Transfer Learning for Nonparametric Contextual Dynamic Pricing
Authors: Fan Wang
Feiyu Jiang
Zifeng Zhao
Yi Yu
Venue: ICML 2025 poster (2025)
Keywords: Dynamic pricing
Transfer learning
Covariate shift
Minimax optimality
Primary area: theory->domain_adaptation_and_transfer_learning

Abstract: 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.

Transfer Learning for Nonparametric Contextual Dynamic Pricing
Fan Wang' Feiyu Jiang"
Zifeng Zhao Yi Yu
Abstract
the rapid growth of online marketplaces and advances in data collection have further enabled sellers to use contex- tual information, such as customer characteristics, product features and market trends, to make more informed pricing decisions. This has fostered research contextual dynamic pricing (e.g. Wang et al., 2023; Luo et al., 2024; Fan et al., 2024), which further incorporates in-depth information to optimize pricing strategies.
Dynamic pricing strategies are crucial for firms to maximize revenue by adjusting prices based market conditions and customer characteris- tics. However, designing optimal pricing strate- gies becomes challenging when historical data are limited, as is often the case when launch- ing new products or entering new markets. One promising approach to overcome this limitation to leverage information from related products or markets to inform the focal pricing decisions. In this paper, we explore transfer learning for non- parametric contextual dynamic pricing under a covariate shift model, where the marginal distri- butions of covariates differ between source and target domains while the reward functions remain the same. We propose a novel Transfer Learn- ing 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 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 sce- nario a special case and is presented for the first time in the literature. Extensive numerical exper- iments validate our approach, demonstrating its superiority over existing methods and highlight- ing its practical utility in real-world applications.
Effective dynamic pricing involves balancing the explo- ration of the unknown revenue model with the exploitation of the estimated revenue model to maximize rewards. The revenue model, representing the relationship between covari- ates, prices and revenue, has been explored under both para- metric and nonparametric settings in the literature. The para- metric settings impose specific assumptions on the revenue model and have been widely studied (e.g. Qiang & Bayati, 2016; Javanmard & Nazerzadeh, 2019; Ban & Keskin, 2021; Wang et al., 202 la; Zhao et al., 2024). The nonparametric settings provide more flexibility, making them more suitable for real-world applications where customer behaviour and market conditions are diverse and unpredictable (e.g. Chen & Gallego, 2021; Chen et al., 2023).
Despite the wide applications of contextual dynamic pric- ing, collecting sufficient data to design an optimal pricing strategy can be challenging, particularly when launching a new product or entering a new market with limited his- torical data. In contrast, abundant data may be available from related products or markets, from which one may lever- age information achieve improved decision-making. For instance, historical data from other platforms cxisting markets can help sellers optimize pricing strategies more efficiently when entering new environments. This naturally falls in the territory of transfer learning (e.g. Pan & Yang, 2009), where datasets from similar but different distribu- tions are utilized to enhance learning of target dataset.
1. Introduction
Dynamic pricing is fundamental strategy used across many industries to maximize revenue by adjusting prices based on market conditions (e.g. Araman & Caldentey, 2009; Bes- bes& & Zeevi, 2009; Wang et al., 2021b). In recent years,
Transfer learning, studied in the machine learning literature, with applications in areas such as recommendation systems (e.g. Pan et al., 2010), language models (c.g. Han et al., 2021) and disease detection (e.g. Maqsood et al., 2019). It has recently also gained attention in statistics and has been explored in vari- ous problems, such as nonparametric regression (Cai & Pu, 2022b), contextual multi-armed bandits (Suk & Kpotufe,
research area, has been extensively
Deepartment of Statistics, University of Warwick "School of Management, Fudan University MMeddzaa College of Busi- ness, University Notre Dame. Correspondence to: Fan Wang <fan. wffn.wang.66.. 6 www.wwwcc..ccc...k..........
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Transfer Learning for Nonparametric Contextual Dynamic Pricing
2021; Cai al., 2024a) and functional data analysis (Cai et al., 2024b). In statistical transfer learning. two common settings are posterior drift, where the conditional distribu- tions of responses given covariates vary (e.g. Reeve et al., 2021; Maity et al., 2022), and covariate shift, where the conditional distributions remain consistent but the marginal distributions of covariates differ (e.g. Hanneke & Kpotufe, 2019; Kpotufe& & Martinet, 2021).
1.2. Notation and organization
Denote X = =00, 1] as the covariate space, P [0, 1] as the price space and 2 = X x P as the joint space of covariates and prices. All balls in this paper are defined with respect to the -nrmm. Let (ss, r), (x(s, and Bp(s, r)) denote the balls in 2, X' and P. respectively, centred at s with radius 7. For any ball B, let r(B) denote its radius and c its centre. For n E E4, let [n] = {l..... n). For any distribution P. let supp(P) be its support.
1.1. List of contributions
The remainder of the paper is organized follows. In Sec- tion 2, we formally define the problem. Section 3 introduces the TLDP algorithm. In Section 4. we present theoretical re- sults on the regret bounds of the proposed method, followed by a matching minimax lower bound. We demonstrate the practical performance of our approach through comprehen- sive numerical experiments in Section and conclude in Section 6.
In this paper, we study transfer learning for nonparametric contextual dynamic pricing under the covariate shift model. The main contributions of this paper are summarized as follows.
Firstly, to the best of our knowledge, this is the first study to explore transfer learning in dynamic pricing. We introduce a novel algorithm, the Transfer Learning for Dynamic Pricing (TLDP) algorithm, which effectively leverages source do- main data to enhance pricing decisions in the target domain.
2. Problem formulation
Secondly, we establish theoretical guarantees for the pro- posed transfer learning algorithm and further derive a mini- max lower bound. We demonstrate that the proposed algo- rithm achieves minimax optimal regret, with the cumulative regret (up to logarithmic factors) given by
In this section, we introduce a nonparametric contextual dy- namic pricing model under the transfer learning framework. Our goal is to minimize the regret for the target data by leveraging pre-collected data from related source domain.
For the target domain, the seller sells product to no Z+ sequentially arriving consumers. At each time i E [nq]] the seller observes contextual information X, E X' drawn from the distribution Qx Based on X,, the seller sets a price Pt E P and then receives a random revenue Y, E [0, 1] with its conditional expectation
ininq + (KΠP)
where no E 24 represents the length of the target hori- zon, np E N represents the number of observations in the pre-collected source dataset, 7 [0, oo] (see Definition 2.3 later) denotes the transfer exponent quantifying the simi- larity between source and target covariate distributions and E [0, 1] (see Definition 2.4 later) represents the explo- ration coefficient quantifying the minimal level to which the source data adequately explores the price space.
E (YY = ((),,)))
(I)
where ʃ: 2 - [0, 1] is the unknown reward function.
In the context of transfer learning, we further assume that the seller has access to pre-collected source dataset
Thirdly, as an important byproduct, our study covers the special case of nonparametric contextual dynamic pricing without transfer learning under the setting of a Lipschitz reward function. Our proposed algorithm achieves the regret upper bound of order
DP = [(X, .P. Y: HEE Z x 10,
(2)
where X E X' is drawn from distribution Px, pf" E P is the price and YPP corresponds to the observed random revenue with its conditional expectation [ , = f"(X,, pP ). As mentioned in the introduction, this paper focuses on the covariate shift model, where the marginal distributions of covariates between the source and target domains are different, i.e. Px + Qx, but the conditional distributions of rewards are identical, i.c.
(d+2)/((+3)
Q
up to logarithmic factors, with an accompanying minimax lower bound. To the best of our knowledge, this the first minimax lower bound result under the Lipschitz condition for nonparametric contextual dynamic pricing with general dimension d > 1.
f(z, p) = ff (2,,), v(z,p) E [0, x 10.
(3)
Finally, we validate the proposed approach through exten- sive numerical experiments, demonstrating its effectiveness compared to existing methods and highlighting its practical utility in real-world applications.
Our primary objective is to design a pricing strategy that minimizes the cumulative regret over the selling horizon in the target domain by effectively utilizing both the source
2

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Transfer Learning for Nonparametric Contextual Dynamic Pricing
data DP and the observed target data. Formally, let II denote the family of all price policies {rr .. Pno } where each price Pi is Fee--meauurable, t E 24. The field Fi-1 is the o-algebra generated by the history target data {(X..Ps, Y=)) all source data {xx _/85 v/ )> and covariate Xg. For any price strategy π E 11, the cumulative regret over ni steps is defined as
The transfer exponent 7 quantifies the extent which the source covariate distribution Px covers the target covariate distribution Qx A smaller y indicates greater overlap, enabling more efficient information transfer. In an extreme case where Px and Qx are identical, 7 = 0 indicating a perfect overlap. This parameter is crucial for analyzing the potential performance improvements achievable by transfer learning in our framework.
==1=
n R= R=(o) = Σ
(4)
To ensure that the source data provides sufficient variability in prices across different covariates, we further introduce the exploration coefficient. This parameter quantifies the minimal extent to which the source data adequately explores the price space. It is essential for accurately estimating the reward function over various price levels.
where (a) = maxper I (z, p) is the maximum expected revenue for any I E X. We further impose the follow- ing assumptions on the reward function and the underlying distributions.
Definition 2.4 (Exploration coefficient), Let μ denote the joint distribution of the source covariate- price pairs and let Px be the marginal distribution of the source covariates. Define the exploration coefficient E [0, 1] as
Assumption 2.1. Assume that the reward function 2:2 + [o, 1] is Lipschitz continuous with respect to the -norm, Le. there exists an absolute constant Ct.p > such that for any (21. P1), (2) E Z.
M(p - r. + r| x Bx(x,r))
K inf #Esupp(Pxx): rE (0,1//22 pE[r.1-r]
(1(x1.P)) p2)1 & CLapll<T .P))(( - (xg 1
2r Px (Bx(
Assumption 2.1 regulates the changes in the reward fune- tion led by the changes in covariates and prices. This is commonly used in the literature of online learning with non- parametric reward function (e.g. Slivkins, 2011; Chen & Gallego, 2021; Chen al., 2023), facilitating theoretical analysis of the estimation and learning processes. Assump- tion 2.2 below is a regularity condition on Qx to ensure the covariate space will be explored sufficiently.
The exploration coefficient s measures the minimal condi- tional probability density of the price given the covariates. A larger к indicates that the source data provides better exploration of the price space for each covariate. This co- efficient is crucial in assessing the usefulness of the source data for learning the reward function in the target domain. The definition of K extends the exploration coefficient intro- duced in Cai et al. (2024a) for multi-armed bandits (MAB) to continuous action spaces in dynamic pricing. To illustrate this concept, consider the following example.
Assumption 2.2. Assume that for the target distribution Qx, there exist constants 0- < cQ < Cq such that for any x E supp(Qx) and r E (0, 1].
Example 2.5. Suppose that the source dataset DP is defined in (2) and the source prices are drawn independently of source covariates from distribution with density hp given by hp(p) 8, if (a.-1, as), for < 0--1 < am = 1 and 5, > 0 such that
cQr" S Q (Bx(x,r)) S Car*.
To quantify the potential benefit of transfer learning from the source domain to the target domain, we introduce the notion of transfer exponent, parameter widely used in the transfer learning literature with covariate shift (e.g. Kpotufe & Martinet, 2021; Suk & Kpotufe, 2021; Cai et al., 2024a). quantifies the degree of similarity between the source and target covariate distributions, regulating the potential of effective knowledge transfer.
[m], where 0 =
0g < 01 Jd hp(p) dp This a piecewise uniform distribution and the exploration coefficient is x mingg(m) o5 which holds for any source covariate distribution Px.
3. Transfer learning algorithm
In this section, we introduce the Transfer Learning for Dy- namic Pricing (TLDP) algorithm, detailed in Algorithm 1. TLDP borrows the idea of contextual zooming from Slivkins (201 1) proposed for MAB problems with only target data, and addresses the unique challenge of leveraging source data to refine the exploration of the covariate-price space in the target domain. At a high level, TLDP is an upper confidence bound (UCB) type algorithm that handles the nonparametric setting through an adaptive partitioning strat-
Definition 2.3 (Transfer exponent). The transfer exponent 7 E [0, oo] of the source covariate distribution Px with respect to the target covariate distribution Qx is defined as
7 inf {~/ 2 0/ 3 constant 0 < Cy S 1 such that Px (Bx(z,r)) 2 2cr Qx (x((xxz)))),
Yo E supp(Qx),re(0.1) 10.
3

