In contextual dynamic pricing, a seller sequentially prices goods based on
contextual information. Buyers will purchase products only if the prices are
below their valuations. The goal of the seller is to design a pricing strategy
that collects as much revenue as possible. We focus on two different valuation
models. The first assumes that valuations linearly depend on the context and
are further distorted by noise. Under minor regularity assumptions, our
algorithm achieves an optimal regret bound of $\tilde{\mathcal{O}}(T^{2/3})$,
improving the existing results. The second model removes the linearity
assumption, requiring only that the expected buyer valuation is
$\beta$-H\"older in the context. For this model, our algorithm obtains a regret
$\tilde{\mathcal{O}}(T^{d+2\beta/d+3\beta})$, where $d$ is the dimension of the
context space.

在此研究中，我们探讨了上下文动态定价的关键问题，提出了两种估值模型，并针对线性性假设和期望买家估值问题分别设计了算法，以实现较低的后悔率。

改进的上下文动态定价算法

Improved Algorithms for Contextual Dynamic Pricing

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.

研究了上下文动态定价问题，探讨了需求模型、遗憾上限的最优化和隐私保护对动态定价的影响。

上下文动态定价：算法、优化和局部差分隐私约束

Contextual Dynamic Pricing: Algorithms, Optimality, and Local  Differential Privacy Constraints

Contextual dynamic pricing aims to set personalized prices based on
sequential interactions with customers. At each time period, a customer who is
interested in purchasing a product comes to the platform. The customer's
valuation for the product is a linear function of contexts, including product
and customer features, plus some random market noise. The seller does not
observe the customer's true valuation, but instead needs to learn the valuation
by leveraging contextual information and historical binary purchase feedbacks.
Existing models typically assume full or partial knowledge of the random noise
distribution. In this paper, we consider contextual dynamic pricing with
unknown random noise in the valuation model. Our distribution-free pricing
policy learns both the contextual function and the market noise simultaneously.
A key ingredient of our method is a novel perturbed linear bandit framework,
where a modified linear upper confidence bound algorithm is proposed to balance
the exploration of market noise and the exploitation of the current knowledge
for better pricing. We establish the regret upper bound and a matching lower
bound of our policy in the perturbed linear bandit framework and prove a
sub-linear regret bound in the considered pricing problem. Finally, we
demonstrate the superior performance of our policy on simulations and a
real-life auto-loan dataset.

本研究旨在探讨基于上下文的动态定价，采用一种新颖的扰动线性 bandit 框架来平衡市场噪音与当前知识的利用，学习上下文函数和市场噪音，证明了所述策略在扰动线性 bandit 框架下的遗憾上界和匹配下界，并且在模拟和现实场景中表现出卓越的性能。