SDEs, considering unknown parameters and using nonparametric estimators.

Contribution

It introduces a new methodology for deriving lower bounds on excess risk and analyzes minimax convergence rates for plug-in classifiers with nonparametric drift and diffusion estimators.

Findings

Established convergence rates for drift coefficient estimators.

Proposed a novel approach for lower bound analysis on excess risk.

Validated theoretical results with simulated data experiments.

Abstract

In the context of binary classification of trajectories generated by time-homogeneous stochastic differential equations, we consider a mixture of two diffusion processes characterized by a stochastic differential equation (SDE) whose drift coefficient depends on the class and whose diffusion coefficient is independent of the class. We assume that the drift and diffusion coefficients are unknown as well as the law of the discrete random variable that models the class. In this paper, we study the minimax convergence rates for the excess risk of the resulting plug-in classifier under different sets of assumptions on the diffusion model. As the plug-in classifier is based on nonparametric estimators of drift and diffusion coefficients, we established rates of convergence for projection estimators of drift coefficients on the real line. We propose a new methodology for the study of the lower bound on the excess risk. The theoretical study is completed with a numerical experiment over simulated data.