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Transfer Learning for Nonparametric Contextual Dynamic Pricing
Algorithm 1 Transfer Learning for Dynamic Pricing (TLDP)
Dynamic pricing via UCB. At each time [nq]] af- ter observing the covariate Xx, TLDP sets a price by first selecting ball B in the covariate-price space from the current candidate set Ar that maximizes the revenue poten- tial (B) defined below, then randomly chooses a price such that the covariate-price pair (X,, p) is in B, to be spe- cific, in dom At) as defined (7).
Input: horizon length ni. source dataset D and the
smallest radius to explore ñ Initialize: t +- 1, A, < {2). nB( + ng(DP),
(B) - (DP), YB E A
while S ng do
Observe Xt
rel, < :B E [0, (X,,,) dom( B, B.)
Definition of 4. Let ni (B) and (B. denote the cumula- tive count and revenue of B from the target data before time and the source data DP, defined ni (B) = (DP) + B3 and re,(B)
Bsel uniformly choose from arg Bard (B)
Pt ~ (p:formmpp(x:::::
E dom (Bad A.))
n @ (Bmd)) (Bal) n (D (DP)
(D*) + E Y,l((Xx,,P)) E B), with nD(D") and re (B) de- fined in (5) and (6), respectively. For any C 2, let its UCB uncertainty level be
Σ-1 1((x.,
while @ (Brrt) 2 The
Bes and r(Brd)) 2 2F do
B' + B
r(Bal)))))
A + Ae U (B'), (B') t ng. (DP)
re,(B') + fef (DP), Bsel + B'
end while
conl, = 2/ log {no V (KΠP) Jis
for B E ( do
(8)
nr(B)
n (B) c n4 n4( (B), ret+1 (B) rB:
end for 1i++1 (Bar)) f- no( (Bddd +1
As n4 (B) grows, i.e. more samples are collected in B, conf, (B) decreases, reflecting increasing in confidence. The revenue potential index is then defined as
re:+1 (B%d) f re; (Boo)) + Y,, t t+1
end while
T(BB)=Cr(B) + min (B') + Cr]le((B c(B'))lo). (9)
B'E.A,
egy. Unlike approaches relying on predefined partitions, TLDP sequentially refines partitions of the covariate-price space in response to observed data. The key components of TLDP are outlined following Algorithm 1.
where C₁ > 0 is a constant and the pre-index (B²((B) is given by
A head start by the source data. At the core, the pric- ing is conducted by sequentially discretizing the contin- uous covariate-price space Z. To be specific, this is to produce a collection of active -alls in Z, namely A, at step E [ma]- The source data provide a head start for such partitioning, starting with the entire space A = (2). TLDP sequentially refines the partition and adds smaller balls into Ae. This partitioning is to be detailed later. We first introduce some necessary notation. For a ball B C 2, let its count and cumulative revenue utilizing the source dataset DP be
= (B) (B) + Cir(B) (B), (10)
with vy = (B) (B).
The index (B) is adapted from its MAB counterpart proposed in Slivkins (2011) to balance the estimated re- ward (B),, the radius of the ball r(B), the uncertainty level (B) and its distance to its neighbours [[c(B) c( ((B')l loc - the choice of the norm is reflected in the Lips- chitz condition imposed in Assumption 2.1. The selection of inputs & and C₁ is further discussed in Section 5.1.
- Σ (EDD
1((xp) e B}
(5)
Partitioning. The last ingredient is the sequential parti- tioning. A ball is further refined when enough samples are collected and its radius is not too small. The refinement is conducted by adding a smaller sub-ball to the candidate set A.. To be specific, the partitioning is summoned when the (6) target sample count 8 (B) exceeds T& where
and
(DPP) Σ F
(X.PY))D
0, (WBB,,
wBB (D)") (1I)
For any ball B Ar, let its domain be
T : T&"("")
w(B) 2 n (DD),
dom( B, A.) (U (B))) B'),
with
excluding any overlaps caused by strictly smaller balls in As. thereby preventing redundant coverage.
wBB) log [nQ V (kmP)
2+3
(12)
r(B)2

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Transfer Learning for Nonparametric Contextual Dynamic Pricing
This threshold incorporates the contributions from the source data. Note that when there is sufficient source data within the ball, no additional target data is required for ex- ploration.
we conjecture the TLDP algorithm (Algorithm 1) can achieve a regret bound
R Sno { no + (kMP)) 241420-7
log Joga {no + (kmP)
d+2428
4. Minimax optimality
In this section, we establish theoretical guarantees for the TLDP algorithm presented in Algorithm 1. Specifically, derive upper bound the cumulative regret and matching minimax lower bound, demonstrating that TLDP achieves minimax optimality under the given assumptions. We also compare our findings with existing literature to highlight the contributions of our approach. We begin by presenting the upper bound on the regret achieved by TLDP.
The smallest radius to explore F mediates exploration- exploitation trade-off. A smaller F leads to finer partitions and more precise local estimates but increases exploration cost, while a larger i reduces the exploration cost at the expense of estimation accuracy. The choice of F in (13) strikes a balance in this trade-off, ensuring the algorithm achieves optimal regret bounds.
The regret bound in Theorem 4.1 elucidates the dependence of the cumulative regret on the transfer exponent and the exploration coefficient K. A lower transfer exponent indicates higher similarity between the source and target co- variate distributions Px and Qx leading to a larger adjusted source data size (Kmp)))(d (dd3) ((d+))/(4++++) and thus reducing the regret.
Theorem 4.1. Suppose that the source dataset 'DP ((F pr y is defined in (2) with triplets indepen- dent across time. Assume that the target dataset, defined in (1), satisfies (3), and that Assumptions 2.1 and 2.2 hold, with C₁ 2 CLip where C1, Clap > O are constants defined in (9) and Assumption 2.1, respectively. Let Algorithm have input satisfying that
Remark 4.3. As an alternative to the global exploration coefficient Definition 2.4, one can instead define a scale- dependent exploration coefficient Ky [0, 1] for any radius (0, 1/2]
Cr log {nq+ + (KMP) 313ET
d+3
ar3
(13)
no + (KMP) 24347
and C.c.cQ 2 8, where Cr > is a constant and C, > are constants defined in Desinition 2.3 and Assumption 2.2, respectively. Let ı denote the TLDP pol- icy given by Algorithm 1. Recall the cumulative regret R defined in (4) and it holds that
u(lp r' + 11 x (x(
Kr inf #Esupp(Px). ). //e p-
2r' 2r' Px (Bx(I.T))
Note that K, is non-decreasing in and satisfies that K inf/e(0.1/2] Ky. It quantifies how well the source data ex- plores the covariate-price space at a given scale. Under the conditions of Theorem 4. for any choice of the smallest radius to explore i E (0, 1/2), with the local coefficient Kr, the regret satisties that
R <Cna { na + (knP)
loggts {no + (kmp) 243+7
where C C>0 0 is a constant only depending on constants C1, Cr. CLip- Cyy and cq-
nof + log(ne); not,
no 2 (MSTP)) J.31T,
Remark 4.2. Theorem 4.1 establishes the regret bound under the assumption that the reward function f is Lipschitz con- tinuous with respect to the lo- -nrm (see Assumption 2.1). This assumption can be relaxed to accommodate more gen- eral smoothness conditions. In particular, one may consider Hölder continuity assumption (e.g. Kleinberg, 2004; Cai & Pu, 2022a), where there exists an absolute constant Cn > 0 such that for any (I1, P1), (I2, P2) E Z,
no (KGMP)) ) I:31T and F > [8log (mmpp)
RS
24347
(d+2) log(n,mp);
otherwise.
These three regimes illustrate how both the sample size and the exploration radius 7 jointly influence the overall regret. While we focus on the global coefficient i: in this paper for simplicity, the local coefficient K, characterizes tighter per- formance guarantees in scenarios where exploration quality varies across different scales, particularly useful when the source data do not uniformly cover the entire price space.
(() f(22 p2)1 < C= .P)
for some smoothness parameter 3 E (0, 1]. We believe that the techniques developed in this paper can be extended to this setting. Choosing the smallest exploration radius as
We now show that the regret upper bound in Theorem 4.1 is minimax optimal (up to a logarithmic factor) by presenting the following matching lower bound.
log (no + (kMp) 24142847
d+1+30
F= Cr
d-1428
no + (kMp) कー1
5

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Transfer Learning for Nonparametric Contextual Dynamic Pricing
Theorem 4.4. Let I(7,, Cry Ri CLip; (Q) denote the class of nonparametric dynamic pricing problems such that (i) the source dataset D {(X]', , YY )() defined in
satisfy (1), Assumptions 2.1 and 2.2. It holds that
inf sup T (EI(CCip.eq)
( 2 @
(2), satisfies Definition 2.4 with triplets independent across time; (ii) the target dataset, defined in (1), satisfies Assump- tion 2.2; and (iii) both datasets satisfy (3), Assumption 2.1
where e > 0 is a constant only depending on constants CLip and cQ.
and Definition 2.3. holds that
It is worth noting that even without considering transfer learning (i.e. when only target data are available), the min- imax lower bound in Corollary 4.5 is, to the best of our knowledge, the first result established under the Lipschitz assumption stated in Assumption 2.1. Previous works have only established minimax lower bounds either for the case with a semi-parametrie form of the expected revenue func- tion (e.g. Luo et al., 2022: Xu & Wang, 2022) or with addi- tional assumption beyond Lipschitz condition on the reward function (e.g. Chen & Gallego, 2020).
inf sup
(
* 1e
2 cn (nQ + (kmp)aa-tt d=2 4t3
where c > i a constant only depending on constants CLip Cy and cq-
Theorems 4.1 and 4.4 together exhibit phase transition depending the relative sizes of the adjusted source data 4-3
24335 and the target data no. When (xmp) ननकтт > no, the algorithm significantly benefits from transfer learn- ing, otherwise the regret of the same rate as when only the target data are used.
5. Numerical experiments
In this section, we conduct numerical experiments to support our theoretical findings. Synthetic and real data analysis are in Sections 5.1 and 5.2, respectively. The code and datasets are available online!
The most relevant work is Cai et al. (2024a), which studied transfer learning for nonparametric contextual MAB under covariate shift. In their setting, actions (i.e. arms) are dis- crete and for simplicity are treated as a constant (i.e. the number of arms K= 1). They established a regret bound of order
5.1. Simulation studies
We conduct simulation studies under the covariate shift model as stated in (3). The target covariates (X,) are independently and identically distributed (i.i..) accord- ing to Qx, uniform distribution over [0. 1].. For the source domain, covariate-price pairs (X," .p d Jes are gen-
(14) with: 3 denoting the smoothness parameter of the reward function (3 in our setting). The simplification K results in their regret bound losing explicit control of K, making their approach inapplicable to infinite or uncount- able action spaces. In contrast, dynamic pricing presents a unique challenge due to its continuous action space (i.e. prices), requiring more intricate methodologies than that in Cai et al. (2024a). Our approach effectively ad- dresses this by adaptively partitioning the joint covariate- price space. This adaptation ensures optimal regret scaling while accounting for both the covariates (of dimension d) and price (of dimension 1). Substituting d ++ 1 for d in (14) shows that their regret bound aligns with the one established in Theorem 4.1.
no (n + (Kmp) 212
4428
erated i.i.d. from a joint distribution p.. We let the source marginal covariate distribution Px be chosen such that the density function px (x) obeys px (a) = cllx - x= l1%, with x² (1/2, 1/2) E Idd and normalization constant 27 /d. This choice of px(xx) ensures that Px satisfies Definition 2.3. The generation of the source prices is detailed in each scenario below. We consider the source data size np E {10000(k - 1): k E [5]]. the target data size no E {10000k: & E [5]} and the transfer exponent 7 E (0.5k: l E [5]].
To assess the performance of TLDP (Algorithm 1) in reduc- ing target-domain regret, compare it with TLDP utilising the target data only (Target-Only TLDP), the Adaptive Bin- ning and Exploration (ABE) algorithm proposed in Chen & Gallego (2020) and the Explore-then-UCBB (EXUCB) algo- rithm developed in Luo et al. (2022).
When only target data are utilized, several existing studies have developed methods and analyzed regret bounds for dy- namic pricing under semi-parametric forms (e.g. Luo al., 2022; Xu & Wang, 2022; Fan et al., 2024) additional shape constraints beyond Lipschitz continuity (e.g. Chen & Gallego, 2020), with detailed comparisons in Appendix A.
Through this subsection. TLDP refers to Algorithm I with nr 2nq. Note that both TLDP and Target-Only TLDP require the smallest exploration radius F as input and the
We conclude with a minimax lower bound for the scenario where only target data is available.
Corollary 4.5, Let [(Ciip <Q) denote the class of non- parametric dynamic pricing problems such that target data
httpp:://itthii com/ ctrr/crriii// fawwangg dynami e-prici cing
6

