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 $\beta$.

我们研究了需求函数为非参数化和 H"older 平滑的动态定价问题，重点关注对需求函数的未知 H"older 平滑参数 β 的可适应性。在没有 β 的知识的情况下，我们证明了没有定价策略能够达到这个极小化后悔的最佳性能。受到这个不可能结果的启发，我们提出了一个自相似条件来实现可适应性。我们开发了一种平滑自适应动态定价算法，并在理论上证明该算法实现了这个最佳性能的后悔边界，而无需先验知识 β。