Learning to Price and Stock Under Contextual and Censored Demand

# Learning to Price and Stock Under Contextual and Censored Demand
Source: [https://arxiv.org/html/2609.06083](https://arxiv.org/html/2609.06083)
Zean HanZezhen Ding††thanks:Corresponding author\.Affiliation:The Hong Kong University of Science and TechnologyEmail:[zdingah@connect\.ust\.hk](mailto:[email protected])Jiheng ZhangAffiliation:The Hong Kong University of Science and TechnologyEmail:[jiheng@ust\.hk](mailto:[email protected])

###### Abstract

To make optimal joint pricing and inventory control decisions is a critical challenge for modern retailers\. In practice, retailers face changing market conditions where demands are influenced by various contextual factors, while simultaneously dealing with the difficulty of lost sales that obscure true demand information\. However, existing approaches often fail to account for both contextual information and censored demand observations\. We address this gap by presenting a framework where we model demand as a linear combination of basis functions with unknown coefficients, allowing for adaptive pricing and inventory decisions that respond to changing contexts\. We propose an efficient algorithm to achieve regret bound𝒪⁡\(K​T​log⁡T\)\\mathcal\{O\}\(K\\sqrt\{T\}\\log T\)under concave revenue conditions and𝒪⁡\(K2/3​T2/3​\(log⁡T\)1/2\)\\mathcal\{O\}\(K^\{2/3\}T^\{2/3\}\(\\log T\)^\{1/2\}\)for the general case, with matching lower bounds confirming optimality\. Extensive numerical experiments across diverse scenarios demonstrate our algorithm’s effectiveness\.

## 1Introduction

In recent years, joint pricing and inventory control has gained significant attention from both academia and industry\. Leading retailers increasingly leverage data science to drive innovation in pricing strategies, inventory management, and supply chain optimization\. This growing interest is reflected in comprehensive surveys\([Chen and Simchi\-Levi, 2012](https://arxiv.org/html/2609.06083#bib.bib38);[Yano and Gilbert, 2004](https://arxiv.org/html/2609.06083#bib.bib39);[Petruzzi and Dada, 1999b](https://arxiv.org/html/2609.06083#bib.bib40)\)\. However, a critical gap remains: most existing models overlook the role of censored demand and contextual information, which are essential for adaptive decision\-making\. In this paper, we address this gap by studying the optimal joint pricing and inventory control strategy with lost sales that incorporates contextual information, aiming to bring theory closer to real\-world applications\.

In this paper, we study a joint pricing and inventory control problem with contextual information and lost sales over a finite horizon ofTTperiods\. At the start of each period, the seller observes contextual information that influences customer demand\. The demand function is modeled as a linear combination ofKKbasis functions with unknown coefficients, plus an independent random noise term\. Using the observed context and an estimated demand function, the seller simultaneously sets the selling price and inventory level, with replenishment taking effect immediately\. After making the decisions, only the censored demand, i\.e\., the minimum between actual demand and available inventory, is observed as unmet demand are lost\. Leftover inventory carries over to the next period, incurring a linear holding cost, while unmet demand results in a linear lost penalty cost\. The objective is to maximize profit, defined as sales revenue minus inventory holding and lost sales costs, across theTT\-period horizon\.

Our problem introduces several interconnected challenges that complicate both theoretical analysis and practical implementation\. First, the presence of censored demand obstructs direct observation of true demand signals, necessitating the development of unbiased estimators for the demand function while mitigating inherent biases from incomplete data\. Second, the inclusion of contextual factors result in the problem being dynamic, thus traditional optimization methods which lead to static policies become inapplicable\. Both the pricing and inventory decisions must adapt to contextual information in order to achieve optimality\. Finally, when applying our methodology to models beyond the combination of the basis functions, approximation errors arise due to structural mismatches between idealized model and real\-world dynamics, demanding rigorous error quantification and robustness guarantees\. Together, these challenges necessitate a novel integration of censored data correction, context\-aware online optimization, stochastic inventory control, and error\-bounded approximation techniques\.

Note:The regrets listed above omit constant factors andpoly⁡\(log⁡T\)\\mathrm\{poly\}\(\\log T\)terms\. Here,ν=13​ln⁡T\+0\.25ln⁡T\\nu=\\frac\{1\}\{\\sqrt\{3\\ln T\}\}\+\\frac\{0\.25\}\{\\sqrt\{\\ln T\}\}\.

Table 1:Comparison of Regret Bounds under Joint Pricing and Inventory Control ProblemsOur contributions can be summarized as followings:

1. 1\.Context\-Aware Pricing and Inventory Optimization\.We propose a framework for joint pricing and inventory management \(see[Table1](https://arxiv.org/html/2609.06083#S1.T1)\) where decisions adapt to contextual information, with both pricing and inventory policies responding to changing contexts\.
2. 2\.Algorithmic Framework and Regret Analysis\.We propose an easy\-to\-implement algorithm addressing censored demand, dynamic contexts, and stochastic inventory dynamics\. Under concave revenue, our method achieves𝒪⁡\(K​T​log⁡T\)\\mathcal\{O\}\(K\\sqrt\{T\}\\log T\)regret, matching the lower bound\. Without concavity, we derive𝒪⁡\(K2/3​T2/3​\(log⁡T\)1/2\)\\mathcal\{O\}\(K^\{2/3\}T^\{2/3\}\(\\log T\)^\{1/2\}\)regret and prove a lower boundΩ⁡\(T\(m\+1\)/\(2​m\+1\)\)\\Omega\(T^\{\(m\+1\)/\(2m\+1\)\}\)in[Theorem4\.4](https://arxiv.org/html/2609.06083#S4.Thmmythm4)formm\-th differentiable revenue functions, establishing minimax optimality \(see[Table1](https://arxiv.org/html/2609.06083#S1.T1)\)\.
3. 3\.Inventory Feasibility\.To address infeasible target inventory levels from contextual information, we develop a queueing\-theoretic method using stochastic recursion to bound regret from inventory mismatches, ensuring stable performance under stochasticity\.

### 1\.1Related Work

Dynamic pricing\.Initial research focused on non\-contextual dynamic pricing\([Besbes and Zeevi, 2015](https://arxiv.org/html/2609.06083#bib.bib22);[Cesa\-Bianchi et al\., 2019](https://arxiv.org/html/2609.06083#bib.bib18)\)\.[Wang et al\. \(2021b\)](https://arxiv.org/html/2609.06083#bib.bib21)achieved𝒪~​\(T\(m\+1\)/\(2​m\+1\)\)\\tilde\{\\mathcal\{O\}\}\(T^\{\(m\+1\)/\(2m\+1\)\}\)regret formm\-th smooth demand functions with matching lower bounds\.[Bu et al\. \(2022\)](https://arxiv.org/html/2609.06083#bib.bib24)extended this to additive linear contextual effects, establishing instance\-dependent bounds\. Other models assume Bernoulli purchase decisions\([Javanmard and Nazerzadeh, 2019](https://arxiv.org/html/2609.06083#bib.bib16);[Golrezaei et al\., 2019](https://arxiv.org/html/2609.06083#bib.bib17);[Choi et al\., 2023](https://arxiv.org/html/2609.06083#bib.bib19);[Xu and Wang, 2022](https://arxiv.org/html/2609.06083#bib.bib14);[Luo et al\., 2024](https://arxiv.org/html/2609.06083#bib.bib12);[Luo et al\., 2022](https://arxiv.org/html/2609.06083#bib.bib13);[Fan et al\., 2024](https://arxiv.org/html/2609.06083#bib.bib15)\)\. Related studies\([Luo et al\., 2024](https://arxiv.org/html/2609.06083#bib.bib12);[Luo et al\., 2022](https://arxiv.org/html/2609.06083#bib.bib13);[Fan et al\., 2024](https://arxiv.org/html/2609.06083#bib.bib15)\)share similar settings but differ in noise distribution assumptions, with regret bounds ranging from𝒪~​\(d02​T2/3\)\\tilde\{\\mathcal\{O\}\}\(d\_\{0\}^\{2\}T^\{2/3\}\)to𝒪~​\(\(d0​T\)\(2​m\+1\)/\(4​m−1\)\)\\tilde\{\\mathcal\{O\}\}\(\(d\_\{0\}T\)^\{\(2m\+1\)/\(4m\-1\)\}\)\([Luo et al\., 2022](https://arxiv.org/html/2609.06083#bib.bib13);[Fan et al\., 2024](https://arxiv.org/html/2609.06083#bib.bib15)\)\.[Gong and Zhang \(2024\)](https://arxiv.org/html/2609.06083#bib.bib11)achieved𝒪~​\(d01/3​T2/3\)\\tilde\{\\mathcal\{O\}\}\(d\_\{0\}^\{1/3\}T^\{2/3\}\)with minimax optimality\.