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Transfer Learning for Nonparametric Contextual Dynamic Pricing
450
2000
1234
350
320-
Method
500
Teges-Day TLDP
12300
10300
19/6
42.00
30003
10003
23000
30000
4000E
05
2.5
625
125
200
339
0.50 3.05
633 0.55 3.0
339-
Figure 2. Results for Scenario 3 under posterior drift. The drift level r varies, with the target data size no = 10000, the source data size up 2ng. the transfer exponent 7 the exploration coefficient A = 1. the index constant Cj = and the exploration radius constant C. 1/4.
04
32
2115
03.00
Method (Pane
Mathod (Panels B, & DJ
1LCe on in lcace will corles aolooatt
the observed rewards for both target and source datasets are independently and uniformly distributed in the interval (f.. - v, f(x,, p) + v/., where f(:..) denotes the expected reward function and v = 0.1. The expected reward function f is constructed as f p) = An + A xp + 6p², for any E [0, 1744 p [0. where do, 0 IR and 0 E Rd are specified for each configuration.
Figure 1. Results for Configuration 1 in Scenario 1. Panel (A) and (B): varying source data size np and target data size no- respectively. Panel (C) varying the transfer exponent (top axis) and the exploration coefficient y (bottom axis). Panel (D): varying the index constant C2 (top axis) and the exploration radius constant C, (bottom axis). For Panels (B). (C) and (D), we fix ni = 10000.
TTzp
We further consider two configurations. Configuration 1. Let the covariate space dimension be d = 2. The parameters of the reward function are set as 9g 0.2, 8 = (0.3, 0.2) and 0 0.11. Configuration 2. Let the covariate space dimension be d 3. The parameters of the reward function are set as 0g 0.3, 0 (0.1, 0.4, -0.2) and 0 -0.1.
constant C, to construct the index defined in (9). Specif- ically, we set C₁ = 1 and compute i from (13) using the true values of the exploration coefficient K, the transfer ex- ponent y and C, = 1/4. To evaluate robustness, we also conduct simulations using a list of mis-specified values of and y when computing F. The corresponding simulation results can be found in Appendix D.1. Furthermore, to ex- amine the sensitivity of TLDP to the choice of the constants, additional simulation studies are conducted for varying C₁ and C,.. For the ABE algorithm, we set M = 0.1, the constant used to define the maximal number observed in a level-k bin in the partition, as suggested by Chen & Gal- lego (2020). For ExUCB, parameter choices include the phase exponent 3 2/3, the UCB exponent 7 1//6, the exploration phase constant C1 = 1, the discretization con- stant C2- = 20 and the regularization parameter in UCB = 0.1, as suggested by Luo et al. (2022). In addition, we fix the maximum price Pmax = and the maximum possible revenue B- for the ExUCB implementation.
Scenario 2. The conditional density of the source price, given the source covariates, is defined as follows for any x E [0, 1]d and p E [0, 1]: hp(plx) = {1 ((-rr"))//" if p [p* - r2/ + /2] and hp(plz) wise, with p² 1/2, r" 1/4 and K E {0.2k: k [5]] Further, conditioned on the covariate and price p. the observed rewards for both target and source datasets are independently distributed Y ~ (), with a = 0.01. For the expected reward function f. we adopt the setting from Cai et al. (2024a). Define a function Φ: R. - [0. by d(=) = for 0 < z < 1/12, (2)) = 2 12x for 1/12 S < 1/6 and ((2) =
K other-
otherwise. The reward function f: 1] x 10,1] [0.1] is then set to be f(x, = 1/44 >m 3/40(1(2 .p) )
((x;) p..) Tl))) r' E (0. 1] and (((x)) .2f)" ===1 are specified for each configuration.
Bx )))). where m > 0.
Scenario 1. The conditional density of the source price, given the source covariates, is defined as follows for any x E [0, 1jd and E [0, 1]. hp(())) = K if E [p- r2 + r /2] and hp(p))) = [1 kr")
r²) oth-
Configuration 1. Let the covariate space dimension be = 2. Let
We further consider two configurations.
erwise, with p² = 1/2, r" = 1/4 and x E {0.2k: & E [5]]. Furthermore, conditioned on the covariate x and price p.
m 4, r = 1/4 and centres be {(xx)) ( =
7

---

Transfer Learning for Nonparametric Contextual Dynamic Pricing
10//4,1/4,/14) (3/4,1/411/4) 31/4, (3/4, 3/4,1/4)) Configuration 2. Let the covariate space dimension be d 3. Let m = 8, r- 1/4 and centres be .pt")""pp))) = /1/44.1/4.144, 1/ 1/4), (3/4, 1/4, / (14433/4 1/4), 1144444444443/4433/443 3/343 3/4,3/4), (3/4,3/4
Table I. Results for the auto loan dataset with East South Central data as the target division. Columns correspond to the source divisions utilized in TLDP. Here, np represents the number of the source data utilized in TLDP and n denotes the total number of source observations. Each cell reports the mean and standard deviation over 100 simulations.
(3/4,3/4,
METHODS
MOUNTAIN
PACIFIC
Scenario 3. Building on the setup of Configuration 1 in
Scenario 1, we fix the target reward parameters as 0g- 0.2, 0 (0.3, 0.2)T and 0 = -0.1. We then modify the true source reward function to fx,p) = 0g + x = + 8'p², where 86 ~ N6.. )²), 8' ~ NN0, (1)) and 0' o"). We set o = r min[[0ol. [|0 lco. [0], where the drift severity parameter r E {0.05(k: - 1): k [5).
ABE
70.81 (1.40)
70.81 (1.40)
ExUCB
64.91 (5.31)
64.91 (5.31)
TLDP(np = 00)
71.95 (2.87)
71.95 (2.87)
TLDP(nP = -..55n)
55.63 (8.04)
51.61 (8.89)
TLDP(1P = -0.5m)
54.29 (7.06)
51.44 (7.12)
TLDP(1F -0.75m)
51.38 (6.92)
49.40 (7.40)
TLDP(n> = -n)
50.23 (7.34)
48.72 (8.27)
In Scenarios and 2. we consider the covariate shift set- ting, where the reward functions identical between the source and target domains. Specifically, in Scenario 1, the reward function is linear in the covariates and quadratic in the price, while in Scenario 2, it is fully nonparametric. In contrast, Scenario considers a posterior drift, where the reward functions differ between source and target do- mains, allowing us assess the robustness of TLDP under model misspecification. The simulation results for Config- uration 1 of Scenario 1 are presented in Figure I while results for Scenario 3 are shown in Figure 2, Additional results are provided in Appendix D. Some key observations are highlighted in order.
butions, the regret of TLDP also increases. This indicates less efficient information transfer from the source to the target domain. Moreover, as the exploration coefficient x increases, the source data provide better exploration of the price space for each covariate. Consequently, the regret of TLDP decreases, indicating more effective information transfer. Panel (C) is consistent with Theorem 4.1, demon- strating the impact of these parameters on the efficiency of the transfer learning framework. It is evident from Panel (D) that the performance of TLDP remains relatively stable across different values of the index constant C₁ and the ex- ploration radius constant C,, indicating that TLDP is robust to the choice of the two constants.
In Scenarios 1 and 2, TLDP outperforms the alternatives. In Scenario 1, ExUCB outperforms ABE and Target-Only TLDP due to the linear structure of the reward function in the covariates, The ExUCB algorithm is specifically de- signed to leverage such linear relationships, thereby achiev- ing superior performance. In Scenario 2, ABE and Target- Only TLDP demonstrate better performance than ExUCB. These findings align with the discussion in Section and Appendix A.
In Scenario 3, Figure 2 demonstrates that TLDP continues to perform well under moderate posterior drift, indicating robustness beyond the covariate shift setting.
5.2. Real data analysis
In this section, we evaluate the practical utility of our pro- posed algorithm using the auto loan dataset (Phillips et al., 2015), which consists of 208, 085 applications submitted to a major online lender in the United States between July 2002 and November 2004. This dataset has been extensively studied in previous works, such Phillips et al. (2015), Luo et al. (2024) and Zhao et al. (2024), to assess various dy- namic pricing algorithms. The performance of TLDP is still compared against ABE and ExUCB.
In Panel (A), as the target data size no increases, the regret for all four methods increases. TLDPP, however, exhibits a significantly slower rate of regret increase compared to its competitors, due to its ability to leverage information from the additional source data. Moreover, both TLDP and Target- Only TLDP demonstrate sublinear growth in regret as no increases, aligning with the theoretical regret order estab- lished in Theorem 4. In Panel (B), as the source data size up increases, the regret of TLDP decreases, reflecting im- proved performance with larger source datasets. This trend aligns with the theoretical results in Theorem 4.1, where the regret is expected to decrease at the rate np -1/(d+3+1) further validating the algorithm's 's performance.
The loan price calculated as the net present value of future payments adjusted by the loan amount, as follows: Price
Monthly Payment x ΣTerm (1 Rate) - Loan Amount, where Rate = 0. 12% represents the average monthly Lon- don interbank offered rate during the study period. The reward metric is defined as the product of the consumer's decision (whether the loan is accepted) and the computed price, capturing the total revenue generated for the lender.
In Panel (C), as the transfer exponent ? increases, reflecting reduced overlap between source and target covariate distri-
8

---

Transfer Learning for Nonparametric Contextual Dynamic Pricing
For this analysis, five covariates identified as significant in prior studies (e.g. Luo al., 2024; Zhao al., 2024) are included: the loan amount approved, the approved term, the prime rate, the competitor's 's rate and the customer's s FICO score. All features, including rewards, prices and covariates, are normalized to the range [0, 1]. Additionally, U.S. states are grouped into nine divisions following the United States Census Bureau classification: East North Cen- tral, East South Central, Middle Atlantic, Mountain, New England, Pacific, South Atlantic, West North Central and West South Central. Table 2 in Appendix D provides sample of the processed dataset.
lyzing the target data and to adaptively update an estimate of y by measuring empirical overlaps between the source and target covariate distributions. Second, in many applica- tions, multiple source datasets (K > 1) may be available, each with distinct transfer exponents */k and exploration coefficients Kig. A straightforward approach is to combine all sources into a single dataset before running TLDP. A more refined approach to weight sources adaptively by estimating Kg and 7/- Establishing regret bounds for such methods would be more challenging. Lastly, it is intriguing to explore transfer learning for nonparametric contextual dynamic pricing under posterior drift. In this framework, we need to quantify the discrepancies between the reward fune- tions in the source and target domains and to develop novel methods that effectively leverage transferable information.
For this study, the East South Central division (8, 062 appli- cations) is designated as the target domain due to its smallest data size. The Mountain (12, 527 applications), East North Central (20, 686 applications), West South Central (25, 695 applications) and Pacific (34, 870 applications) divisions are used as source domains for the TLDP algorithm. The true re- ward function is approximated using a random forest model implemented in R (R Core Team, 2021), leveraging the randomF orest package (Liaw & Wiener, 2002), trained on the target data. Optimal prices for each observation are determined by numerically optimizing the approximated reward function.
Impact Statement
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
References
The results are averaged over 100 simulations, with each simulation randomly selecting 90% of the target data as the test data. All results can be found in Appendix D.2. Table presents selected results with the key findings as follows. When no source data are utilized (np- = 0), TLDP incurs slightly higher regret compared to ABE and ExUCB. While ExUCB achieves the lowest average cumulative regret, it exhibits significantly higher variance. Conversely, ABE demonstrates low variance but relatively high regret. As the source data size (np) increases, TLDP's cumulative regret consistently decreases, indicating its ability to effectively leverage data from other divisions.
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Transfer Learning for Nonparametric Contextual Dynamic Pricing
Wang, Y., Chen, B., and Simchi-Levi, D. Multimodal dy- namic prieing. Management Science, 67100)66136-6155, 2021b.
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Transfer Learning for Nonparametric Contextual Dynamic Pricing
Appendix
All technical details of this paper can be found in the Appendix. A detailed comparison of our regret bound Theorem 4.1 with those from coantransfer learning studies is presented in Appendix A. The proofs of Theorem 4.1 and Theorem 4.4 are included in Appendices B and respectively. Additional details and results for Section 5 are collected in Appendix D.
A. Comparisons with non-transfer learning regret bounds
We compare the regret bound established for TLDP in Theorem 4.1 with those from relevant studies on dynamic pricing and contextual bandits that do not incorporate transfer learning:
Slivkins (2011) studied contextual bandits with similarity information, where both contexts and actions are embedded in metric space equipped with distance function. The regret bounds in Slivkins (2011), 1), adapted to notation, of order
4+3
J+3
log(ne),
matching our result up to logarithmic factor;
Chen & Gallego (2020) investigated nonparametric dynamic pricing with covariates and proposed the ABE algorithm, which adaptively partitions the covariate space based on observed data to balance exploration and exploitation. They showed that the ABE algorithm achieves a regret bound of order
log 2 (nQ);
which grows more slowly than ours due to additional assumptions beyond the standard Lipschitz condition stated in Assumption 2.1, such as the local strong concavity of the reward function;
Several studies on nonparametric dynamic pricing (e.g., Luo et al., 2022; Xu & Wang, 2022; Fan et al., 2024) consider specific form of the expected revenue function,
f(x, = - F(p 0)},
where F is nonparametric cumulative distribution function (CDF) of noise influencing customer valuations and 0 E Rd is an unknown parameter vector representing customer sensitivity. For instance, Luo et al. (2022) developed the ExUCB algorithm and derived regret bound of order
3/4 "QQ
independent of in the exponent. This independence arises from the linear parametric form of x which reduces the intrinsic complexity of the problem and mitigates the curse of dimensionality often encountered in fully nonparametric settings. In contrast, our results address a more general nonparametric reward function, which introduces additional complexity in capturing the covariate-price relationship without structural simplifications. Consequently, this leads to dimensional dependence, reflected in the d appearing in the exponent of our regret bound;
Bu et al. (2022) studied dynamic pricing under two partially linear demand models:
f(z,p) pfbp + g(x)} and f(x,p) plg' (p) + +x
where 6 < 0, a E Rd, and g, g' are unknown functions. They derived minimax regret bounds for both models. For the first model, assuming that g is 3-Hölder continuous, they established minimax regret bound of order
VT TT
In comparison, our setting considers a reward function that is nonparametric in both the covariates and the price, leading to a regret bound with d + 1 in the exponent instead of d, thereby capturing the additional complexity introduced by the price dimension. For the second model, assuming that g' is kth-order smooth with smoothness parameter 8, the minimax regret is of order
VT V 1))
which independent of in the exponent, as the model assumes a linear parametric structure in the covariates.
12

