Certifiable Robustness for Nearest Neighbor Classifiers

TL;DR

This paper investigates the computational complexity of certifying the robustness of k-Nearest Neighbors classifiers in the presence of inconsistent datasets with functional dependencies, revealing a dichotomy in problem complexity.

Contribution

It establishes a complexity dichotomy for certifying robustness of k-NN classifiers under data inconsistencies with functional dependencies, including polynomial-time and coNP-hard cases.

Findings

Complexity dichotomy for robustness certification problems.

Polynomial-time algorithms exist for certain FDs.

Certifying robustness is coNP-hard in some cases.

Abstract

ML models are typically trained using large datasets of high quality. However, training datasets often contain inconsistent or incomplete data. To tackle this issue, one solution is to develop algorithms that can check whether a prediction of a model is certifiably robust. Given a learning algorithm that produces a classifier and given an example at test time, a classification outcome is certifiably robust if it is predicted by every model trained across all possible worlds (repairs) of the uncertain (inconsistent) dataset. This notion of robustness falls naturally under the framework of certain answers. In this paper, we study the complexity of certifying robustness for a simple but widely deployed classification algorithm, $k$-Nearest Neighbors ($k$-NN). Our main focus is on inconsistent datasets when the integrity constraints are functional dependencies (FDs). For this setting, we establish a dichotomy in the complexity of certifying robustness w.r.t. the set of FDs: the problem either admits a polynomial time algorithm, or it is coNP-hard. Additionally, we exhibit a similar dichotomy for the counting version of the problem, where the goal is to count the number of possible worlds that predict a certain label. As a byproduct of our study, we also establish the complexity of a problem related to finding an optimal subset repair that may be of independent interest.