We consider the Continuum Bandit problem where the goal is to find the optimal action under an unknown reward function, with an additional monotonicity constraint (or, markdown constraint) that requires that the action sequence be non-increasing. This problem faithfully models a natural single-product dynamic pricing problem, called markdown pricing, where the objective is to adaptively reduce the price over a finite sales horizon to maximize expected revenues. Jia et al '21 and Chen '21 independently showed a tight $T^{3/4}$ regret bound over $T$ rounds under *minimal* assumptions of unimodality and Lipschitzness in the reward (or, revenue) function. This bound shows that the demand learning in markdown pricing is harder than unconstrained (i.e., without the monotonicity constraint) pricing under unknown demand which suffers regret only of the order of $T^{2/3}$ under the same assumptions (Kleinberg '04). However, in practice the demand functions are usually assumed to have certain functional forms (e.g.

This problem faithfully models a natural revenue management problem, called "markdown pricing",

We consider the Continuum Bandit problem where the goal is to find the optimal action under an unknown reward function, with an additional monotonicity constraint (or, "markdown" constraint) that requires that the action sequence be non-increasing. This problem faithfully models a natural single-product dynamic pricing problem, called "markdown pricing", where the objective is to adaptively reduce the price over a finite sales horizon to maximize expected revenues. This bound shows that the demand learning in markdown pricing is harder than unconstrained (i.e., without the monotonicity constraint) pricing under unknown demand which suffers regret only of the order of T {2/3} under the same assumptions (Kleinberg '04). However, in practice the demand functions are usually assumed to have certain functional forms (e.g. We investigate two fundamental questions, assuming the underlying demand curve comes from a given parametric family: (1) Can we improve the T {3/4} regret bound for markdown pricing, under extra assumptions on the functional forms of the demand functions?

Consider a single-product revenue-maximization problem where the seller monotonically decreases the price in $n$ rounds with an unknown demand model coming from a given family. Without monotonicity, the minimax regret is $\tilde O(n^{2/3})$ for the Lipschitz demand family and $\tilde O(n^{1/2})$ for a general class of parametric demand models. With monotonicity, the minimax regret is $\tilde O(n^{3/4})$ if the revenue function is Lipschitz and unimodal. However, the minimax regret for parametric families remained open. In this work, we provide a complete settlement for this fundamental problem. We introduce the crossing number to measure the complexity of a family of demand functions. In particular, the family of degree-$k$ polynomials has a crossing number $k$. Based on conservatism under uncertainty, we present (i) a policy with an optimal $\Theta(\log^2 n)$ regret for families with crossing number $k=0$, and (ii) another policy with an optimal $\tilde \Theta(n^{k/(k+1)})$ regret when $k\ge 1$. These bounds are asymptotically higher than the $\tilde O(\log n)$ and $\tilde \Theta(\sqrt n)$ minimax regret for the same families without the monotonicity constraint.