Price of Fairness in Bandits: A Tight Minimax Characterization

2026-07-14

AI Summary

This work addresses the challenge of regret minimization under strict Rawlsian fairness in multi-armed bandits, where conventional algorithms lack tight theoretical guarantees and often impose excessive losses on early users. The authors introduce a generalized p-mean fairness criterion (with \( q = -p > 0 \)) and establish, for the first time, an information-theoretic lower bound under stringent fairness constraints. They propose the UCB-HARE algorithm, which integrates inverse-weighted harmonic ranking scheduling with a positive-mean anchor protection mechanism, achieving a fairness-aware regret bound of \( \tilde{O}(\sigma \sqrt{k^{\max(1,q)}/T}) \)—within a logarithmic factor of the derived lower bound. The analysis proves that a fairness cost of \( k^{q/2} \) is unavoidable when \( q > 1 \). Empirical results demonstrate that UCB-HARE substantially outperforms uniform exploration baselines, with performance gains increasing as \( q \) grows.

Abstract

In bandit problems, standard regret-minimizing algorithms treat exploration as an amortized cost, which can expose early participants to unfair ex-ante losses in settings such as clinical trials. Recent work addresses this by evaluating the sequence of per-round expected rewards through the generalized $p$-mean, interpolating between utilitarian welfare ($p=1$), Nash welfare ($p\to0$), and Rawlsian fairness ($p\to-\infty$). Although tight guarantees are known for $p\ge0$, the strictly fair regime $q=-p>0$ remains unresolved because negative-power means are dominated by the smallest per-round rewards. For $σ$-sub-Gaussian rewards with nonnegative means, the best prior algorithm relied on uniform early exploration and achieved regret $O(k^{(q+1)/2}/\sqrt{T})$, while the only general lower bound was the classical $Ω(σ\sqrt{k/T})$. Thus it was unclear whether the extra dependence on $k$ was intrinsic to strict fairness or an artifact of uniform exploration. We close this gap by identifying the exact polynomial price of strict fairness. Using a needle-in-haystack construction, we prove an algorithm-independent lower bound $Ω(σ\sqrt{k^{\max(1,q)}/T})$; for $q>1$, this shows that the penalty $k^{q/2}$ is information-theoretically unavoidable. We then introduce \textsf{UCB-HARE} (Harmonic Anchored Rank Exploration), which replaces uniform exploration with an inverse-weighted harmonic rank schedule protected by a certified positive-mean anchor. Its regret is $\widetilde{O}(σ\sqrt{k^{\max(1,q)}/T})$, matching the lower bound up to logarithmic factors. Experiments on synthetic instances confirm that \textsf{UCB-HARE} improves over uniform-exploration baselines, with gains increasing as $q$ grows.

Problem

• Price of Fairness
• Bandits
• Minimax Regret
• Strict Fairness
• Negative-power Mean

Innovation

• price of fairness
• minimax regret
• strict fairness
• UCB-HARE
• negative-power mean