Inventory control\.Classical inventory models, such as the newsvendor model\([Petruzzi and Dada, 1999a](https://arxiv.org/html/2609.06083#bib.bib25)\), assume i\.i\.d\. demand and use SGD\-based methods for optimization, extended to multi\-product settings, delayed replenishment\([Zhang et al\., 2020](https://arxiv.org/html/2609.06083#bib.bib30)\), and quantity constraints\([Shi et al\., 2016](https://arxiv.org/html/2609.06083#bib.bib29)\)\.[Lyu et al\. \(2024\)](https://arxiv.org/html/2609.06083#bib.bib27)addressed SGD infeasibility using mini\-batch strategies\. Recent work incorporates demand characteristics\([Ding et al\., 2021](https://arxiv.org/html/2609.06083#bib.bib26)\), while SAA methods construct empirical distributions from demand samples\([Lin et al\., 2022](https://arxiv.org/html/2609.06083#bib.bib28)\)\.

Joint pricing and inventory control\.[Whitin \(1955\)](https://arxiv.org/html/2609.06083#bib.bib31)first introduced joint pricing and inventory control with full information\. Subsequent work\([Bensoussan et al\., 2019](https://arxiv.org/html/2609.06083#bib.bib32)\)used dynamic programming, while[Qin et al\. \(2022\)](https://arxiv.org/html/2609.06083#bib.bib33)proposed approximations with sample complexity analysis\. In online learning settings,[Chen et al\. \(2019\)](https://arxiv.org/html/2609.06083#bib.bib34)considered data\-driven models without historical data\.[Chen et al\. \(2021\)](https://arxiv.org/html/2609.06083#bib.bib36)addressed censored demand using spline interpolation and sample average methods\.[Chen et al\. \(2024\)](https://arxiv.org/html/2609.06083#bib.bib35)proposed ternary/binary search methods for convex/non\-convex demand functions, establishing theoretical lower bounds\. However, existing models ignore contextual information; once contexts are considered, optimal prices and inventory change over time, making prior algorithms inapplicable\. We summarize comparisons in[Table1](https://arxiv.org/html/2609.06083#S1.T1)\.

Contextual bandit\.Our policy connects to bandit algorithms\([Lattimore and Szepesvári, 2020](https://arxiv.org/html/2609.06083#bib.bib1);[Foster and Rakhlin, 2020](https://arxiv.org/html/2609.06083#bib.bib9);[Abbasi\-Yadkori et al\., 2011](https://arxiv.org/html/2609.06083#bib.bib4);[Takemura et al\., 2021](https://arxiv.org/html/2609.06083#bib.bib10);[Auer, 2002](https://arxiv.org/html/2609.06083#bib.bib2);[Simchi\-Levi and Xu, 2021](https://arxiv.org/html/2609.06083#bib.bib6)\), including linear[Abbasi\-Yadkori et al\. \(2011\)](https://arxiv.org/html/2609.06083#bib.bib4)and generalized linear bandits[Li et al\. \(2017\)](https://arxiv.org/html/2609.06083#bib.bib5);[Chu et al\. \(2011\)](https://arxiv.org/html/2609.06083#bib.bib3)\. Recent work[Foster et al\. \(2018\)](https://arxiv.org/html/2609.06083#bib.bib8);[Agarwal et al\. \(2012\)](https://arxiv.org/html/2609.06083#bib.bib7);[Foster and Rakhlin \(2020\)](https://arxiv.org/html/2609.06083#bib.bib9)achieved optimal regret bounds for general function classes with regression oracles\.

Notations\.Throughout the paper, we use the following notations\. For any positive integernn, we denote the set\{1,2,⋯,n\}\\\{1,2,\\cdots,n\\\}as\[n\]\[n\]\. The cardinality of a setAAis denoted by\|A\|\|A\|\. We use𝐈\{E\}\\mathbf\{I\}\_\{\\\{E\\\}\}to represent the indicator function of the eventEE\. Specifically,𝐈\{E\}\\mathbf\{I\}\_\{\\\{E\\\}\}takes the value11ifEEhappens, and00otherwise\. For norms, we utilize the notation∥⋅∥p\\\|\\cdot\\\|\_\{p\}where1≤p≤∞1\\leq p\\leq\\inftyto denote theℓp\\ell\_\{p\}norm\. Throughout the analysis, the notation𝒪~\\tilde\{\\mathcal\{O\}\}is employed to hide the dependence on absolute constants and logarithmic terms\. It allows us to focus on the dominant behavior of the quantities involved\.

## 2Basic Setting

Consider a company selling a single type of products over a time period ofTTrounds\. At the beginning of each roundtt, the firm observes a contextxtx\_\{t\}drawn i\.i\.d\. from an unknown distribution in a compact domain𝒳\\mathcal\{X\}\. Using this context, the company determines a pricingptp\_\{t\}and inventory order\-up\-to decisionyty\_\{t\}\. The demand is modeled asDt=λ⁡\(xt,pt\)\+ϵtD\_\{t\}=\\lambda\(x\_\{t\},p\_\{t\}\)\+\\epsilon\_\{t\}, whereλ⁡\(⋅,⋅\)\\lambda\(\\cdot,\\cdot\)is a deterministic function capturing the demand\-context\-price curve, andϵt\\epsilon\_\{t\}is an i\.i\.d\. noise random variable with mean00\. The distribution function and density function ofϵt\\epsilon\_\{t\}are denoted asFFandff, respectively\. The company has no prior knowledge of the functionλ⁡\(⋅,⋅\)\\lambda\(\\cdot,\\cdot\)or the distributionFF, and it must make its pricing and inventory decisions sequentially, relying solely on historical data to maximize theTTperiod total profit\.

Decision dynamics\.LetItI\_\{t\}denote the inventory level before replenishment at the start of the roundtt\. An admissible policy is represented by\{\(pt,yt\),t≥1\}\\\{\(p\_\{t\},y\_\{t\}\),t\\geq 1\\\}, wherept∈\[pm​i​n,pm​a​x\]p\_\{t\}\\in\[p\_\{min\},p\_\{max\}\]is the price, andyt≥Ity\_\{t\}\\geq I\_\{t\}is the order\-up\-to inventory level\. The decision\(pt,yt\)\(p\_\{t\},y\_\{t\}\)depends only on the observable contexts and historical data prior to the roundtt\.

