Price of Fairness in Bandits: A Tight Minimax Characterization

• URL: http://arxiv.org/abs/2607.13402v1
• Date: Wed, 15 Jul 2026 02:58:57 GMT
• Title: Price of Fairness in Bandits: A Tight Minimax Characterization
• Abstract summary: In bandit problems, standard regret-minimizing algorithms treat exploration as an amortized cost, exposing early participants to unfair ex-ante losses.<n>Recent work addresses this by evaluating the sequence of per-round expected rewards through the generalized $p$-mean.<n>We prove an algorithm-independent lower bound $(sqrtkmax(1,q)/T)$; for $q>1$, this shows that the penalty $kq/2$ is information-theoretically unavoidable.
• Score: 9.360852329235477
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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