---

Transfer Learning for Nonparametric Contextual Dynamic Pricing
B. Proof of Theorem 4.1
The proof of Theorem 4.1 is in Appendix B.1 with all necessary auxiliary results Appendix B.2.
B.1. Proof of Theorem 4.1
Proof. For any S niq, let A₁ denote the set of active balls at the beginning of time t. For any B E At, let re, and (B( denote the cumulative revenue and the cumulative count of from the target data before time and the source data DP. Further, define
E = {VB A:: (B)(B) f(B) (Bi + (onf,(B)}
and
E - {VB As: En holds} with En - 1a=-(B(( (B((D)) (ERMpr(B)(tt*
where Cg c. and f(B) fc((B). Intuitively speaking, Ei represents the good event where the empirical average revenue (B) for every active ball B Ar lies within the upper confidence bound around the true expected revenue / (BB,, while E states that the source data (measured by the number) can still provide sufficient information up to time t.
Consequences of Ef: Suppose for any time point 1, El holds. Then by (10), the condition C₁ 2 CLip and the event Ef, for any B At, we have
1. (B) (B) 2 f(B).
(15)
For any x E X, denote p- (xx) min(p' E arg maxpe [(x,, p)}. By Lemma B.3, there exists a E A, such that (Xe, p" (Xx)) E dom(B. B, At). Let B denote the selected ball at time t, then we derive that for such B,
((Batt) 21(BB
=Cr((B)+ + min + - A'EA:
20r((B)+ B'E.,
min
+Crr(B)+ 2 (Xi),
(16)
where the first inequality follows from the selection rule in Algorithm I (Step 4 therein), the first equality follows from (9), the second inequality follows from (15), and the third and the last inequalities follow from Assumption 2.1 and Cy 2 CLip-
Let BTtr be the parent of Bgel, we have that (due to step 9 in Algorithm
[la(B - of (. Iooo I (BBaar)
(17)
Further note that B!ar must satisfy the condition in Step 7 in Algorithm 1 (in order to produce Beet), we have ni (B("a))
Egue (DP) T plor (DP). Then
log {nQ (KT,)) 44345 }
log {no (KTT)) atst7 }
conf, conf.(B.ar) 21
T
+ n.
nag (DP)
443 log { (n V (KHp) w (B."r))
a43t5
$2
< 2r 2r (B,ur))
(18)
where the second inequality follows from (11) and the last form (12). Note that
pre (..Tr)
=v (B. + Crr((,"") + conf, (B!"r)
<B( + (Cr + CLip)r(BB.)) (B!ar) + 2conf. (B.ar)
<f(D + (4 + C7 + ((ip)r(BB"") < (Bxt) + (4+ + Cr (C
(19)
13

---

Transfer Learning for Nonparametric Contextual Dynamic Pricing
where the first equality follows from (10), the first inequality follows from the event Ej, the second inequality follows form (18) and the last inequality follows from Assumption 2.1 and (17). Note that
(B,)
ICrr(B,t) + 1. (B("r) + CrIlo(B,)) - of (B 11
2Crr (B:e)) 11 (B(ar) (Br(
<Cpr (Bxe)) (B,)) (4 20, 2Cp (B.t)))
(B,") + (8 + 50, + 4CLip) )r (B,t.) S f(X,, + (8 - 5C, + 50LLp)) (((t))
(20)
where the first inequality holds due to (9), second inequality follows from (17), the third inequality follows from (19), the
forth holds due to r (B r (B) /2, and the last inequality follows from Assumption 2.1 and that (Xt,pt) E B
Combining (16) and (20), we have that
S SCr((Br)) (((),) (Cr
(21)
where C₁ = (8 + 50, + 5.
413 Regret decomposition: We then decompose the regret into several terms
Step 1: Consider the case where no 2 (knp)aisii)
n2 ( ((x.s.))
(n() =R
ng
no
- ((x.))) +
Σ
-(xx)
= (1) + (II).
(22)
In the following, we deal with the above two terms separately.
Step 1.1: Bound for (I). By union bound argument and Lemma B.1, we have that
P((8)) <
Σ BEA.
18 2-- log(nq) 18 2 llo((n)) S 188 18--17-3 llo((n))
(23)
where the second inequality holds by noting |A| S tF since the smallest radius is lower bounded by 7 and hence most F-1 balls be activated each time point; and the third inequality holds by S nq.
Therefore, recall that 0 S f(X,, pr) I (X,) < 1. it holds that
nQ
(I) <
P(E < 187 log(no) S "
(24)
t=1
(B Ano r(B)= r). For any B C Z, let (9) be the set of times s E [no] when ball B was selected for the target data. By Lemma B.3, we have that
Step 1.2: Bound for (II). For any r E
2
d+1
F < PPPck < (2) <
(25)
where N.Pack(2 (2) and (2) denote r-packing number and r-covering number of Z, respectively, and the second inequality follows from the fact that N.Pck (2) S (2).
14

---

Transfer Learning for Nonparametric Contextual Dynamic Pricing
Then, note that
nQ
1
Σ
(Xx
E
E
(x)) ((x (E))
Sf BEFr (ES)(B)
=E
Σ
Σ
- (f ((X.
BEF; (ES
+E
Σ3Σ
(F(()) f(X,, (()))
rI (ES
E Carlee 1
SCinor +E
Σ ΣED
(26)
MPRr BET, (B)
where the second inequality holds due (21).
d+3
When no 2 (knP) 443+7, by (11) and (12), for any r(B) > f, we have
log(no) r(B)²
S (B) S TT S w (B) <
+1.
(27)
as otherwise a new ball within B will be activated due to Step 6 in Algorithm 1. As consequence, by (25), (26) and (27),
we have that
(II s0inff+ +
Σ Cir
(2)
2
(log(n)) y2
+1
MPcrSI
+ 1
rr- (0 + r
<Cinoi + 2 C { (d+-2) log(ne) + tra)
(28)
where the last inequality follows from the fact that
[- log, (1)
2 (olgg
2d -1
2d
Σ
Σ
2kd
7
<
27 d
2d
2d
1
2d
k=
Combining (22), (24) and (28), we have that the case where no (kNp) a43+5
(x(n S4C
$4C(F=>lo((n)) llg(ne)) + Cinof + 20+2 ((d+2) log(nq) + +
<Cinor 2 g (d+2) log(nq).
(29)
d43 Step Consider the case where no < (kNp)
d43+7 and
d+3 8 log { (kRP) 4+3+7
(30)
Ceknr
where Ce = C.
15

---

Transfer Learning for Nonparametric Contextual Dynamic Pricing
Regret decomposition:
We then decompose the regret into several terms
ng
(xx) - s(x.,
=E
n4 (X,) F(Xe.
(x))) p(X.. ( (cEEp)))
+
RQ
(xx)) -
(T) + (II) + (III.
(31)
In the following, we deal with the above three terms separately.
Step 2.1: Bound for (I). Similar to (23), we have that
(((E)) - Σ
([(p))
}-2 313 [((np) 1-347 }-2 I 3nqr 3
{(tnp) 3-3-7 }-2
(32)
BEAt
Therefore, recall that 0 S f(X,, pi) S (Xi) s 1. it holds that
(1) S no
P(E1) s 3n2r {(kmp) 7+J+7 }-2
(33)
Step 2.2: Bound for (II). By similar arguments as (26), we have
(I) <Cnnq +
Σ Σ
Σ
( ((rres)
(34)
riFcrrl BEF, (ES)(B)
Note that under no < (knp) (0mp))aaa काअा5 and (30), if E holds, by Lemma B.4, we have that T = for any B A.. This implies that (B( = 0. Therefore, the second term in (34) vanishes and thus we have,
413
MII
Cinof.
(35)
Step 2.3: Bound for (III). Note that by Lemma B.4 and union bound argument, under no < (kRp) d+3-7 and (30), it
4+3
holds that
d-g ((()))) s nor ((mpp) x-3+y}
}
Therefore,
nQ > P (E))
(III
2n {(kp)) बाउन -(s0)p))3331} 3-2
(36)
Combining (31), (33), (35) and (36), we can conclude that
( Rs (nq) <3n [(kp)) - } 12 + Cinqi + n22r ( ((NM)) a13t5 3-2
d+3
d+3
<4n [(kpp) - ) } 12 Cinoi - 2Cinqi
(37)
where the last inequality follows from (30).
4+3
Step 3: Consider the case where no < (knP)
243+5 and
{
8 log (kMP) गरकन. कमकन.
d+3
F
Cermp
16

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Transfer Learning for Nonparametric Contextual Dynamic Pricing
Regret decomposition: We then decompose the regret into several terms
n
(nq) =E-
(Xx))
ng
no
((x.))
+
Σ
((xx) ((x.
(T) + (IT).
(38)
In the following, we deal with the above two terms separately.
Step 3.1: Bound for (I). By the same argument as (23), we can show that
P((E)) < F log { (kH,)) Jगउ1.
d13
}.
(39)
which implies that
(1) S nQ P((E))) s 18F-3 log (KTP) }.
2+3
(40)
Step 3.2: Bound for (II). By similar arguments (26), have
(I)) <Cingf + +E
Σ mPrrs BEF, (ESR(B)
Σ
Σ
Σ
Cirle8)
(41)
When no < (KNp) 44347, similar to (27), we have
log (KTIp) 24315
[B) < TT sw(B)
r(B)2 + 1.
(42)
As consequence, by (25), (41) and (42), we have that
d+1 log (kAp)
(II) <Cinof+ +
Σ Cir
+1
n.
nFergi
<Cnor + 20+1C
Σ
r (d++2) log (kmM)) デデデन
+ r r-
MFcnSI
<Cinor + 2 { (d+22)
log (KAP))
2+3+7
d+g
+ 8
(43)
Combining (38), (40) and (43), in the case where no < (knP) नtन (snp) and
4+3
log (KMP) 243+7
(44)
have that
Ry (nQ) 518 3 log (NT,)) at3+5
Cinor + 2d42 2
(- (d+2) log (kMp) क+उ45 + z-d]
d13 < F- (d+2) (d(+2) log (KmH,)) 24347
},
(45)
where the last inequality follows from (44).
17

