We initiate the study of contextual dynamic pricing with a heterogeneous population of buyers, where a seller repeatedly posts prices (over $T$ rounds) that depend on the observable $d$-dimensional context and receives binary purchase feedback. Unlike prior work assuming homogeneous buyer types, in our setting the buyer's valuation type is drawn from an unknown distribution with finite support size $K_{\star}$. We develop a contextual pricing algorithm based on optimistic posterior sampling with regret $\widetilde{O}(K_{\star}\sqrt{dT})$, which we prove to be tight in $d$ and $T$ up to logarithmic terms. Finally, we refine our analysis for the non-contextual pricing case, proposing a variance-aware zooming algorithm that achieves the optimal dependence on $K_{\star}$.

We investigate the problem of learning a Lipschitz function from binary feedback. In this problem, a learner is trying to learn a Lipschitz function $f:[0,1]^d \rightarrow [0,1]$ over the course of $T$ rounds. On round $t$, an adversary provides the learner with an input $x_t$, the learner submits a guess $y_t$ for $f(x_t)$, and learns whether $y_t > f(x_t)$ or $y_t \leq f(x_t)$. The learner's goal is to minimize their total loss $\sum_t\ell(f(x_t), y_t)$ (for some loss function $\ell$). The problem is motivated by \textit{contextual dynamic pricing}, where a firm must sell a stream of differentiated products to a collection of buyers with non-linear valuations for the items and observes only whether the item was sold or not at the posted price.

We study a pricing setting where each customer is offered a contextualized price based on customer and/or product features that are predictive of the customer's valuation for that product. Often only historical sales records are available, where we observe whether each customer purchased a product at the price prescribed rather than the customer's true valuation. As such, the data is influenced by the historical sales policy which introduces difficulties in a) estimating future loss/regret for pricing policies without the possibility of conducting real experiments and b) optimizing new policies for downstream tasks such as revenue management. We study how to formulate loss functions which can be used for optimizing pricing policies directly, rather than going through an intermediate demand estimation stage, which can be biased in practice due to model misspecification, regularization or poor calibration. While existing approaches have been proposed when valuation data is available, we propose loss functions for the observational data setting. To achieve this, we adapt ideas from machine learning with corrupted labels, where we can consider each observed customer's outcome (purchased or not for a prescribed price), as a (known) probabilistic transformation of the customer's valuation. From this transformation we derive a class of suitable unbiased loss functions. Within this class we identify minimum variance estimators, those which are robust to poor demand function estimation, and provide guidance on when the estimated demand function is useful. Furthermore, we also show that when applied to our contextual pricing setting, estimators popular in the off-policy evaluation literature fall within this class of loss functions, and also offer managerial insights on when each estimator is likely to perform well in practice.

We investigate the problem of learning a Lipschitz function from binary feedback. In this problem, a learner is trying to learn a Lipschitz function $f:[0,1] d \rightarrow [0,1]$ over the course of $T$ rounds. On round $t$, an adversary provides the learner with an input $x_t$, the learner submits a guess $y_t$ for $f(x_t)$, and learns whether $y_t f(x_t)$ or $y_t \leq f(x_t)$. The learner's goal is to minimize their total loss $\sum_t\ell(f(x_t), y_t)$ (for some loss function $\ell$). The problem is motivated by \textit{contextual dynamic pricing}, where a firm must sell a stream of differentiated products to a collection of buyers with non-linear valuations for the items and observes only whether the item was sold or not at the posted price.

We investigate the problem of learning a Lipschitz function from binary feedback. In this problem, a learner is trying to learn a Lipschitz function $f:[0,1]^d \rightarrow [0,1]$ over the course of $T$ rounds. On round $t$, an adversary provides the learner with an input $x_t$, the learner submits a guess $y_t$ for $f(x_t)$, and learns whether $y_t > f(x_t)$ or $y_t \leq f(x_t)$. The learner's goal is to minimize their total loss $\sum_t\ell(f(x_t), y_t)$ (for some loss function $\ell$). The problem is motivated by \textit{contextual dynamic pricing}, where a firm must sell a stream of differentiated products to a collection of buyers with non-linear valuations for the items and observes only whether the item was sold or not at the posted price. For the symmetric loss $\ell(f(x_t), y_t) = \vert f(x_t) - y_t \vert$, we provide an algorithm for this problem achieving total loss $O(\log T)$ when $d=1$ and $O(T^{(d-1)/d})$ when $d>1$, and show that both bounds are tight (up to a factor of $\sqrt{\log T}$). For the pricing loss function $\ell(f(x_t), y_t) = f(x_t) - y_t {\bf 1}\{y_t \leq f(x_t)\}$ we show a regret bound of $O(T^{d/(d+1)})$ and show that this bound is tight. We present improved bounds in the special case of a population of linear buyers.

We investigate the problem of learning a Lipschitz function from binary feedback. In this problem, a learner is trying to learn a Lipschitz function $f:[0,1]^d \rightarrow [0,1]$ over the course of $T$ rounds. On round $t$, an adversary provides the learner with an input $x_t$, the learner submits a guess $y_t$ for $f(x_t)$, and learns whether $y_t > f(x_t)$ or $y_t \leq f(x_t)$. The learner's goal is to minimize their total loss $\sum_t\ell(f(x_t), y_t)$ (for some loss function $\ell$). The problem is motivated by \textit{contextual dynamic pricing}, where a firm must sell a stream of differentiated products to a collection of buyers with non-linear valuations for the items and observes only whether the item was sold or not at the posted price. For the symmetric loss $\ell(f(x_t), y_t) = \vert f(x_t) - y_t \vert$, we provide an algorithm for this problem achieving total loss $O(\log T)$ when $d=1$ and $O(T^{(d-1)/d})$ when $d>1$, and show that both bounds are tight (up to a factor of $\sqrt{\log T}$). For the pricing loss function $\ell(f(x_t), y_t) = f(x_t) - y_t {\bf 1}\{y_t \leq f(x_t)\}$ we show a regret bound of $O(T^{d/(d+1)})$ and show that this bound is tight. We present improved bounds in the special case of a population of linear buyers.