A Query-Optimal Algorithm for Finding Counterfactuals

TL;DR

This paper introduces a query-efficient algorithm for finding optimal counterfactuals in monotone models, with theoretical guarantees close to the proven lower bounds, improving over brute-force methods.

Contribution

The paper presents a novel algorithm with near-optimal query complexity for identifying nearest counterfactuals in monotone models, supported by tight theoretical bounds.

Findings

Algorithm makes S(f)^{O(Δ_f(x*))}·log d queries

Achieves optimality within proven lower bounds

Outperforms previous brute-force approaches

Abstract

We design an algorithm for finding counterfactuals with strong theoretical guarantees on its performance. For any monotone model $f : X^d \to \{0,1\}$ and instance $x^\star$, our algorithm makes \[ {S(f)^{O(\Delta_f(x^\star))}\cdot \log d}\] queries to $f$ and returns {an {\sl optimal}} counterfactual for $x^\star$: a nearest instance $x'$ to $x^\star$ for which $f(x')\ne f(x^\star)$. Here $S(f)$ is the sensitivity of $f$, a discrete analogue of the Lipschitz constant, and $\Delta_f(x^\star)$ is the distance from $x^\star$ to its nearest counterfactuals. The previous best known query complexity was $d^{\,O(\Delta_f(x^\star))}$, achievable by brute-force local search. We further prove a lower bound of $S(f)^{\Omega(\Delta_f(x^\star))} + \Omega(\log d)$ on the query complexity of any algorithm, thereby showing that the guarantees of our algorithm are essentially optimal.