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Transfer Learning for Nonparametric Contextual Dynamic Pricing Step 4: Combining all results. Combining (29), (37) and (45), we have that
Cinoř + 2 (d422 log(no),
4+3 if no 2 (KMP) d45+7,
(8lo (Kmp)) 2チ
} गामा
Rz(nq) <
20nn (+++)))-(++2)) log (kMp)) 245+7
if n < (kmp) #43+9 and f 2
CeKnp
otherwise.
By (13), Coe,co 2 8, we can conclude that
(nq)) & Cna ni + (kmp) *43+7
443 a43
logats ats {no + (kmpp) 243+7 },
where C is constant only depending constants C1, Cr. CLips by and cq. We complete the proof.
B.2. Auxiliary results
Lemma B.1. For Algorithm at the beginning of the round t S no, if ball As, then under Assumptions 2.1 and 2.2, have that
(B) N (B) conf 2 1 EF,
where Liip > 0 is defined in Assumption 2.1 and
18n-27-2logh log(nn),
iq 2 (kMP) ननअं.
iQ < (kΠp) गोड and F 8log ( 252
4
3((KMp) Āすअ1 J-2
(46)
18n 2 log { (kNp) 4-3-1 }, otherwise.
Here Cg c,,Q with constants Cy defined in Definition 2.3 and Assumption 2.2, respectively.
Proof. Fix time and a ball E Ar, and recall its centre as cBB). Let S (B) be the set of times E [t - 1] when ball was selected for the target data, with |St (B) = n (B). Let S (B) be the set of times & E [ur] when ball was fallen for the source data, with |S = (DPP) Note that ny [B) = n, (B) + n mg (DP) and
vB( = ne(B)
Σ BES? (B)
Y, + Σ
YP #ESP(B)
Denote
(B) (B)
Σ
f(X,,P.)+
Σ
f(x! p(x))
#ES (m)
SESP(B)
By Lemma B.2, it holds that
(B( Un! (B), " 82)
< <85179 282,
where 7 is defined in (11) and
2lg(11/8) n ((B)+n((OP)
if (f > 0,
U
if =
(D""
Let 61 = no² and 82 = {nQ v (kпp) オ+3+7 } -2 By (8), we have that
P (B) (B) conf. ((F of 1
s S + 2(nQ V (kmp) 21mp))
d13
(47)
18

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Transfer Learning for Nonparametric Contextual Dynamic Pricing
Furthermore, note that
(B) ((B)) - n4 (B)
Σ (B)
- +
Σ
.ESr(B)
(B),
(48)
where the second inequality follows form Assumption 2.1. Finally, combining (47) and (48), have that
P [1((B) - f(B)) (Lipr(B) + conf, (B) {x, p Ma
T + 2 V (kΠp) - }-2
(49)
In what follows, consider different cases in (46).
Case 1. In this case, consider no 2 (kmpp) #4347. Then, have that
SnQ "T + 2fnQ v (kΠp) ( = (B) + 2n -2 1 (nqrr(B) 2 log(nn)) + 10n
2 log(n2) + 10mm
118m2272 log(n)),
(50)
where the first inequality follows from (11), the second inequality follows from (12), and the third inequality follows from r(B) 2 F. Then combining (49) and (50), we have that
ur (B) f(B) > (B + conf, (B)}
=E | (B) f(B) (BB) conf, (on] ((x, ..fff.pp')) -))))=]]]]
$18n 2-2 2 log(nq)-
(51)
Case 2. In this case, consider no < (knp) 2ists, and
443 8 log { (KmP) 443+7
45345
>
CeKnr
Define the event
ΣB n (DPP)) Ce)npr(B))tttt
By Lemma B.4, we have that
((mm))=fsi7) 413 }- (( < {kmp))
(52)
and that under the event Ep, we have T 0. Then holds that
P [B) f(B) (B) + (B)
-E
{ (B) f(B) > (Bapr(B) + conf, (B) ((XP) pr
( [B(( f(B) > Crar((B) conf, ((XF' pf ((X,,
E
+ +R(D/ (B) [(B)
1P(E ((x())()))))
<B <E(p{ [B{ - f(B) > (B) + cont ,p2
EB + +1(8)}
<3((km)) -
(53)
19

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Transfer Learning for Nonparametric Contextual Dynamic Pricing
where the second inequality follows from (49) and (52).
Case 3. In this case, we consider ni < (kmp) #43+7 and
log (NTP) CεкпΩ
443+7
a+347
Then, we have that
4+3 T + 2nn V (kΠp)
443+7 }-2
<an0" wBB) + 2 2[ (NTP) 24347
1-2
<snQ 2 (BB) log (kiAp)
d+3+7
+ SnQ= 2 + (xmP 24317 3-2
< <16n@² 2 log { (kmpp)
143+7
4+3
+ (
4+3 2+3+7 }-
d13 118m22 log (kTP))
01315
(54)
where the first inequality follows from (1I), the second inequality follows from (12), the third inequality follows from r(B) 2 f and the last inequality follows from no < (knp))+3+>. <43+>. Then combining (49) and (54), we have that
p [0((B) (1(B))
(( [(B) > CLipr(B) cont,
- { [B()) f(B)] rLp conf, (B) conf,
( .p. p3) }]
$18 27 2 log (KTRP)
243+7
4+3
(55)
Finally, combining (51), (53) and (55), we complete the proof.
Lemma B.2. Let {21)21 be bounded martingale difference sequence with X; E [-1, 1]. Then for any 81,82 > and
integers T 2 0 and n 2 holds that
n+t
X, +
Σ 1mm++1
X,
> U(t.n,d1,bb) s 8T8, + +
where
22 t) log (1/8)
ift 0, ift = 0.
Ut,, (1,
2m 1og(//2))
Proof. The proof here is minor modification of Lemma 12 in Cai et al. (2024a). For completeness, we provide the details.
We start with the case T = 0. Following the Azuma-Hoeffding inequality (e.g. Corollary 2.20 in Wainwright, 2019), it holds that
n
X, >
/2n 2onlog( S 282.
(56)
i=1
We consider the case T O. Note that
P
30
S
Σ i=1
X,
Σ X,
> U(I,n,01,055
1=
att
SP
X;
> v 2n log(1/62)
PP
70 =0< T:
Σ
X,
2(n 1) log(11 1/81)
1=1
i=1
Rt1
<26 P
3 [T]:
Σ x,
(2(m+ ()
(57)
20

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Transfer Learning for Nonparametric Contextual Dynamic Pricing
where the second inequality follows from union bound argument and the final inequality follows from (56).
It remains to control the second term in (57). Note that for any and T" > 0, by Azuma-Hoefiding inequality (e.g. Corollary 2.20 in Wainwright, 2019), we have that
n+t Σ
8²
< 27" exp {
8
X
2 exp
(58)
2(n + t)
2(n + T
Note that
P 3 [T]
Σ X
> 222nn + 1og((//1)
1=1
[log (T)
ntt
P 727 < 2152111
Σ
X; >
2(n + 1) (1/()
1=0
==1
(T)] (T)
n +
2/+2 exp 2
log(1/61)
n2 2/+1
(T))
20+281 < 210 -870,
(59)
where the second inequality follows from (58) and the third inequality follows from the fact that (n+ ( 2 1/2 for any integers j 2 0.
Combining (57) and (59), we complete the proof.
Lemma B.3. For any I. E [nn]
Z
(60)
Moreover, for any two different balls B1, B2 Ang with the same radius their centres are at a distance of at least
Proof. Note that for any E [no].
UBeA.dom(B. (B, At) =UBEA, B,
By construction, Z C UBEA, B. This completes the proof of (60).
For the second part, consider two distinct balls B1, B- E Ang with radius By construction, they cannot be activated at the same time E [nq]- Without loss of generality, let Bj be activated time with parent Bear, and Bg be activated earlier. Denote the centre of B₁ (x: :p) T. We have that (x: , E dom (B A.) A.)(( Since r(Bpar) > r(B2) and B2 is activated earlier, we have that (x ,p) $ B2, which establishes the required separation between B and B2, and completes the proof.
Lemma B.4. For any t E [bq] and B A, denote
Eg = (n=")) Ce((mp)r(B))ttt
where Ce = C.PQ- Under Assumption 2.2, it holds that
PpEEn} 2 1 - -exp { -
(61)
Furthermore, ifng < (knP) 243+7 and
8 log { (KMP) 243+5
(62)
21

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Transfer Learning for Nonparametric Contextual Dynamic Pricing
have that
P(εB} {(Kmp) オ33ドス } }-2
(63)
and under the event EB,
T 0,
(64)
where T defined (11).
Proof. Fix time and a ball B E Ar, and recall its centre as c(B) = (z (B..PB)". By Step 8 of Algorithm 1, we have that UB supp(QQx). Note that n (D²) is sum of i..d zero mean Bernoulli random variables with
r(B), r(B)) (x(=B. E [DnE(DD)) =npu(B) npp (pp (DD
2
(x 2 ( Qx (x((xxx_r(B))) cQr (B) = 2Ccxmpr(B))dtt))
where the first inequality follows from Definition 2.4, the second inequality follows from Definition 2.3 and the third inequality follows from Assumption 2.2. As a consequence, by Chernoff's bound, we have that
P } =P (DP) < Cenpar(B)
<P B(n - >((">))))))))
sexpl - -4 C
Since r(B) 2 F, it holds that
P(E sexp { 4- Cgnn -4-CCgnn C 1
(65)
which completes the proof of (61).
Now we consider the case where
8 log (Km, d13 21887
273
4+3 nQQ < (kNp)
4+3+9
and
Cgxnp
Note that
r(B) 2 [8log (KAP)
0+349
{
8log (kMP)
>
(66)
Cexmp
Cgxup
Combining (65) and (66), it holds that
2+3 P(E < {kmpp)
24347 (x3p) }-2
which proves (63). Under the event Eg, we have that
w(B) n (D*)
log (Km,)) प+गF.
- CEKMpr (B)
r(B)2
log (mi))) x+3+7
1 - CekmpoAtHH
62
2
Css
243+7
(AΠ') शाड log 2137 (KT2p) 2ंड } -
(
8 82131
d+1+y
C
(kNP) पा3 log {KTH,) )
s+|+3
<1 21-21/2 22 1+3+7
(KΠP) ममें5 log {(T, . } <
#15
22

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Transfer Learning for Nonparametric Contextual Dynamic Pricing
where the first inequality follows from (12) and the event &g, the second inequality follows from r(B) f and the third inequality follows from (62). Thus by (11), we have under the event EB, T = O, which proves (64).
C. Proof of Theorem 4.4
The proof of Theorem 4.4 is in Appendix C.I with all necessary auxiliary results in Appendix C.2.
C.1. Proof of Theorem 4.4
Proof. Let F E 10, be specified later and define
Sx,F (I), ................................ .... ma) E X xi (k: 1/2)2F, & E (21/22]]]), e [d]]
with cardinality [1/(2)) jd. We define grid set of covariates
Sx
{ ( [1((2))1))
Next, define the integer m /F] for s constant 0 < Gm < 1/2. Using Varshamov-Gilberr bound (e.g. Lemma 2.9 in Tsybakov, 2009), we can find a set of well-separated vectors 2m = (w) M i= C [+1)" such that
m and p(w(1) wl)
m
log2 (M)
vo <i<js
(67)
where p(w,w") = [k& E [m]] wk # w H is the Hamming distance between and w'. In the same spirit of Sx,,, we define the grid set for price as
Sp, = (k- 1/2)F, [LL/FI]] =
= {ps pi...... ................................ Pivai
In the following proof, we first construct a collection Ho.,. = 110%, ,1, fr) | w E Im J of probability distributions, where Q% is a probability distribution over X' representing the covariate distribution target data, μM is a probability distribution over Z representing the joint distribution of covariate-price pairs in source data, and f Z + [0, is the reward function for both target and source data. We then verify the constructed Ha... C I(7, Cy, CLip. eq), from which the lower bound follows by applying Proposition C.I, which Proposition 1 in Kpotufe & Martinet (2020).
Step 1. Constructing the distribution collection Ho... 1" w E 2m
Constructing the target covariate distribution. Let the target covariate distribution be independent of w, with the density denoted by qx, defined for any x E X' as follows
91. if L Bx(2,,
qx (x) =
go.
i E X UR Bx
(68)
0, otherwise,
where
cQ Leb( Bx (x1,
mcai
41
and
90 Lob U Bx(xt,f))
with cQ defined in Assumption 2.2.
Constructing the source covariate-price pair distribution. independent of w. The density of the source covariate, denoted by px. is defined for any x X' follows
Let the distribution of the source covariate-price pairs be
if i E U Bx(a,, F/4), if x E uR JE F) Bx F/2),
5,
(x)
(69)
g0
if E u (x)
0,
otherwise,
23