Given any admissible policy\{\(pt,yt\),t≥1\}\\\{\(p\_\{t\},y\_\{t\}\),t\\geq 1\\\}, the sequence of events for each roundttis described as follows:

1. 1\.At the beginning of each roundtt, the firm observes the contextxt∈𝒳⊆ℝdx\_\{t\}\\in\\mathcal\{X\}\\subseteq\\mathbb\{R\}^\{d\}and the starting inventory levelItI\_\{t\}\.
2. 2\.The firm decides to place an order to bring the inventory level up toyt≥Ity\_\{t\}\\geq I\_\{t\}, and also sets the selling pricept∈\[pm​i​n,pm​a​x\]p\_\{t\}\\in\[p\_\{min\},p\_\{max\}\]\. We assume that newly ordered items arrive instantly, meaning that the ordering lead time is zero\.
3. 3\.The demandDtD\_\{t\}realizes and is satisfied up to the available inventory\. Any unsatisfied demand is lost and unobservable\. Therefore, the lost\-sales quantity\(Dt−yt\)\+\(D\_\{t\}\-y\_\{t\}\)^\{\+\}is not observable, and the firm only observes the sales quantity \(censored demand\)ot=min⁡\{Dt,yt\}o\_\{t\}=\\min\\\{D\_\{t\},y\_\{t\}\\\}, rather than the full realizedDtD\_\{t\}\.
4. 4\.At the end of roundtt, the firm incurs a profit of qt\\displaystyle q\_\{t\}=pt​min⁡\{Dt,yt\}−b​\(Dt−yt\)\+−h​\(yt−Dt\)\+\\displaystyle=p\_\{t\}\\min\\\{D\_\{t\},y\_\{t\}\\\}\-b\(D\_\{t\}\-y\_\{t\}\)^\{\+\}\-h\(y\_\{t\}\-D\_\{t\}\)^\{\+\}=pt​Dt−\(b\+pt\)​\(Dt−yt\)\+−h​\(yt−Dt\)\+\.\\displaystyle=p\_\{t\}D\_\{t\}\-\(b\+p\_\{t\}\)\(D\_\{t\}\-y\_\{t\}\)^\{\+\}\-h\(y\_\{t\}\-D\_\{t\}\)^\{\+\}\.\(1\)wherehhandbbare the per\-unit holding and lost\-sales penalty costs, respectively\.
5. 5\.Products may be perishable, with an unknown perishability rateρ∈\[0,1\]\\rho\\in\[0,1\], which impacts the inventory carried over to the next period\. The state transition is given by It\+1=max⁡\{ρ⁡\(yt−Dt\),0\}\.I\_\{t\+1\}=\\max\\\{\\rho\(\{y\}\_\{t\}\-D\_\{t\}\),0\\\}\.

Full information benchmark\.The firm’s goal is to maximize theTT\-period expected total profit∑t=1T𝔼⁡\[qt\]\\sum\_\{t=1\}^\{T\}\\mathbb\{E\}\[q\_\{t\}\], by constructing an admissible policy\{\(pt,yt\),t≥1\}\\\{\(p\_\{t\},y\_\{t\}\),t\\geq 1\\\}\. This is a fundamental model for joint pricing and inventory control, with potential applications in various realistic retail scenarios\.

Assuming the demand curveλ⁡\(⋅,⋅\)\\lambda\(\\cdot,\\cdot\)and noise distributionFFare known, we define the myopic pricing and replenishment policy as a benchmark\. This policy is represented as

\(pt∗,yt∗\)=arg⁡maxp,y⁡Q⁡\(xt,p,y\),\(p^\{\*\}\_\{t\},y^\{\*\}\_\{t\}\)=\\mathop\{\\arg\\max\}\_\{p,y\}Q\(x\_\{t\},p,y\),fort=1,…,Tt=1,\\dots,T, where

Q⁡\(x,p,y\)=\\displaystyle Q\(x,p,y\)=p​𝔼ϵ​\[min⁡\{λ⁡\(x,p\)\+ϵ,y\}\]\\displaystyle p\\mathbb\{E\}\_\{\\epsilon\}\\left\[\\min\\left\\\{\\lambda\(x,p\)\+\\epsilon,y\\right\\\}\\right\]−b​𝔼ϵ​\[\(λ⁡\(x,p\)\+ϵ−y\)\+\]\\displaystyle\\quad\-b\\mathbb\{E\}\_\{\\epsilon\}\\left\[\(\\lambda\(x,p\)\+\\epsilon\-y\)^\{\+\}\\right\]−h​𝔼ϵ​\[\(y−λ⁡\(x,p\)−ϵ\)\+\]\\displaystyle\\quad\-h\\mathbb\{E\}\_\{\\epsilon\}\\left\[\(y\-\\lambda\(x,p\)\-\\epsilon\)^\{\+\}\\right\]=\\displaystyle=p​λ​\(x,p\)−\(b\+p\)​𝔼ϵ​\[\(λ⁡\(x,p\)\+ϵ−y\)\+\]\\displaystyle p\\lambda\(x,p\)\-\(b\+p\)\\mathbb\{E\}\_\{\\epsilon\}\\left\[\(\\lambda\(x,p\)\+\\epsilon\-y\)^\{\+\}\\right\]−h​𝔼ϵ​\[\(y−λ⁡\(x,p\)−ϵ\)\+\]\.\\displaystyle\\quad\-h\\mathbb\{E\}\_\{\\epsilon\}\\left\[\(y\-\\lambda\(x,p\)\-\\epsilon\)^\{\+\}\\right\]\.
The regret at timettis defined as the loss in reward resulting from setting the priceptp\_\{t\}and the inventory levelyty\_\{t\}compared to the optimal price and inventory level\. The cumulative regret over the horizon ofTTperiods, denoted asℛ⁡\(T\)\\mathcal\{R\}\(T\), is given by the expression:

ℛ⁡\(T\)=∑t=1T\[Q⁡\(xt,pt∗,yt∗\)−Q⁡\(xt,pt,yt\)\]\.\\mathcal\{R\}\(T\)=\\sum\_\{t=1\}^\{T\}\\left\[Q\(x\_\{t\},p\_\{t\}^\{\*\},y\_\{t\}^\{\*\}\)\-Q\(x\_\{t\},p\_\{t\},y\_\{t\}\)\\right\]\.To assess performance, we consider the expected cumulative regret𝔼⁡\[ℛ⁡\(T\)\]\\mathbb\{E\}\[\\mathcal\{R\}\(T\)\], which accounts for the randomness in covariates and the potential randomness in the pricing policy\.

Assumptions\.We make the following assumptions, most of which are rather standard in the literature\.

First, we consider the linear additive model as the demand function, which means thatλ⁡\(x,p\)\\lambda\(x,p\)is a linear combination of the basis function\.

###### Assumption 1\.

There exists a set of basis functions\{λi​\(x,p\)\}i=1K\\\{\\lambda\_\{i\}\(x,p\)\\\}\_\{i=1\}^\{K\}and a unique unknown parameterθ∗=\(θ1∗,θ2∗,⋯,θK∗\)⊤\\theta^\{\*\}=\(\\theta\_\{1\}^\{\*\},\\theta\_\{2\}^\{\*\},\\cdots,\\theta\_\{K\}^\{\*\}\)^\{\\top\}such thatλ⁡\(x,p\)=∑i=1Kθi∗​λi​\(x,p\)\.\\lambda\(x,p\)=\\sum\_\{i=1\}^\{K\}\\theta\_\{i\}^\{\*\}\\lambda\_\{i\}\(x,p\)\.

This assumption includes many commonly used demand functions, such as\([Cohen et al\., 2020](https://arxiv.org/html/2609.06083#bib.bib23)\)\. Next, we introduce some basic assumptions on the revenue function\.

###### Assumption 2\.

The revenue functionQ⁡\(x,p,y\)Q\(x,p,y\)is continuously differentiable with respect toxx,ppandyy\.

