We study the dynamic pricing problem where the demand function is nonparametric and H\"older smooth, and we focus on adaptivity to the unknown H\"older smoothness parameter $\beta$ of the demand function. Traditionally the optimal dynamic pricing algorithm heavily relies on the knowledge of $\beta$ to achieve a minimax optimal regret of $\widetilde{O}(T^{\frac{\beta+1}{2\beta+1}})$. However, we highlight the challenge of adaptivity in this dynamic pricing problem by proving that no pricing policy can adaptively achieve this minimax optimal regret without knowledge of $\beta$. Motivated by the impossibility result, we propose a self-similarity condition to enable adaptivity. Importantly, we show that the self-similarity condition does not compromise the problem's inherent complexity since it preserves the regret lower bound $\Omega(T^{\frac{\beta+1}{2\beta+1}})$. Furthermore, we develop a smoothness-adaptive dynamic pricing algorithm and theoretically prove that the algorithm achieves this minimax optimal regret bound without the prior knowledge $\beta$.

翻译：我们研究需求函数为非参数且Hölder光滑的动态定价问题，重点关注对未知需求函数Hölder光滑参数$\beta$的自适应性。传统上，最优动态定价算法高度依赖$\beta$的先验知识以实现$\widetilde{O}(T^{\frac{\beta+1}{2\beta+1}})$的极小化最优遗憾值。然而，我们通过证明任何定价策略在未知$\beta$的情况下均无法自适应地达到该极小化最优遗憾值，揭示了动态定价问题中的自适应性挑战。受此不可能性结论的启发，我们提出自相似条件以实现自适应性。值得注意的是，我们证明自相似条件不会降低问题的固有复杂度，因为它保留了遗憾下界$\Omega(T^{\frac{\beta+1}{2\beta+1}})$。此外，我们开发了一种自适应平滑度的动态定价算法，并从理论上证明该算法无需$\beta$先验知识即可达到该极小化最优遗憾界。