Motivated by privacy concerns in sequential decision-making on sensitive data, we address the challenge of nonparametric contextual multi-armed bandits (MAB) under local differential privacy (LDP). We develop a uniform-confidence-bound-type estimator, showing its minimax optimality supported by a matching minimax lower bound. We further consider the case where auxiliary datasets are available, subject also to (possibly heterogeneous) LDP constraints. Under the widely-used covariate shift framework, we propose a jump-start scheme to effectively utilize the auxiliary data, the minimax optimality of which is further established by a matching lower bound. Comprehensive experiments on both synthetic and real-world datasets validate our theoretical results and underscore the effectiveness of the proposed methods.

Dynamic pricing strategies are crucial for firms to maximize revenue by adjusting prices based on market conditions and customer characteristics. However, designing optimal pricing strategies becomes challenging when historical data are limited, as is often the case when launching new products or entering new markets. One promising approach to overcome this limitation is to leverage information from related products or markets to inform the focal pricing decisions. In this paper, we explore transfer learning for nonparametric contextual dynamic pricing under a covariate shift model, where the marginal distributions of covariates differ between source and target domains while the reward functions remain the same. We propose a novel Transfer Learning for Dynamic Pricing (TLDP) algorithm that can effectively leverage pre-collected data from a source domain to enhance pricing decisions in the target domain. The regret upper bound of TLDP is established under a simple Lipschitz condition on the reward function. To establish the optimality of TLDP, we further derive a matching minimax lower bound, which includes the target-only scenario as a special case and is presented for the first time in the literature. Extensive numerical experiments validate our approach, demonstrating its superiority over existing methods and highlighting its practical utility in real-world applications.

We study the contextual dynamic pricing problem where a firm sells products to $T$ sequentially arriving consumers that behave according to an unknown demand model. The firm aims to maximize its revenue, i.e. minimize its regret over a clairvoyant that knows the model in advance. The demand model is a generalized linear model (GLM), allowing for a stochastic feature vector in $\mathbb R^d$ that encodes product and consumer information. We first show that the optimal regret upper bound is of order $\sqrt{dT}$, up to a logarithmic factor, improving upon existing upper bounds in the literature by a $\sqrt{d}$ factor. This sharper rate is materialised by two algorithms: a confidence bound-type (supCB) algorithm and an explore-then-commit (ETC) algorithm. A key insight of our theoretical result is an intrinsic connection between dynamic pricing and the contextual multi-armed bandit problem with many arms based on a careful discretization. We further study contextual dynamic pricing under the local differential privacy (LDP) constraints. In particular, we propose a stochastic gradient descent based ETC algorithm that achieves an optimal regret upper bound of order $d\sqrt{T}/\epsilon$, up to a logarithmic factor, where $\epsilon>0$ is the privacy parameter. The regret upper bounds with and without LDP constraints are accompanied by newly constructed minimax lower bounds, which further characterize the cost of privacy. Extensive numerical experiments and a real data application on online lending are conducted to illustrate the efficiency and practical value of the proposed algorithms in dynamic pricing.