The differentiability ofQQis a common assumption in the literature, as seen in works such as\([Chen et al\., 2019](https://arxiv.org/html/2609.06083#bib.bib34);[Qin et al\., 2022](https://arxiv.org/html/2609.06083#bib.bib33);[Chen et al\., 2024](https://arxiv.org/html/2609.06083#bib.bib35)\)\. For the dynamic pricing component, this assumption is particularly relevant, as it encompasses many practical demand functions, including those discussed in\([Cohen et al\., 2020](https://arxiv.org/html/2609.06083#bib.bib23);[Wang et al\., 2021b](https://arxiv.org/html/2609.06083#bib.bib21);[Wang et al\., 2021a](https://arxiv.org/html/2609.06083#bib.bib20)\)\. In inventory control, several papers, such as\([Shi et al\., 2016](https://arxiv.org/html/2609.06083#bib.bib29);[Ding et al\., 2021](https://arxiv.org/html/2609.06083#bib.bib26)\), adopt the newsvendor objective function, which is differentiable with respect to the inventory levelyy\. These properties are crucial for deriving closed\-form solutions for the optimal inventory levely∗​\(x,p\)y^\{\*\}\(x,p\)given the contextxx\.

###### Proposition 1\.

The optimal inventory level enjoys the following structure:y∗​\(x,p\)=λ⁡\(x,p\)\+z∗​\(p\)y^\{\*\}\(x,p\)=\\lambda\(x,p\)\+z^\{\*\}\(p\), wherez∗​\(p\)=F−1​\(b\+pb\+p\+h\)z^\{\*\}\(p\)=F^\{\-1\}\\left\(\\frac\{b\+p\}\{b\+p\+h\}\\right\)\.

A context\-free version of this result has appeared in\([Ding et al\., 2021](https://arxiv.org/html/2609.06083#bib.bib26);[Chen et al\., 2021](https://arxiv.org/html/2609.06083#bib.bib36);[Chen et al\., 2024](https://arxiv.org/html/2609.06083#bib.bib35)\)\. To extend this to contextual dynamic pricing, we follow the process outlined below\. To obtain the optimal inventory level\. We begin by noting that∂Q∂y=b\+p−\(b\+p\+h\)​F​\(y−λ⁡\(x,p\)\)\.\\frac\{\\partial Q\}\{\\partial y\}=b\+p\-\(b\+p\+h\)F\(y\-\\lambda\(x,p\)\)\.The optimal order\-up\-to level thus is given byy∗​\(x,p\)=λ⁡\(x,p\)\+z∗​\(p\),y^\{\*\}\(x,p\)=\\lambda\(x,p\)\+z^\{\*\}\(p\),wherez∗​\(p\)z^\{\*\}\(p\)satisfiesz∗​\(p\)=F−1​\(b\+pb\+p\+h\)\.z^\{\*\}\(p\)=F^\{\-1\}\\left\(\\frac\{b\+p\}\{b\+p\+h\}\\right\)\.Finally, we assume some basic properties of the distribution noise, which is rather standard in the literature\.

###### Assumption 3\.

The random errorϵt\\epsilon\_\{t\}is bounded on the interval\[ϵ¯,ϵ¯\]\[\\underline\{\\epsilon\},\\overline\{\\epsilon\}\], and there exist positive constantsκ1\\kappa\_\{1\}andκ2\\kappa\_\{2\}such that for anyp∈\[pm​i​n,pm​a​x\]p\\in\[p\_\{min\},p\_\{max\}\]andϵ0∈\[F−1​\(b\+pb\+p\+h\)−κ1,F−1​\(b\+pb\+p\+h\)\+κ1\]\\epsilon\_\{0\}\\in\\left\[F^\{\-1\}\\left\(\\frac\{b\+p\}\{b\+p\+h\}\\right\)\-\\kappa\_\{1\},F^\{\-1\}\\left\(\\frac\{b\+p\}\{b\+p\+h\}\\right\)\+\\kappa\_\{1\}\\right\], we havef⁡\(ϵ0\)≥κ2\.f\(\\epsilon\_\{0\}\)\\geq\\kappa\_\{2\}\.

The first assumption onϵt\\epsilon\_\{t\}ensures the boundedness of the demand, which is a mild and common assumption in real\-world applications\([Chen et al\., 2024](https://arxiv.org/html/2609.06083#bib.bib35)\)\. It is important to note that we do not require the decision maker to know the exact bounds of the noise or the demand function\. The second assumption is satisfied by many continuous random variables, including uniform and truncated Gaussian distributions\. And the third assumption needs the demand to be larger than a certain range of the noise ensuring robustness\. In the following sections, we propose our algorithm and further discuss the approximation error of the demand function\.

## 3Algorithm Design

We present our algorithmic design and summarize the key challenges in this setting\.

The first challenge is*censored demand*: lost sales are unobservable, so we cannot directly estimate the true demand parameterθ∗\\theta^\{\*\}\. To mitigate this, we set the inventory level to scale withlog⁡T\\log T, improving demand observability without knowing the true demand upper bound\.

The second challenge is the*period\-varying optimal inventory level*due to changing contexts and perishable inventory constraints, which may make the ideal up\-to\-order level infeasible\. We address this using a queue\-based approach to control deviations from the recommended target inventory\.

Recall from[proposition1](https://arxiv.org/html/2609.06083#Thmproposition1)that the optimal inventory level isy∗​\(x,p\)=λ⁡\(x,p\)\+z∗​\(p\)y^\{\*\}\(x,p\)=\\lambda\(x,p\)\+z^\{\*\}\(p\), wherez∗​\(p\)z^\{\*\}\(p\)is the safety stock level\. The corresponding revenue function is

G⁡\(x,p\)=maxy⁡Q⁡\(x,p,y\),G\(x,p\)=\\max\_\{y\}Q\(x,p,y\),wherexxis the context andppis the price\. The decision\-maker estimatesG⁡\(x,p\)G\(x,p\)to determine the optimal price and then computes the corresponding target inventory\.

Algorithm 1Joint Pricing and Inventory Control Algorithm with Censored Demand1:the time horizon

TT, the length of the exploration phase

T0T\_\{0\}, cost parameter

hhand

bb, basis function

ϕ⁡\(x,p\)=\(λ1​\(x,p\),λ2​\(x,p\),⋯,λK​\(x,p\)\)⊤\\phi\(x,p\)=\(\\lambda\_\{1\}\(x,p\),\\lambda\_\{2\}\(x,p\),\\cdots,\\lambda\_\{K\}\(x,p\)\)^\{\\top\}
2:forround

t=1,2,3,⋯,T0t=1,2,3,\\cdots,T\_\{0\}do

3:Observe a context

xtx\_\{t\}and current inventory level

ItI\_\{t\}
4:Uniformly choose

pt∈\[pm​i​n,pm​a​x\]p\_\{t\}\\in\[p\_\{min\},p\_\{max\}\]
5:Set the inventory level

yt=log⁡Ty\_\{t\}=\\log T
6:Observe the sale

ot=min⁡\{Dt,yt\}o\_\{t\}=\\min\\\{D\_\{t\},y\_\{t\}\\\}
7:Estimate the parameter

θ∗\\theta^\{\*\}by

θ^=arg⁡minθ⁡∑t=1T0\(ot−θ⊤​ϕ​\(xt,pt\)\)2\\hat\{\\theta\}=\\mathop\{\\arg\\min\}\_\{\\theta\}\\sum\_\{t=1\}^\{T\_\{0\}\}\(o\_\{t\}\-\\theta^\{\\top\}\\phi\(x\_\{t\},p\_\{t\}\)\)^\{2\}
8:Collect the samples for residuals

