A Query-Optimal Algorithm for Finding Counterfactuals

07/14/2022

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by   Guy Blanc, et al.

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