# Orthogonal Statistical Learning

## TL;DR

This paper establishes non-asymptotic excess risk bounds for statistical learning with nuisance parameters, leveraging Neyman orthogonality to achieve second-order impact of nuisance estimation errors, applicable to complex models.

## Contribution

It introduces a general framework for excess risk guarantees in nuisance-robust learning, accommodating arbitrary algorithms and complex classes with weaker assumptions.

## Key findings

- Second-order impact of nuisance estimation errors.
- Applicable to complex nonparametric classes.
- Achieves oracle rates under certain entropy conditions.

## Abstract

We provide non-asymptotic excess risk guarantees for statistical learning in a setting where the population risk with respect to which we evaluate the target parameter depends on an unknown nuisance parameter that must be estimated from data. We analyze a two-stage sample splitting meta-algorithm that takes as input arbitrary estimation algorithms for the target parameter and nuisance parameter. We show that if the population risk satisfies a condition called Neyman orthogonality, the impact of the nuisance estimation error on the excess risk bound achieved by the meta-algorithm is of second order. Our theorem is agnostic to the particular algorithms used for the target and nuisance and only makes an assumption on their individual performance. This enables the use of a plethora of existing results from machine learning to give new guarantees for learning with a nuisance component. Moreover, by focusing on excess risk rather than parameter estimation, we can provide rates under weaker assumptions than in previous works and accommodate settings in which the target parameter belongs to a complex nonparametric class. We provide conditions on the metric entropy of the nuisance and target classes such that oracle rates of the same order as if we knew the nuisance parameter are achieved.