ηt=ot−θ^⊤​ϕ​\(xt,pt\)\\eta\_\{t\}=o\_\{t\}\-\\hat\{\\theta\}^\{\\top\}\\phi\(x\_\{t\},p\_\{t\}\)with

t∈\[T0\]t\\in\[T\_\{0\}\]
9:Estimate the distribution

FFas

F^\(ϵ\)=1T0∑t=1T0𝐈\{ηt≤ϵ\}\\hat\{F\}\(\\epsilon\)=\\frac\{1\}\{T\_\{0\}\}\\sum\_\{t=1\}^\{T\_\{0\}\}\\mathbf\{I\}\_\{\\\{\\eta\_\{t\}\\leq\\epsilon\\\}\}
10:Estimate the quantity

z⁡\(p\)z\(p\)as

z^​\(p\)=inf\{u:F^​\(u\)≥b\+pb\+p\+h\}\\hat\{z\}\(p\)=\\inf\\left\\\{u:\\hat\{F\}\(u\)\\geq\\frac\{b\+p\}\{b\+p\+h\}\\right\\\}
11:Construct the function

G^​\(x,p\)\\hat\{G\}\(x,p\)using:

G^​\(x,p\)=p​θ^⊤​ϕ​\(x,p\)\\displaystyle\\hat\{G\}\(x,p\)=p\\hat\{\\theta\}^\{\\top\}\\phi\(x,p\)−b\+pT0∑t=1T0\[ηt−z^\(p\)\]\+−hT0∑t=1T0\[z^\(p\)−ηt\]\+\\displaystyle\-\\frac\{b\+p\}\{T\_\{0\}\}\\sum\_\{t=1\}^\{T\_\{0\}\}\[\\eta\_\{t\}\-\\hat\{z\}\(p\)\]^\{\+\}\-\\frac\{h\}\{T\_\{0\}\}\\sum\_\{t=1\}^\{T\_\{0\}\}\[\\hat\{z\}\(p\)\-\\eta\_\{t\}\]^\{\+\}
12:forround

t=T0\+1,T0\+2,⋯,Tt=T\_\{0\}\+1,T\_\{0\}\+2,\\cdots,Tdo

13:Observe a context

xtx\_\{t\}and current inventory level

ItI\_\{t\}
14:Set

pt∈arg⁡max⁡G^​\(xt,p\)p\_\{t\}\\in\\mathop\{\\arg\\max\}\\hat\{G\}\(x\_\{t\},p\)
15:Set the inventory level

yt=\(θ^⊤​ϕ​\(xt,pt\)\+z^​\(pt\)\)∨Ity\_\{t\}=\(\\hat\{\\theta\}^\{\\top\}\\phi\(x\_\{t\},p\_\{t\}\)\+\\hat\{z\}\(\{p\_\{t\}\}\)\)\\vee I\_\{t\}
16:Observe the sale

ot=min⁡\{Dt,yt\}o\_\{t\}=\\min\\\{D\_\{t\},y\_\{t\}\\\}

Our method splits the time horizon into two phases: exploration and commitment\. During exploration, we sample prices uniformly at random and set the inventory level high enough to fully observe demand, avoiding censoring and ensuring reliable feedback\. The inventory level is tuned to balance learning accuracy and regret\. We estimate the demand function using least\-squares, leveraging the linear structure of the basis functions\. Least\-squares is chosen because its estimation error is negligible compared to the overall regret\. Next, we estimate the total cost \(including lost\-sales and holding costs\) by accurately estimating the optimal inventory functionz⁡\(p\)z\(p\)and the residual error\. We approximate the demand distribution empirically and computez⁡\(p\)z\(p\)via the inverse CDF\. With these estimates, we construct an empirical revenue function\. Overall, our algorithm is simple, efficient, and practical for joint pricing and inventory control under censored demand and perishable constraints\.

## 4Regret Analysis

The main result of this section is that the proposed algorithm converges to the full information benchmark at a rate of𝒪⁡\(K​T​log⁡T\)\\mathcal\{O\}\(K\\sqrt\{T\}\\log T\)under the strongly concave assumption, and at a rate of𝒪⁡\(K2/3​T2/3​\(log⁡T\)1/2\)\\mathcal\{O\}\(K^\{2/3\}T^\{2/3\}\(\\log T\)^\{1/2\}\)without this assumption\. We provide a detailed regret analysis in the following two subsections\.

### 4\.1Strongly Concave Revenue Function

In this subsection, we present several additional assumptions that are standard in the literature \(e\.g\.,[Chen et al\. \(2021\)](https://arxiv.org/html/2609.06083#bib.bib36),[Chen et al\. \(2024\)](https://arxiv.org/html/2609.06083#bib.bib35),[Bensoussan et al\. \(2019\)](https://arxiv.org/html/2609.06083#bib.bib32)\)\. These assumptions are crucial for establishing the theoretical framework and ensuring the robustness of our model\.

First, we assume that the full information optimal pricept∗p\_\{t\}^\{\*\}lies in the interior of the price interval\[pm​i​n,pm​a​x\]\[p\_\{min\},p\_\{max\}\]for any timett\. This assumption is necessary to ensure that optimal pricing decisions can be made without hitting the boundaries of the price range, which could limit the effectiveness of pricing strategies\.

###### Assumption 4\.

The full information optimal pricept∗p\_\{t\}^\{\*\}lies in the interior of\[pm​i​n,pm​a​x\]\[p\_\{min\},p\_\{max\}\]for anytt\.

Building on this, we further assume that the functionG⁡\(x,p\)G\(x,p\), which represents the relationship between the state variablexxand pricepp, is strongly concave inppfor any givenxx\. This concavity condition is crucial, as it guarantees that the revenue generated from pricing is maximized, leading to unique optimal pricing strategies\. Specifically, there exists a parameterκ3\>0\\kappa\_\{3\}\>0such that the second derivative ofGGwith respect toppis always less than or equal to−κ3\-\\kappa\_\{3\}\.

###### Assumption 5\.

G⁡\(x,p\)G\(x,p\)is strongly concave inppfor anyxx\. In other words, there exists a parameterκ3\>0\\kappa\_\{3\}\>0such that∂p2G⁡\(x,p\)≤−κ3\\partial^\{2\}\_\{p\}G\(x,p\)\\leq\-\\kappa\_\{3\}for anyxx\.

Additionally, we require that the revenue functionQ⁡\(x,p,y\)Q\(x,p,y\)is twice continuously differentiable with respect to the variablesxx,pp, andyy\. This differentiability condition facilitates the application of optimization techniques and allows for smooth adjustments in response to changes in the underlying variables, ensuring a well\-behaved revenue landscape\.

###### Assumption 6\.

The revenue functionQ⁡\(x,p,y\)Q\(x,p,y\)is twice continuously differentiable with respect topp, andyyand continuous with respect toxx\.

Together, these assumptions create a robust foundation for analyzing the optimal pricing strategies within our model, ensuring both theoretical validity and practical applicability\.