---

Transfer Learning for Nonparametric Contextual Dynamic Pricing
where
8 == 1- qi (x(x) - LoLeb U (x(x),
U (xxx2,7))
Let S be defined later. The density of the conditional distribution of the source price p [0,1] given the source covariate x E X' is defined as
i E - otherwise.
pr(pl=)
(70)
1-fx I-F
Constructing the reward distributions. For both target and source data, let the random reward, conditional on the covariate I and price p. be a Bernoulli random variable with parameter (x,p). The reward functions { jwe are constructed as follow. First, define the function Φ: R4+ + [0, by
<0< 1/4, 2 4z, 1/ < 1/2,
1,
(62) =
0,
otherwise.
Next, define the function ip: Z [0, 1/4] via
((x, p) CoF _ l1////))
where Cp (CLip A 1)/4. For any w E 2n, the reward function f 2 [0, 1] is defined as follows
(z,p) 1/2 + Σ wipls 2; Bx(x),
(71)
By construction, we note that if (x] F/4) for some i [m], then f" p) only depends on the value of p and w; and if X U Bx F/2), ("((, = 1/2 for any p and
For any w E Im and & E X, p (x) min {p' E arg maxpe](,1] f (xx be the optimal price. Furthermore, recall that ğ E Sp,r, we define the function h: [0, 1] + {0, 13 as
h(p) 1 pp E (p //2
We now make the following remarks:
For any w E Pm, I E U. (x F/4) and E U!!/1 11/F (p: - 8/4.2t + F/44,
(z, (x)) - (I, = (h + h(p)}.
(72)
To this, first note that E Bx F/4) for some i [m]. If -1, we have (2) 0 & (p
+ and h(p*** (z)) = O. Therefore, f (x, (=.p (x)) (r, = C.r1(lp - p S F/2] = C,rh(p)
C.f1 (p # h(p)} where the first equality holds since p E Ull U1/F |p: - -/4, + f/4]. Next, if = we
have p (x) = -- F/4, and (x)) 1. Hence f (z. (0 - (= = C.fl1 - 1(pp p S fF/2]]
- C = F1 (1( (x)) + h(p))
For any w Pm. p E [0, 11, if x E U (x(2;,7/2),
(x, p) = 1/2.
(73)
For any Qm and X, if E (x, = 1/2.
(74)
24

---

Transfer Learning for Nonparametric Contextual Dynamic Pricing
For any two different w + w/ Qm if F//1) for some E [mm] such that wi # w/.
hxp"*(x)) # h(p=' (x))).
(75)
Step Verifying Ha... CI(T, Gy, N, Ctap: cQ).
Transfer exponent. Recall that the support of Qx is the union of the set U Bx(2,, and X u Bx 2x(;;;7 R).
Therefore, only need to check Definition 2.3 for in these sets.
First, i Bx(x; F/44 for some i [m]] and - 37/4 by (68), it holds that
Qx Qx = LIbcob n 8/44) (/4))
By (69), it follows that
Px (x((x(e,r) 20, Bx
2x, (B)
Second, for any x E X U Bx F) and r - 37/4, by (68), it holds that
Qx (x((x((,r)) = (x) U UT (x(x),)))
By (69), it holds that
(x(Bx(a,r) 2 (xx (x(2,,))) - Qx (Bx(2,))).
Thus, for small S 3F/4, the transfer exponent defined in Definition 2.3 equals y with respect to the constant 0 <, S 1.
Furthermore, since we set
Px Px(Bx(e(, -=(
Vi E [m),
verify that the above inequalities hold for 3F/4 <r 1. Thus, the transfer exponent of the constructed source covariate distribution with respect to the target covariate distribution is 7, with the corresponding constant Cy.
Exploration coefficient. For any a E X, r E (0,1) and [r, 1 r), we have that
m(w r. Bx(x,r)) 2 Px (xx((xx min { ( 1 1-F
1- FR = Px (B),
where the first inequality follows from (70) and the final equality holds beacuse
1- Fk K 5.
As result, the exploration coefficient defined in Definition 2.4 for the constructed source covariate-price pair distribution equals k.
Lipsehitz condition. By the construction in (71), it suffices to show that the function pp: Z [o, 1/4] satisfics Assump- tion 2.1 with respect to the Lipschitz constant CLip > For any (x, (a', ,p) 2, we have that
(p(x.p) - p((x',')) = C=Flø(1(=T,, .p) _)TT /P) <ll(<= (l)
(76)
Since & is a Lipschitz function with Lipschitz constant 4, it follows that
|4(x, p) - (pla..p)) <4Gxf|1(x>] .p) 1l//F 11(x "TaTTp''TTT 1bo///
<4C,11<x ,p) (a' .') lbo s Cupll(=) _p) -(xT pp) 1bo
25

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Transfer Learning for Nonparametric Contextual Dynamic Pricing
where the first inequality follows from the triangle inequality and the last inequality holds since C₁ (CLip/ A (1/4). Thus, for any Am. the reward function f satisfy Assumption 2.1 with respect the Lipschitz constant Cl.p 0.
Lower-bounded density. It is straightforward to see that
CQro Q1 (7/2)d 22cqq.
and
1 mcqi"
Cm// // 2 cq) Ccm/ (2))
j [Cm/Fi [ dpd (2 cQ) Coml /r] [ (2
qo m (27)
Thus, confirm that the constructed target covariate distribution satisfies Assumption 2.2.
Step 3. Applying Proposition C.1. We now verify the conditions required for applying Proposition C.I.
and for Step 3.1. Verification of condition (i) in Proposition C.1. For any admissible price policy ......., .PPi....PP... pig
1m let
( =
(X),)))))
where Ey, denotes the expectation under the distribution (Qx nº (u). Note that for any price policy æ, if there exists time point t' [[o]] such that
[L8J]
p
U l0 - -02-7/2pp 7/2.11 - F/4. F/44,
then there exists a policy n' with
[1/F]
Pe E U -/4, +/44.
and vi =p for all t + L', which satisfies
( S RI (ng); for any w E 9.n
To see this, suppose Pf E 1p: - p + - chooses pp 1oj- +7/4| for some # i; otherwise if w; = 1, then one chooses the same index i. Furthermore, if E X U (x f). then (x, 1/2 is universal for all p [0, 1] due (73).
- 7/4,p; + F/4], for some i E [LL/F]] If = -1, then one simply
[1/F1 Therefore, it suffices to consider price policies where p. E 1= 1 - 7/4, pt + F/4] for all t E [nq]. By (72), we have that
nQ
Ri
-Cif
B
(((pDi((x))) # (11)) E E Bx(,
RQQ
Cpp
Bes
1(h(p=**(00 # MbT) Bx(x) F/4))
Let II denote the space of all price policies. For any pd, p² E II, define the semi-metric
no
plp" ,p'') -C,F
(1h6o) + M60 (x(x2,
26

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Transfer Learning for Nonparametric Contextual Dynamic Pricing
For any w E Qms denote your
Eirer (X1), ................................ p p0 (Xma) (na)): }}. For any w, w' 22 we have that
n
m
CpP
Bus 1fh(p"00 (Xi)) # ((x))(X Bx(x2,
ni
Bes 1(x, Bx(o2
Cpconorati plw,w') 2 Coconoiatt 7d+1 m
8 n n
Coco
rult [cm cm/Fj/ Canor
(77)
where (a) holds by (75), (b) follows from (68), (c) is derived from (67), and Cp > 0 is constant only depending constant Cipy EQ and Cm.
Step 3.2. Verification of condition (ti) in Proposition C.1.
Fix policy n, for any u E [LT/r]],
n
O₂ = Σ 1p lpp - -f/2.00 + F/2/3.
(78)
Since
M [1/R]
Σ
E Eg (On) S Mnq.
follows that for at least [M/2] elements in (w) = there must exist at least one index u E [L1/F]] such that
2n [1/F]
(Ow) =
(79)
Therefore, we can choose one such and without loss of generality, assume that { satisfy (79). Define ý P to construct the density of the conditional distribution of the source price given the source covariate in (70) in Step 1.
Denote the target data and source data by Do = (X,,Pf Yoh DP = (XP, WE respectively. For each (+)) E Pm, let O' be the joint distribution of the random variables in D& and D induced by (Qx") ,1060) ,w(()) i.e. under O!:
the target data DN, follow the target covariate distribution Qx with density qx, the price policy is represented by its conditional density 7 F(-1) and rewards drawn from Bernoulli (p (X,,p7)), whose likelihood denoted by 0 p
the source data D follow the source covariate distribution Px (the marginal from ()) with density px, the source price distribution with conditional density denoted by " (-IX,) and rewards drawn from Bernoulli ((w)) (XP, ,f)) again with likelihood denoted by 0 (-)).
We denote by 0¹ the density corresponding O¹, then
no
PD 2 npン Duo) = II
ax (x.) X,, Ft- Xt,pi
nr
px (X (p | pwico (Y:P .p.)
(80)
27

---

Transfer Learning for Nonparametric Contextual Dynamic Pricing
Note that the densities involving the policy a (x; Fi-1) are identical since ι is fixed. For any i E [LM/2]]. and
w0))) E Pm
de' den (D
0 |- (Yr Xe,
8 8 (V :P.)
log
D
log
+
log
O yuicos (Yr X1, pi
1 (10000 (Y, XP ipp
1=1
Taking the expectation with respect to O¹, we obtain
ng
pulti (Ye X2 )
ap
fuitt) (X.') xfx .p!))
KL(O', 10
(0)
log
log
poios (Vr] Xe.Pf
polos (Y,P) xx =Pr
no
pulti (Yr) X2:PF
}
mp
0 p (X,²) X .ff
log
+ IE 1
log
pollo (Yi Xe.
0 foll (X, (x pff)
t=1
-KL Q + KI."
(81)
Note that
no Σ log
KL
Ewit
(x(x)d
x
-/ -/ Fx10
no
{pF E [pt 182-8/20 + log ( (Yelo, pf))
{
=
E
Bur
0 f0 (Yr p?) {
9x (x)dx
5
Σ
Qx
(O.)) (..,)(/))ELL...,XK.KKBBrrrrol,,, + CiFF), Bernoulli CiFF)
jE[m]: w o Aug
<3202r2
Σ
Qx
jE(m]: af" 20
<6AC
no CQ
m² d+2
< CQnatd+3 m,
(82)
[1/l]
where
the second equality follows from (68), (73), (74) and p(z) p
the first inequality follows from (71) and (78).
the second inequality follows from Lemma 15 in Cai et al. (2024a),
the third inequality follows from (68) and (79),
and Co > 0 is a constant only depending on constants Cpp and cQ.
We also have that
e pult))
(Y:") z.Pl()
KL P
E x
Σ
log
Px (x)dr
plios (Y: x,
{ 1 (Yo pt.)
Bua
(op E [vG F/2, + F/22] log
Pal (Y: 22. Px px(x))d
Ju (x(
SkrpF
Σ
Px (L)) 1/2 + CDFF, Bernoullil// 2 CxF))
je(m]ud
66)
(
<320 3 10
Σ
Px (Bx
jE(m)un #w
<32C <320%.occn (ec.snnnnpppppmmmmmm 73
(83)
where
28

---

Transfer Learning for Nonparametric Contextual Dynamic Pricing
the second equality follows from (69), (73) and (74),
the first inequality follows from (70) and (71),
the second inequality follows from Lemma 15 in Cai et al. (2024a),
the third inequality follows from (69),
and Cp 0 is constant only depending constants Cp., eq and Cy.
Combining (81), (82) and (83), it holds that
[1M/211
|M/2] Σ
K 0²) EConofett mn + CpKn pf m.
(84)
Now set
i= c. nQ (kΠP)
413
(85)
where & is sufficiently small constant. From (67), it holds that
[LM/22]
[M/2]
Σ
KL(O", 00) s (LM/2)).
where < a </1
Step 3.3. We have verified that all conditions of Proposition C.1 hold for the family (0:)!-m)) By applying Proposition C.I,
[M/2]
Markov's inequality, (77) and (85), we obtain that
( eng(ng (x ) गोड
inf sup Me
where is constant only depending on constants Crips by and cq. This completes the proof.
C.2. Auxiliary results
For completeness, we include Proposition from Kpotufe & Martinet (2020) below, it is a key tool for establishing the minimax lower bound.
Proposition C.1 (Proposition I in Kpotufe & Martinet (2020)). Let (Or)NE be family of distributions indexed by
subset H of a semi-metric space (F, p). Assume that there exists ho.... hM H. with M 2 2, such that
(1) p(hi, hy) 2s 0, 00 <i<jEM,
(it) Oh. & Ohp Vi E [M] and the average KL-divergence to Oh, satisfies
KL(O,,, On.)) S a log M, where 0 < 1/8.
M
Let ~ Ob and let h: Z + F denote any improper learner of h E H. It holds that
sup Or (p (f(Z)..)) 25) 1+ kEW
VM
2a log(M)
3 2/2
1
VM
1
8
29

