Open Problem: Structure-Agnostic Minimax Risk for Partial Linear Model

Abstract

Double machine learning is a theoretically grounded and practically efficient procedure for a variety of causal estimands and functional estimation problems when adopting black-box machine learning models for estimating nuisance parameters. It is known that double machine learning may have sub-optimal performance in the structure-aware settings, e.g., the nuisances are H{ö}lder smooth functions, and recent articles (Balakrishnan et al., 2023) are delivering the message that double machine learning is optimal in structure-agnostic settings. This note claims that whether double machine learning is optimal for black-box machine learning models remains open, even for the simplest linear coefficient estimation in the partial linear model. We argue that the key gap that differentiates structure-agnostic and structure-aware settings, and also the previous lower bound results do not address, is the role of variance – the awareness of well-conditioned structures offers the possibility to mitigate the effects of variance, while that is not clear for structure-agnostic settings. The answer to this question has significant implications both in theory and practice.