Now we are ready to give a sketch of proof of the[Theorem4\.1](https://arxiv.org/html/2609.06083#S4.Thmmythm1): under strongly concave assumption\. To give an upper bound on the regret, we first decompose the regret into the following terms:

ℛ⁡\(T\)\\displaystyle\\mathcal\{R\}\(T\)≤𝔼⁡\[∑t=1T0\(Q⁡\(xt,pt∗,yt∗\)−Q⁡\(xt,pt,yt\)\)\]\\displaystyle\\leq\\mathbb\{E\}\\left\[\\sum\_\{t=1\}^\{T\_\{0\}\}\(Q\(x\_\{t\},p\_\{t\}^\{\*\},y\_\{t\}^\{\*\}\)\-Q\(x\_\{t\},p\_\{t\},y\_\{t\}\)\)\\right\]\+𝔼⁡\[∑t=T0\+1T\(Q⁡\(xt,pt∗,yt∗\)−Q⁡\(xt,pt,y^t\)\)\]\\displaystyle\\quad\+\\mathbb\{E\}\\left\[\\sum\_\{t=T\_\{0\}\+1\}^\{T\}\(Q\(x\_\{t\},p\_\{t\}^\{\*\},y\_\{t\}^\{\*\}\)\-Q\(x\_\{t\},\{p\}\_\{t\},\\hat\{y\}\_\{t\}\)\)\\right\]\+𝔼⁡\[∑t=T0\+1T\|Q⁡\(xt,pt,y^t\)−Q⁡\(xt,pt,yt\)\|\]\.\\displaystyle\\quad\+\\mathbb\{E\}\\left\[\\sum\_\{t=T\_\{0\}\+1\}^\{T\}\|Q\(x\_\{t\},\{p\}\_\{t\},\\hat\{y\}\_\{t\}\)\-Q\(x\_\{t\},p\_\{t\},y\_\{t\}\)\|\\right\]\.\(2\)
wherey^t=θ^T​ϕ​\(xt,pt\)\+z^​\(pt\)\\hat\{y\}\_\{t\}=\\hat\{\\theta\}^\{T\}\\phi\(x\_\{t\},p\_\{t\}\)\+\\hat\{z\}\(p\_\{t\}\)is the recommended inventory level andyty\_\{t\}is the truly implemented inventory level\.

Now we consider two separate situations regarding the demand model: the first is the linear additive model, while the second is the linear additive model with an approximation bias\. We provide upper bounds for the four terms respectively under each demand model\.

###### Theorem 4\.1\.

Under[Assumption3](https://arxiv.org/html/2609.06083#Thmmyasp3),[Assumption1](https://arxiv.org/html/2609.06083#Thmmyasp1),[Assumption4](https://arxiv.org/html/2609.06083#Thmmyasp4),[Assumption5](https://arxiv.org/html/2609.06083#Thmmyasp5),[Assumption6](https://arxiv.org/html/2609.06083#Thmmyasp6), there exists a constantC^\\hat\{C\}such that the regret bound of our algorithm isℛ⁡\(T\)≤C^​K​T12​log⁡T\.\\mathcal\{R\}\(T\)\\leq\\hat\{C\}KT^\{\\frac\{1\}\{2\}\}\\log T\.

In[Equation2](https://arxiv.org/html/2609.06083#S4.E2), the first term represents the regret during the exploration phase, which increases linearly with the total duration of the exploration\. In the exploitation phase, we decompose the regret into two terms, wherey^\\hat\{y\}denotes the recommended inventory level andyty\_\{t\}denotes the actual implemented inventory level\. This decomposition is based on the fact thatyt=max​\{It,y^t\}y\_\{t\}=\\text\{max\}\\\{I\_\{t\},\\hat\{y\}\_\{t\}\\\}\.

Using Taylor expansion, the second term in[Equation2](https://arxiv.org/html/2609.06083#S4.E2)can be bounded as follows:

Q⁡\(xt,pt∗,yt∗\)−Q⁡\(xt,pt,y^t\)≤C^2​‖\(pt,y^t\)−\(pt∗,yt∗\)‖22\.Q\(x\_\{t\},p\_\{t\}^\{\*\},y\_\{t\}^\{\*\}\)\-Q\(x\_\{t\},\{p\}\_\{t\},\\hat\{y\}\_\{t\}\)\\leq\\hat\{C\}\_\{2\}\\\|\(\{p\}\_\{t\},\\hat\{y\}\_\{t\}\)\-\(p\_\{t\}^\{\*\},y\_\{t\}^\{\*\}\)\\\|\_\{2\}^\{2\}\.\(3\)
What remains is to bound the difference between the optimal price decision and the optimal inventory decision using the recommended strategy\. By utilizing the strong concavity of the functionGG, we turn the difference of the price into the maximum of the differential of the functionGG:

\|\|pt∗−pt\|≤1κ3\(maxp∈\[pm​i​n,pm​a​x\]\|∂pG−∂pG^\|\(xt,p\)\)\.\|\|p^\{\*\}\_\{t\}\-\{p\}\_\{t\}\|\\leq\\frac\{1\}\{\\kappa\_\{3\}\}\\left\(\\max\_\{p\\in\[p\_\{min\},p\_\{max\}\]\}\|\\partial\_\{p\}G\-\\partial\_\{p\}\\hat\{G\}\|\(x\_\{t\},p\)\\right\)\.\(4\)
Next, we need to bound the difference between the first derivative of the true functionGGand the empirical estimateG^\\hat\{G\}, which can be bounded by the estimation error\.

Similarly, the difference in inventory levelsyt∗−y^ty\_\{t\}^\{\*\}\-\\hat\{y\}\_\{t\}can be bounded using the definition and the bound of the price decisionpt∗−ptp^\{\*\}\_\{t\}\-\{p\}\_\{t\}\. Thus, we have the following theorem:

###### Theorem 4\.2\.

In the exploitation phase, there exists a constantC~\\tilde\{C\}such that

ℙ⁡\(‖\(pt∗,yt∗\)−\(pt,y^t\)‖2CLOSE\\displaystyle\\mathbb\{P\}\\Bigl\(\\\|\(p\_\{t\}^\{\*\},y\_\{t\}^\{\*\}\)\-\(\{p\}\_\{t\},\\hat\{y\}\_\{t\}\)\\\|\_\{2\}OPEN≤C~​K​T0−12​\(log⁡T0\)12\)\\displaystyle\\leq\\tilde\{C\}KT\_\{0\}^\{\-\\frac\{1\}\{2\}\}\(\\log T\_\{0\}\)^\{\\frac\{1\}\{2\}\}\\Bigr\)\(5\)\>1−18​T0−4\.\\displaystyle\>1\-18T\_\{0\}^\{\-4\}\.

Finally, we need to provide an upper bound for the third term in[Equation2](https://arxiv.org/html/2609.06083#S4.E2)\.

In the caseIt≤y^tI\_\{t\}\\leq\\hat\{y\}\_\{t\}, we haveyt=y^ty\_\{t\}=\\hat\{y\}\_\{t\}, which results in zero regret\. Below, we only consider the case whereIt\>y^tI\_\{t\}\>\\hat\{y\}\_\{t\}:

\|Q⁡\(xt,pt,y^t\)−Q⁡\(xt,pt,yt\)\|\\displaystyle\|Q\(x\_\{t\},\{p\}\_\{t\},\\hat\{y\}\_\{t\}\)\-Q\(x\_\{t\},\{p\}\_\{t\},y\_\{t\}\)\|=\\displaystyle=\|h𝔼\[\(y^t−Dt\)\+\]\+\(b\+pt\)𝔼\[\(Dt−y^t\)\+\]\\displaystyle\|h\\mathbb\{E\}\[\(\\hat\{y\}\_\{t\}\-D\_\{t\}\)^\{\+\}\]\+\(b\+\{p\}\_\{t\}\)\\mathbb\{E\}\[\(D\_\{t\}\-\\hat\{y\}\_\{t\}\)^\{\+\}\]−h𝔼\[\(yt−Dt\)\+\]−\(b\+pt\)𝔼\[\(Dt−yt\)\+\]\|\\displaystyle\\quad\-h\\mathbb\{E\}\[\(y\_\{t\}\-D\_\{t\}\)^\{\+\}\]\-\(b\+\{p\}\_\{t\}\)\\mathbb\{E\}\[\(D\_\{t\}\-y\_\{t\}\)^\{\+\}\]\|≤\\displaystyle\\leqmax⁡\{h,b\+pm​a​x\}​\(yt−y^t\)\.\\displaystyle\\max\\\{h,b\+p\_\{max\}\\\}\(y\_\{t\}\-\\hat\{y\}\_\{t\}\)\.
Now we need to provide an upper bound on the differenceyt−y^ty\_\{t\}\-\\hat\{y\}\_\{t\}\. To do this, we establish the connection between the contextual inventory level and the waiting time process\. Unlike the method in[Chen et al\. \(2021\)](https://arxiv.org/html/2609.06083#bib.bib36), which considers a non\-contextual setting with the same optimal decision across all periods, we need to bound the difference in inventory levelsyt−y^ty\_\{t\}\-\\hat\{y\}\_\{t\}dynamically\.

