Minimax Optimal Regret for Causal Logistic Bandits with Counterfactual Fairness

2026-10-01

AI Summary

This study addresses regret minimization in causal logic bandits under counterfactual fairness constraints, focusing on estimation challenges arising from unidentifiable directions. For settings where only factual rewards are observable, this work proposes a weak full-rank coverage condition and defines an information scale V*. It further develops an adaptive exploration-exploitation algorithm that integrates causal inference with minimax theoretical analysis. The primary contributions lie in overcoming the limitations of existing strong assumptions by deriving matching upper and lower bounds. Specifically, the proposed algorithm achieves a minimax optimal trade-off between regret and cumulative fairness violations at the T^(2/3) rate, with the upper and lower bounds of the dominant terms matching exactly.

Abstract

We study causal logistic bandits with counterfactual fairness constraints. The causal structure is given through known factual and counterfactual feature maps that share an unknown logistic reward parameter, but the learner observes only factual rewards. Consequently, the directions determining counterfactual feasibility need not be identifiable from the available feedback. The closest prior analyses either omit a coverage condition or impose a comparatively strong one, and do not establish matching lower bounds. We first show that some coverage condition is necessary: without a coverage-type restriction, factually indistinguishable environments with different optimal fair actions force $Ω(T)$ expected joint loss. Under a weaker full-rank condition on the factual covariance pooled across actions, we identify a target-specific information scale $V_\star$ that measures the difficulty of estimating rewards and counterfactual effects from factual feedback. We construct worst-case families satisfying this condition on which every policy incurs expected joint loss $Ω\left(\left[V_\star\min\{\log K,d\}\right]^{1/3}T^{2/3}\right)$. We also give an explore--then--exploit procedure tuned using $V_\star$ and an adaptive algorithm that does not require its value. Both algorithms achieve $\max\{R_T,V_T\}=\widetilde{O}\left(\left[V_\star\min\{\log K,d\}\right]^{1/3}T^{2/3}+κd/σ_0^2\right)$, where $R_T$ is regret relative to the best fair action and $V_T$ denotes the cumulative stage-wise positive violations. Thus the upper and lower bounds match in their leading dependence on $T$, $V_\star$, and $\min\{\log K,d\}$, up to logarithmic factors.

Problem

• causal logistic bandits
• counterfactual fairness
• minimax optimal regret
• coverage condition

Innovation

• Causal Logistic Bandits
• Counterfactual Fairness
• Minimax Optimal Regret
• Information Scale
• Explore-then-Exploit