---

Transfer Learning for Nonparametric Contextual Dynamic Pricing
D. Additional details in Section 5
D.I. Additional results Section 5.1
Scenario 1. The simulation results for Configuration 2 of Scenario 1, as described in Section 5.1, are presented in Figure 3.
Scenario 2.
The simulation results for Configurations and 2 of Scenario 2, described in Section 5.1, presented in Figure and Figure 5, respectively.
To assess robustness of TLDP, we conducted additional simulations under Configuration 1 in Scenario 1, using list of mis-specified values tuning parameters (v,, 7), while keeping the true values fixed at = 0.6 and 7 = 1. The results, presented in Figure 6, indicate that our method remains robust under moderate misspecification of these parameters.
A
B
400
Regret
Regret
Method (Panel
2000
300
Ta Target-Onni TLDP
1000
200
ABE
EXUCB
10000
20000
30000
40000
50000
10000
20000
30000
40000
ng
np
C
Y
0.5
1.5
2.5
0.25
0.5
0.75
1.25
190
500
180
170
Regret
400
Method (Panels
160
300
axis)
150
TLDP (varies w/ axis)
200
140
130
0.8
0.15
0.20
0.30
0.35
Figure 3. Results for Configuration 2 in Scenario 1. Panel (A) and (B): varying source data size up and target data size ngg respectively. Panel (C) varying the transfer exponent y (top axis) and the exploration coefficient x (bottom axis). Panel (D): varying the index constant Cy (top axis) and the exploration radius constant C, (bottom axis). For Panels (B), (C) and (D), we fix nq 10000.
D.2. Additional results in Section 5.2
In this subsection, we present samples of the processed auto loan dataset and the corresponding results in Tables 2 and respectively.
30

---

Transfer Learning for Nonparametric Contextual Dynamic Pricing
A
B
7500
1000
Regret
Regret
M Method (P (Panel A)
5000
TLDP
600
Target-Only TLDP
2500
ABE
400
ExUC8
10000
20000
30000
40000
50000
10000
20000
30000
40000
ngg
Пр
C
1,5
0.25
0.5
0.75
1.25
1600
1500
1250
Regret
1200
Regret
Method (P (Panels B, & D)
1000
TLDP (varies bettom axis)
750
TLDP (varies w/ top axis)
500
0.8
1.0
0.30
0.35
C,
Figure 4. Results for Configuration 1 in Scenario 2. Panel (A) and (B): varying source data size and target data size no. respectively.
Panel (C) varying the transfer exponent y (top axis) and the exploration coefficient (bottom axis). Panel (D): varying index constant C1 (top axis) and the exploration radius constant C, (bottom axis). For Panels (B), (C) and (D), we fix nq 10000.
B
5000
700
Regret
4000
Regret
M Method (P (Panel A)
600
3000
TLDP
Target-Only TLDP
2000
500
ABE
1000
ExUC8
10000
20000
30000
40000
50000
10000
20000
30000
40000
nig
Y
D
C
0.25
0.5
0.75
1
1.25
900
800
Regret
Regret
M Method (P [Panels B, & D)
TLDP (varies bettom axis)
700
700
TLDP (veries w/ top axis)
600
0.2
0.5
1.0
0.15
0.20
0.25
0.30
0.35
C,
Figure 5. Results for Configuration 2 in Scenario 2. Panel (A) and (B): varying source data size np and target data size no. respectively.
Pancl (C) varying the transfer exponent y (top axis) and the exploration coefficient (bottom axis). Panel (D): varying the index constant C₁ (top axis) and the exploration radius constant C₁ (bottom axis). For Panels (B). (C) and (D), we fix no = 10000.
31

---

Transfer Learning for Nonparametric Contextual Dynamic Pricing
150-
150-
Method
Melhod
TLOP
TLPP
100
Tages-OrlyyLLP
300
Tages-Orlyyl TLBP
250
100
Figure 6. Results for Configuration I in Scenario 2. Left panel: Varying the estimated with true fixed at K = 0.6. Right panel: Varying estimated y with true value fixed at r = both panels, the target data size is fixed at nq = 10000 source size mp 2ng, and we set the index constant Cj 1 and the exploration radius constant C, 1/4.
32

---

Transfer Learning for Nonparametric Contextual Dynamic Pricing
Table 2. Sample normalized auto loan dataset, including key covariates, prices and reward values.
0. 0.00000000
0.03455228
0. 0.00000000
0.00000000
0. 0.00000000
0,00000000
REWARD RA
1.0000000
0. 0.0000000
0 0.3333333
0.6666667
1.0000000
0.3333333
TERM
PRIME RATE
0.7396567
0.7 0.7396567
0.7396567
0.7396567
0.7396567
0.7396567
AMOUNT APPROVED
0.31578947
0. 1.1442105
0.2 4.2144407
0. 16207000
0.03204972
0.08330877
COMPETITION RATE
0.9421 0.9421965 1965
0.5664740
0. 0.8265896
0.7687861
0.9421965
0.8265896
0. 0.4007937
0. 0.591 1.5912698
0. 7.3022520
14227642
0.2490119
0.5731225
FICO
0.26497654
WEST SOUTH CENTRAL 0.03455226
EAST NORTH CENTRAL 328724
0.08203688
0.09374806
0.04983916
PRICE
WEST SOUTH CENTRAL
EAST NORTH CENTRAL
EAST SOUTH CENTRAL
EAST SOUTH CENTRAL
DIVISION
33

---

Transfer Learning for Nonparametric Contextual Dynamic Pricing
Table Results for auto loan dataset South Central as target division. Columns correspond to source divisions utilized in TLDP, Here, ne represents the number of the source data utilized in TLDP and denotes the total number of source observations. Each cell reports mean and standard deviation over 100 simulations.
METHODS
MOUNTAIN
EAST NORTH CENTRAL
WEST SOUTH CENTRAL
PACIFIC
ABE
70.81 (1.40)
70.81 (1.40)
70.81 (1.40)
70.81 (1.40)
EXUCB
64.91 (5.31)
64.91 (5.31)
64.91 (5.31)
64.91 (5.31)
TLDP(n = 0)
71.95 (2.87)
71.95 (2.87)
71.95 (2.87)
71.95 (2.87)
TLDP(n = 0.25n)
55.63 (8.04)
56.80 (8.59)
53.95 (8.26)
51.61 (8.89)
TLDP( 0.5m)
54.29 (7.06)
55.31 (8.60)
54.07 (7.44)
51.44 (7.12)
TLDP(np 0.75m)
51.38 (6.92)
51.52 (7.43)
50.67 (7.19)
49.40 (7.40)
TLDP(np n1)
50.23 (7.34)
51.92 (7.09)
49.85 (7.97)
48.72 (8.27)
34