To achieve this, we define the following two stochastic processes:

At\+1=\[At−Jt\]\+andBt\+1=\[Bt\+ρqt−Jt\]\+A\_\{t\+1\}=\[A\_\{t\}\-J\_\{t\}\]^\{\+\}\\quad\\text\{and\}\\quad B\_\{t\+1\}=\[B\_\{t\}\+\\frac\{\\rho\_\{q\}\}\{\\sqrt\{t\}\}\-J\_\{t\}\]^\{\+\}\(6\)where

Jt=ϵt\+ϵ¯−ϵ¯\+θ^T​ϕ​\(xt\+1,pt\+1\)−λm​a​x​C1​K​\(log⁡T0\)12T012\.J\_\{t\}=\\epsilon\_\{t\}\+\\underline\{\\epsilon\}\-\\overline\{\\epsilon\}\+\\hat\{\\theta\}^\{T\}\\phi\(x\_\{t\+1\},p\_\{t\+1\}\)\-\\lambda\_\{max\}C\_\{1\}K\\frac\{\(\\log T\_\{0\}\)^\{\\frac\{1\}\{2\}\}\}\{T\_\{0\}^\{\\frac\{1\}\{2\}\}\}\.\(7\)We have:

yt−y^t≤At≤Bty\_\{t\}\-\\hat\{y\}\_\{t\}\\leq A\_\{t\}\\leq B\_\{t\}fort≥T0\+1t\\geq T\_\{0\}\+1\.

In the above stochastic processes,AtA\_\{t\}serves as an intermediate process\. Additionally,BtB\_\{t\}can be interpreted as the waiting time of thett\-th customer in the queuing system, whereJtJ\_\{t\}is the interarrival time between thettth and\(t\+1\)\(t\+1\)\-th customer, andρq/t\\rho\_\{q\}/\\sqrt\{t\}is the service time for thettth customer\.

This idea is similar to the method presented in[Huh and Rusmevichientong \(2009\)](https://arxiv.org/html/2609.06083#bib.bib37), which established a relationship between the amount of inventory exceeding the target level and the waiting time process in a GI/D/1 queue\. However, they do not consider the impact of context and only need to bound the gradients to establish such a connection\. In our analysis, since there are no gradients in the exploitation phase, we only need to exploit the estimation error during the exploration phase and the variability of features\. Since this only involves the case whereIt\>y^tI\_\{t\}\>\\hat\{y\}\_\{t\}, this method is also robust to perishable inventory levels\.

###### Proposition 4\.1\.

For anyt≥T0\+1t\\geq T\_\{0\}\+1,yt−y^t≤Bty\_\{t\}\-\\hat\{y\}\_\{t\}\\leq B\_\{t\}with probability at least1−2T041\-\\frac\{2\}\{T\_\{0\}^\{4\}\}, whereBtB\_\{t\}is defined in[Equation6](https://arxiv.org/html/2609.06083#S4.E6)\.

After providing a bound for the stochastic process defined above, we can upper bound the regret between the actual implemented inventory level and the recommended inventory level with the following proposition:

###### Proposition 4\.2\.

There exists a constantC4C\_\{4\}such that the regret from not achieving inventory targets during the commitment phase is bounded by𝔼⁡\[∑t=T0\+1T\|Q⁡\(xt,pt,y^t\)−Q⁡\(xt,pt,yt\)\|\]≤C4​T12\\mathbb\{E\}\\left\[\\sum\_\{t=T\_\{0\}\+1\}^\{T\}\|Q\(x\_\{t\},\{p\}\_\{t\},\\hat\{y\}\_\{t\}\)\-Q\(x\_\{t\},\{p\}\_\{t\},y\_\{t\}\)\|\\right\]\\leq C\_\{4\}T^\{\\frac\{1\}\{2\}\}which holds with probability at least1−2T04\.1\-\\frac\{2\}\{T\_\{0\}^\{4\}\}\.

Summing up all the inequalities above, we can obtain the bound mentioned in[Theorem4\.1](https://arxiv.org/html/2609.06083#S4.Thmmythm1)\.

Figure 1:Relative regret over time horizonTTfor different values ofhhandbb\.Figure 2:Relative regret over time horizonTTfor different values ofhhandbb\(log\-log scale\)\.
### 4\.2Non\-concave Revenue Function

The non\-concave revenue function has been studied in[Chen et al\. \(2024\)](https://arxiv.org/html/2609.06083#bib.bib35)in a non\-contextual setting\. They show that the penalty term related to inventory levels can destroy the concavity of the objective function, making the lost\-sales model significantly more difficult to analyze compared to the backlogged case\. To address this challenge,[Chen et al\. \(2024\)](https://arxiv.org/html/2609.06083#bib.bib35)propose a search\-based method for identifying the optimal price under the assumption that the revenue function is twice differentiable, withΘ~​\(T3/5\)\\tilde\{\\Theta\}\(T^\{3/5\}\)minimax optimal regret bounds\.

We extend their work in two key directions: we relax the assumption of second\-order differentiability and generalize the problem to a contextual setting\. In our case, the optimal prices vary over time due to changing contexts, and the revenue functionGGmay lack smoothness, making the approach in[Chen et al\. \(2024\)](https://arxiv.org/html/2609.06083#bib.bib35)inapplicable\. Without assuming strong concavity ofGG, we are unable to derive a precise upper bound on the difference between successive price decisions, as was done in the previous section\. Instead, we adopt a greedy strategy to control this difference, which leads to a regret bound that is worse than the𝒪⁡\(K​T​log⁡T\)\\mathcal\{O\}\(K\\sqrt\{T\}\\log T\)rate achieved under more favorable assumptions\.

###### Theorem 4\.3\.

With[Assumption2](https://arxiv.org/html/2609.06083#Thmmyasp2)and[Assumption1](https://arxiv.org/html/2609.06083#Thmmyasp1), there exists a constantC~\\tilde\{C\}such that the regret bound of our algorithm isℛ⁡\(T\)≤C~​K23​T23​\(log⁡T\)12\.\\mathcal\{R\}\(T\)\\leq\\tilde\{C\}K^\{\\frac\{2\}\{3\}\}T^\{\\frac\{2\}\{3\}\}\(\\log T\)^\{\\frac\{1\}\{2\}\}\.

