Are all models wrong? Fundamental limits in distribution-free empirical model falsification

TL;DR

This paper investigates the fundamental limits of establishing lower bounds on model risk in distribution-free settings, revealing inherent hardness in certifying model optimality or inadequacy, especially in interpolation learning.

Contribution

It provides a distribution-free, model-agnostic hardness result for lower bounds on model risk, highlighting fundamental limitations in empirical model falsification.

Findings

Hardness results apply broadly across model classes

Limits on certifying model optimality in distribution-free settings

Implications for model selection and evaluation in machine learning

Abstract

In statistics and machine learning, when we train a fitted model on available data, we typically want to ensure that we are searching within a model class that contains at least one accurate model -- that is, we would like to ensure an upper bound on the model class risk (the lowest possible risk that can be attained by any model in the class). However, it is also of interest to establish lower bounds on the model class risk, for instance so that we can determine whether our fitted model is at least approximately optimal within the class, or, so that we can decide whether the model class is unsuitable for the particular task at hand. Particularly in the setting of interpolation learning where machine learning models are trained to reach zero error on the training data, we might ask if, at the very least, a positive lower bound on the model class risk is possible -- or are we unable to detect that "all models are wrong"? In this work, we answer these questions in a distribution-free setting by establishing a model-agnostic, fundamental hardness result for the problem of constructing a lower bound on the best test error achievable over a model class, and examine its implications on specific model classes such as tree-based methods and linear regression.