source: arxiv statistics ml: price of fairness in bandits: a tight minimax characterization
level: research
in bandit problems, standard algorithms minimize total regret but can treat early participants unfairly. recent work uses the generalized p-mean to balance welfare and fairness, with p=1 for utilitarian, p→0 for nash, and p→−∞ for rawlsian fairness. tight guarantees were known only for p≥0. the strictly fair regime with q=−p>0 was unresolved because negative-power means focus on the smallest per-round rewards, making analysis harder.
for σ-sub-gaussian rewards with nonnegative means, the best prior algorithm used uniform early exploration and had regret o(k^{(q+1)/2}/√t), where k is the number of arms and t is the horizon. the only general lower bound was the classical ω(σ√(k/t)), leaving a gap. it was unclear if the extra dependence on k was necessary for strict fairness or an artifact of the algorithm.
this paper closes the gap by proving a tight minimax regret bound of θ(k^{(q+1)/2}/√t) for all q>0. the upper bound comes from a new algorithm that carefully balances exploration and exploitation under the negative p-mean. the lower bound shows that any algorithm must suffer this dependence on k, confirming that strict fairness inherently costs more in terms of regret. the results apply to clinical trials and other settings where fairness across participants is critical.
why it matters: this work gives practitioners precise trade-offs between fairness and regret in sequential decision making, guiding algorithm choice in sensitive applications like clinical trials.
source: arxiv statistics ml: price of fairness in bandits: a tight minimax characterization