To prove[Theorem4\.3](https://arxiv.org/html/2609.06083#S4.Thmmythm3), we decompose the regret into two parts:

1. 1\.Bounding the revenue gap:G⁡\(xt,pt∗\)−G⁡\(xt,pt\)G\(x\_\{t\},p\_\{t\}^\{\*\}\)\-G\(x\_\{t\},p\_\{t\}\)
2. 2\.Bounding the inventory regret:Q⁡\(xt,pt,yt∗\)−Q⁡\(xt,pt,y^t\)Q\(x\_\{t\},p\_\{t\},y\_\{t\}^\{\*\}\)\-Q\(x\_\{t\},p\_\{t\},\\hat\{y\}\_\{t\}\)

For the first part, we use a decomposition ofG⁡\(xt,pt∗\)−G⁡\(xt,pt\)G\(x\_\{t\},p\_\{t\}^\{\*\}\)\-G\(x\_\{t\},p\_\{t\}\)and apply the estimation error bound forG^​\(xt,p\)\\hat\{G\}\(x\_\{t\},p\), leading to:

G⁡\(xt,pt∗\)−G⁡\(xt,pt\)≤2​‖G⁡\(x,p\)−G^​\(x,p\)‖∞≤C⋅KT0−1/2\(logT0\)2\.\\begin\{split\}G\(x\_\{t\},p\_\{t\}^\{\*\}\)\-G\(x\_\{t\},p\_\{t\}\)&\\leq 2\\\|G\(x,p\)\-\\hat\{G\}\(x,p\)\\\|\_\{\\infty\}\\\\ &\\leq C\\cdot KT\_\{0\}^\{\-1/2\}\(\\log T\_\{0\}\)^\{2\}\.\\end\{split\}\(8\)For the second part, we use an existing result in[Proposition4\.2](https://arxiv.org/html/2609.06083#S4.Thmmyprop2)to bound the inventory regret term\. Finally, combining these bounds results yields the desired results\.

###### Theorem 4\.4\.

For any policyπ\\pi, there exists a non\-contextual problem instance whose revenue function ismm\-th differentiable, such thatℛ⁡\(T\)≥Ω⁡\(T\(m\+1\)/\(2​m\+1\)\)\\mathcal\{R\}\(T\)\\geq\\Omega\(T^\{\(m\+1\)/\(2m\+1\)\}\)for sufficiently largeTT\.

The instance construction is motivated by\([Wang et al\., 2021b](https://arxiv.org/html/2609.06083#bib.bib21)\)\. The construction of the noise distribution starts with a smooth functionu⁡\(x\)u\(x\), which is normalized to define a functionB⁡\(x\)B\(x\), leading to a density functionf⁡\(x\)f\(x\)supported on\[−1,1\]\[\-1,1\]with mean 0\. Using this, a demand functionλj​\(p\)\\lambda\_\{j\}\(p\)is defined, which perturbs the base revenue functionr0​\(p\)r\_\{0\}\(p\)using a small parameterη\\etaand a bump function\. The reward functionQj​\(p,y\)Q\_\{j\}\(p,y\)is shown to bemm\-th differentiable, with bounds on the Hellinger distance between different environments\. A lower bound on expected regret is derived using the pigeonhole principle, showing that for some intervali∗i^\{\*\}, the regret for any policy is at leastΩ⁡\(Tm\+12​m\+1\)\\Omega\(T^\{\\frac\{m\+1\}\{2m\+1\}\}\)\.

Combing the above upper and lower bounds, we conclude thatΘ⁡\(T23\)\\Theta\(T^\{\\frac\{2\}\{3\}\}\)is an achievable minimax optimal regret bounds only under the differentiabilty assumption\.

## 5Numerical Experiments

We conduct a numerical study to evaluate the empirical performance of our algorithms\.

### 5\.1Test on parametershhandbb

The experiments are based on a linear demand model:

dt=α\+xt⊤​β\+ϵt,d\_\{t\}=\\alpha\+x\_\{t\}^\{\\top\}\\beta\+\\epsilon\_\{t\},where the parameter vector\(α,β\)∈ℝ4\(\\alpha,\\beta\)\\in\\mathbb\{R\}^\{4\}is sampled from a standard normal distribution and normalized to unit norm\. The noise termϵt\\epsilon\_\{t\}is drawn independently from the uniform distribution over\[−1,1\]\[\-1,1\]\. Prices are constrained to the rangepmin=0\.1p\_\{\\min\}=0\.1andpmax=2p\_\{\\max\}=2\.

For each experimental setting, we perform 50 independent runs\. At each round, context vectorsxtx\_\{t\}are drawn from a standard normal distribution and normalized to unit length\. We evaluate performance using relative regret, defined asℛ⁡\(T\)T\\frac\{\\mathcal\{R\}\(T\)\}\{T\}, whereℛ⁡\(T\)\\mathcal\{R\}\(T\)is the cumulative regret up to timeTT\. Results are reported for time horizonsT∈\{100,200,500,1000,2000,5000\}T\\in\\\{100,200,500,1000,2000,5000\\\}, including the mean and standard deviation across trials\.

The numerical experiments in[Figure1](https://arxiv.org/html/2609.06083#S4.F1)show that relative regret decreases as the time horizonTTincreases\. This trend holds across all three subplots with different parameter settings, indicating that the algorithm improves over time by leveraging more data for better decision\-making and a balanced exploration\-exploitation trade\-off\.

The error bars, representing standard deviation across trials, highlight increased variability at smallerTT, due to insufficient data for reliable convergence\. AsTTgrows, the decreasing variability reflects improved stability\. In[Figure2](https://arxiv.org/html/2609.06083#S4.F2), the log\-log plot shows a linear trend, suggesting a power\-law relationship between regret andTT, which supports the algorithm’s robustness across different parameter settings\.

### 5\.2Test on dimensionKK

We investigate how the algorithm’s performance depends on the dimension of the basis functions, denoted byKK\. We varyKKfrom44to1212to assess its impact on regret\. To ensure a fair comparison across different values ofKK, we fix the total time horizon atT=1000T=1000, and set the cost parameters ash=1h=1andb=1b=1\. All other experimental settings remain consistent with the base configuration described earlier\.

[Figure3](https://arxiv.org/html/2609.06083#S5.F3)shows the relative regret across differentKKvalues\. In summary, our results highlight key insights into the trade\-offs involved in contextual joint pricing and inventory control:

Figure 3:Relative regret across differentKKvalues with standard deviation error bars\.- •Regret Increases with Model Complexity:The relative regret increases consistently withKK, indicating that higher\-dimensional models are more difficult to learn under a fixed time horizon\. Intuitively, increasingKKenhances the expressive power of the model but requires more exploration\.
- •Stable Performance Across Trials:Despite the increased complexity, the algorithm maintains controlled variance, with standard deviations remaining moderate across different values ofKK, indicating robust and reliable performance\.

## 6Conclusion

This paper studies the joint pricing and inventory control problem under a linear additive model with contextual information and censored demand observations\. We address the critical gap where existing approaches fail to account for both contextual factors and the difficulty of lost sales that obscure true demand information\.

Our main contributions include: \(1\) a context\-aware framework for joint pricing and inventory management where both pricing and inventory policies adapt to changing contexts; \(2\) an easy\-to\-implement algorithm that achieves optimal regret bounds of𝒪⁡\(K​T​log⁡T\)\\mathcal\{O\}\(K\\sqrt\{T\}\\log T\)under concave revenue conditions and𝒪⁡\(K2/3​T2/3​\(log⁡T\)1/2\)\\mathcal\{O\}\(K^\{2/3\}T^\{2/3\}\(\\log T\)^\{1/2\}\)for the general case, with matching lower bounds confirming minimax optimality; and \(3\) a queueing\-theoretic method to handle infeasible target inventory levels, ensuring stable performance under stochasticity\. Extensive numerical experiments demonstrate the algorithm’s effectiveness across diverse scenarios\.

For future work, it would be valuable to investigate this problem under alternative modeling assumptions beyond the linear additive framework\. It is also of interest to develop UCB\-style analyses tailored to censored demand, which could offer principled exploration under partial observability\.

## Acknowledgments

This work was supported by the Theme\-based Research Project T32\-615/24\-R and the General Research Fund 16500023 from the Hong Kong Research Grants Council\.

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