Reviews:
[{"review_id": "EiegnCnygj", "reviewer": "Reviewer_pvZ8", "summary": "This paper introduces a novel Transfer Learning for Dynamic Pricing (TLDP) algorithm designed to effectively utilize pre-collected data from a source domain to improve pricing decisions in a target domain. The regret upper bound of TLDP is established under a straightforward Lipschitz condition on the reward function. To demonstrate the optimality of TLDP, we also derive a matching minimax lower bound, which encompasses the target-only scenario as a special case and is presented for the first time in the literature.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your appreciation, especially in acknowledging our presentation and our novelty.", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "Yes", "theoretical_claims": "No", "experimental_designs_or_analyses": "Yes", "relation_to_broader_scientific_literature": "This paper considers a transfer learning scenario in the dynamic pricing problem.", "essential_references_not_discussed": "No", "other_strengths_and_weaknesses": "1. The presentation is good. The authors clearly state their problem background as well as their contributions. \n2. The paper proposes a novel algorithm for transfer learning in dynamic pricing , and provide regret upper/lower bounds of the problem, which is shown to be minimax optimal.", "other_comments_or_suggestions": "-", "questions_for_authors": "-", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "CMU6MWmGEG", "reviewer": "Reviewer_USgz", "summary": "The paper studies contextual dynamic pricing with nonparametric demands, a critical application in revenue management. The authors consider how transfer learning techniques can be applied for this problem, and achieve minimax optimal regret by devising a provably optimal online dynamic pricing algorithm while also providing matching information-theoretic lower bounds. They show the regret exhibits a phase transition phenomenon: when the source data size is small, the optimal regret rate is the same as the case where only target data is used, and the rate will benefit a lot from transfer learning when the source data size is larger. The authors also conduct numerical experiments to study the empirical behavior of their TLDP algorithm.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the detailed and insightful comments. \n\n**Comparing our work with Cai et al (2024).** Cai et al (2024) study transfer learning for nonparametric contextual MAB under covariate shift. In their setting, actions (i.e. arms) are discrete and for simplicity are treated as a constant (i.e. the number of arms $K \\asymp 1$).  The simplification $K \\asymp 1$ results in their regret bound losing explicit control of $K$, making their approach inapplicable to infinite or uncountable action spaces. In contrast, dynamic pricing presents a unique challenge due to its continuous action space (i.e. prices), requiring more intricate methodologies than that in Cai et al (2024). Our approach effectively addresses this by adaptively partitioning the joint covariate-price space.  This adaptation ensures optimal regret scaling while accounting for both the covariates (of dimension $d$) and price (of dimension $1$). We will include the discussion in our revision.\n\n\n**Nonparametric assumption.** By not assuming a specific functional form for the demand curve, a nonparametric approach allows for more adaptability in capturing complex relationships between price and demand, especially when the true underlying demand structure is unknown or difficult to specify a priori. Therefore, compared with parametric model, the nonparametric model offers greater flexibility, which motivates our work. \n\n**Partially linear models in Bu et al (2022).** In terms of the partially linear model as you mentioned in Bu et al (2022), it covers two different demand models, $d(x,p)=bp+g(x)$ and $d(x,p)=f(p)+a^\\top x$. For the first one, we believe that similar techniques developed in this paper, namely the adaptive exploration/partition of the covariate space, could be particularly useful for handling the nonparametric part $g(x)$ of the demand function and its transfer learning. On the other hand, due to the linearity in price, we conjecture a local linear demand model should be fitted within each bin to leverage the structure of the problem and achieve the optimal regret (instead of a local constant revenue function as in our paper). Since there is no interaction between $p$ and $x$ and there is linearity structure in $p$, we believe sharper rates than the ones in our paper can be achieved.\n\nFor the second one, we conjecture a different upper bound strategy other than our adaptive exploration/partition should be used to leverage the linearity in $a^\\top x$ and the smoothness in the one-dimensional function $f(p)$. In addition, for the linear part, under the covariate shift framework, we believe that techniques developed in existing literature, such as those in Lei et al (2021) and He et al (2024), can be applied for its transfer learning.\n\nA thorough investigation of transfer learning under the partially linear demand models is indeed an interesting avenue for future research. Following your suggestion, we will discuss these aspects in the revision.\n\n**References**\n\n[1] Cai, Changxiao, T. Tony Cai, and Hongzhe Li. \"Transfer learning for contextual multi-armed bandits.\" The Annals of Statistics 52.1 (2024): 207-232.\n\n[2] Bu, Jinzhi, David Simchi-Levi, and Chonghuan Wang. \"Context-based dynamic pricing with partially linear demand model.\" Advances in Neural Information Processing Systems 35 (2022): 23780-23791.\n\n[3] Lei, Qi, Wei Hu, and Jason Lee. Near-optimal linear regression under distribution shift. In International Conference on Machine Learning, pages 6164–6174. PMLR, 2021.\n\n[4] He, Zelin, Ying Sun, and Runze Li. Transfusion: Covariate-shift robust transfer learning for high-dimensional\nregression. In International Conference on Artificial Intelligence and Statistics, pages 703–711. PMLR, 2024", "claims_and_evidence": "Yes. The paper's contributions are mainly on the theoretical side, and all assumptions are clearly stated and all claims are rigorously proved.", "methods_and_evaluation_criteria": "The experiment section is easy to follow. The benchmark selection of the ABE and the ExUCB algorithms is reasonable.", "theoretical_claims": "I did not check the proofs carefully, but I did follow the main steps regarding the minimax optimality of regret. The proofs seem to be well-written.", "experimental_designs_or_analyses": "The experimental results look sound.", "relation_to_broader_scientific_literature": "The authors are the first to present a theoretical characterization of the benefits of applying transfer learning to the nonparametric contextual dynamic pricing setting.", "essential_references_not_discussed": "None.", "other_strengths_and_weaknesses": "The strength of the paper lies in the strong technical content involved in characterizing the minimax optimal regret for nonparametric contextual dynamic pricing with transfer learning. The analysis is quite skillful and the obtained results are strong.\n\nIn particular, the paper can be seen as an extension of [1], to a contextual bandit with continuous actions (specialized to pricing), since the major setup and assumptions are the same. A minor suggestion is that the authors need to comment on the additional technical challenges of analyzing the discretized version of the algorithm in [1] for the dynamic pricing setting (lack of such discussion is a current weakness of the paper).\n\n[1] Cai, Changxiao, T. Tony Cai, and Hongzhe Li. \"Transfer learning for contextual multi-armed bandits.\" The Annals of Statistics 52.1 (2024): 207-232.", "other_comments_or_suggestions": "None.", "questions_for_authors": "It is more common for researchers to study parametric demand models in dynamic pricing. Can you comment on why they solve the non-parametric problem first while not look into the parametric problem? In particular, how will your result look like or what are the challenges of characterizing the benefits of transfer learning for contextual dynamic pricing with (partially) linear demands (as in [1])?\n\n[1] Bu, Jinzhi, David Simchi-Levi, and Chonghuan Wang. \"Context-based dynamic pricing with partially linear demand model.\" Advances in Neural Information Processing Systems 35 (2022): 23780-23791.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "lSFkyd0y9n", "reviewer": "Reviewer_WLhP", "summary": "The authors study the problem of transfer learning in the context of dynamic pricing. Given a pre-collected source dataset, the authors propose an algorithm that exploits such a dataset to learn a partitioning of the joint context-price space in order to propose a price for each user with the goal of maximizing the cumulative revenue. The authors also study a lower bound of the problem and derive, as a consequence, both an upper and a lower bound in the case in which no source dataset is available.\n\n---\n\nPost rebuttal: The authors adequately answer my questions. Given that, I increased my score from 2 to 3.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the detailed and helpful comments. The additional simulation results are given in this anonymized link. https://docs.google.com/document/d/e/2PACX-1vRBfGzJo3ETCltTWfOi_0p4RjLbsUJo6g9z0J-Ckm2m6fL0fahJWSrrptiFwOGCyxhtHNuyHsQP0tOh/pub\n\n**W1)** The classic $0$ or $p$ revenue model is a special case of our model. Formally, revenue $Y$ is $p\\cdot D(x,p)$, where $D(x,p)$ is the random demand given context $x$ and price $p$ with $\\mathbb E(D(x,p))=d(x,p)$. This leads to $\\mathbb E(Y)=f(x, p) = p d(x, p)$. Thus we can directly model $f(x,p)$. When $D(x,p)$ is Bernoulli, it gives the 0 or $p$ revenue. However, demand $D(x,p)$ can be continuous (e.g. when sold by weight/volume like rice/gas). Also, we clarify in simulation scenario 2, we use [0,1]-truncated Gaussian centred at $f(x, p)$.\n\n**W2)&3)** Covariate shift is a widely used scheme in transfer learning (Suk&Kpotufe2021, Cai et al 2024).  The context $X$ can encode product specifications and consumer features, thus allowing revenue function to remain roughly stable across products or markets when conditioned on a rich $X$. In Section 5.2, loans differ in their terms (e.g. duration). The source and target data differ in market conditions (i.e. different US regions). Despite the discrepancies, our method still improves performance, showing its robustness.\n\nWe further test robustness under posterior drift. Following the setup of Configuration 1 in Scenario 1, we modify the true source revenue function as $f(x, p) = \\theta_0' + x^{\\top} \\theta' p + \\tilde{\\theta}' p^2,$ where $\\theta_0' \\sim \\mathcal{N}(\\theta_0, \\sigma^2)$, $\\theta' \\sim \\mathcal{N}(\\theta, \\sigma^2 I_d)$, and $\\tilde{\\theta}' \\sim \\mathcal{N}(\\tilde{\\theta}, \\sigma^2)$, with $\\theta_0, \\theta, \\tilde{\\theta}$ being the target revenue parameters. We set $\\sigma = r \\cdot \\min(|\\theta_0|, \\|\\theta\\|_\\infty, |\\tilde{\\theta}|)$ and control the drift severity by varying $r$. Results (Figure 1 in the link above) show our method performs well and is robust under moderate posterior drift.\n\n**Q1)** The independence assumption (i.e. source data generated by a fixed behavior policy) is a standard assumption in bandit and RL literature, which simplifies theoretical analysis by enabling concentration inequalities for independent data. Our analysis can be extended to allow temporally dependent observations, such as those satisfying mixing conditions. Note that one must impose some structural conditions on the adaptivity of source data, as without them, data with arbitrary dependence can be uninformative. It would be interesting to allow source data to be collected via an adaptive policy. Doing so introduces additional complexities in terms of dependence structures and requires careful handling in the analysis to ensure valid results. We leave this for future research.\n\n**Q2)** The constant $C_r$ is used to define the smallest exploration radius $\\tilde{r}$, which intuitively controls a bias-variance trade-off: $\\tilde{r}$ small enough to yield fine local estimates v.s. large enough to avoid excessive partitioning and exploration. We do not have a single optimal value for $C_r$ in closed form as it depends on unknown quantities (e.g. $c_Q$ and $c_{\\gamma}$). However, any constant $C_r$ in a range that satisfies the bounding inequalities leads to optimal regret (in order $O(\\cdot)$). This is inline with bandit and RL literature, where one often settles for an order-wise optimal choice rather than a numerically unique constant.\n\n**Q3)** You are correct that $Q_X$ is a probability distribution, so Assumption 2.2 implies $c_Q\\leq 1$. The constant 8 in $C_r^4c_{\\gamma}c_Q\\geq 8$ is merely sufficient for technical simplicity, and by no means necessary and sharp. While we conjecture that it may be technically possible to sharpen $8$ to some smaller constant to cover the choice of our experimental design, the magnitude of these constants does not affect the order of regret. In fact, our experiment shows our method is robust when the sufficient condition $C_r^4c_{\\gamma}c_Q\\geq 8$ is violated.\n\n**Q4)** Misspecification of $\\gamma$ and $\\kappa$ affects the optimal choice of $\\tilde{r}$, and consequently, the regret bound, as discussed in Remark 4.2. In practice, $\\kappa$ can be estimated from source data, while $\\gamma$ can be estimated by measuring the empirical overlap between the source and target covariate distributions using a small portion of target data (e.g., $\\sqrt{n_{Q}}$ samples). In our experiment, we use true $\\kappa$ and $\\gamma$ to compute $\\tilde{r}$. To assess robustness, we conduct additional simulations under Configuration 1 in Scenario 1, using a list of mis-specified values of $(\\kappa, \\gamma)$, while fixing true values at $\\kappa = 0.6$ and $\\gamma = 1$. Results (Figure 2 in the link above) indicate our method remains robust under moderate misspecification of these parameters.\n\nWe hope our responses help resolve your concerns and positively influence your evaluation.", "claims_and_evidence": "The theorems proposed by the authors are supported by proofs. However, my concern is with respect to the assumptions required for the theorems to hold, as I will discuss later.", "methods_and_evaluation_criteria": "The authors propose both a synthetic experiment and an experiment on real data. Regarding the synthetic experiment (i.e., Section 5.1), the simulation setting is crafted to match the assumptions required by the algorithm, which do not seem likely to hold in a real-world scenario, thus subtracting from the representativeness of the experiment. The experiment on real data (i.e., Section 5.2), on the other hand, seems to be much more promising, however some concerns remain, which will be discussed later.", "theoretical_claims": "I have not thoroughly checked the proof in the appendix.", "experimental_designs_or_analyses": "I am not completely convinced by the validity of the experiments, but I am looking forward to the discussion with the authors to confirm or clear my concerns.", "relation_to_broader_scientific_literature": "This work tackles an interesting problem, namely trying to exploit previous data to reduce the number of samples required to learn the pricing strategy of a new product, however, to the best of my understanding, some peculiarities of the specific setting have not been correctly addressed.", "essential_references_not_discussed": "The references cited in this paper are sufficient to understand the motivation and the contributions of this work. To the best of my knowledge, no further references are necessary.", "other_strengths_and_weaknesses": "**Strengths**  \nThis work in written in a clear and understandable way, guiding the reader through the authors' reasoning. Also, I believe the problem tackled by the authors to be of practical importance.\n  \n**Weaknesses**  \nMy main concerns are on the problem formulation and the assumptions made in this work.  \nW1) The formulation of the random revenue seems to be incorrect. Although I understand the provenance of such a formulation, I believe it does not fit in the dynamic pricing setting, as the seller either observes a revenue of 0 (i.e., the customer does not buy the product) or of $p_t$ (i.e., the customer buys the product), and not a random revenue which can, in general, be in $[0,1]$. The simulation in Section 5.1 shows that a Gaussian is used to model the random revenue, which again does not seem correct.  \nW2) The assumption in Equation (3) (i.e., that the unknown reward functions are identical between the source problem and the target problem) seems to me as too strong, and unverifiable in the majority of applications. Such an assumption would require that every possible customer has the same propension to buy the two (in general, different) products for all the prices in the range. This is not in general true for different products and/or for different markets. The real-world experiment in Section 5.2 works well since the same product (i.e., loans) in the same market (i.e., the USA) is considered. However, I find this assumption very limiting.  \nW3) Considering weakness W2), it seems to me that the alternative setting (i.e., posterior drift) could have been a more applicable approach.\n\nThe weaknesses stated above have guided my judgement, however, in case the authors address them satisfactorily, I could reconsider my evaluation.", "other_comments_or_suggestions": "Here I list some minor modifications that I believe could increase the readability of the work:\n\n1. Line 57, right column, $\\mathcal{Z} = \\mathcal{X} \\times \\mathcal{P}$ is the more unambiguous definition of $\\mathcal{Z}$;\n2. The authors should formalize what is $Q_X (B)$ when $B$ is a ball (see Assumption 2.2);\n3. Although understandable from the text, Algorithm 1 is demanding to read, and I would suggest the authors to write a higher-level algorithm in the main paper, reviewing and rewriting Algorithm 1 in the appendix;\n4. In Theorem 4.1, it is not clear what the authors mean by writing \"Assume that the target dataset, defined in (1)\", as Equation (1) represents the expected reward.", "questions_for_authors": "I would like the authors to answer the following questions:  \nQ1) In Theorem 4.1, the authors write that the source dataset comprises \"triplets independent across time\". Does this mean that a dataset obtained from the observations collected in a different dynamic pricing application would not work, since each price would then be dependent on past observations? If yes, does removing such an assumption have an impact on the resulting regret bound?  \nQ2) In lines 264-266, the authors write \"The choice of $\\widetilde{r}$ in (13) ...\". However, since $\\widetilde{r}$ depends on $C_r$, and $C_r$ is, in turn, only defined in terms of two inequalities, is there a way to define a single optimal value of $C_r$?  \nQ3) From my understanding, $Q_X$ is a probability distribution, and as such $Q_X (\\cdot) \\in [0,1]$. Thus, following Assumption 2.2, I infer that $0 < c_Q < C_Q <= 1$. However, from the experiment in Section 5.1, at line 305, right column, a choice of $C_r = 1/4$ together with the conditions of line 247, left column, would result in $c_Q \\ge 2048$. Can the authors clarify this point?  \nQ4) $P_X$ and $Q_X$ are not known in a real-world scenario, and $\\gamma$ and $\\kappa$ depend on these distributions. The value of $\\widetilde{r}$, in turn, depends on $\\gamma$ and $\\kappa$. Given these observations and what the authors write in the conclusions, what could be the impact of misspecification, or of an interactive refinement, of these values on the regret? Moreover, what values have the authors considered for $\\gamma$ and $\\kappa$ in the experiment in Section 5.2?\n\nFinally, I ask the authors to address my comments from the \"Weaknesses\" section of the review.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "J2itH0Ui9K", "reviewer": "Reviewer_uzaF", "summary": "The paper studies dynamic pricing problems using transfer learning techniques. The objective is to maximize expected total rewards and to minimize regret. The problem setup is stylized (monopoly scenario, time homogeneous demand), yet, reasonable. Numerical experiments are performed and results are compared to other baseline solutions. Overall, the paper is well-executed.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the detailed and helpful comments. The additional simulation results are given in this anonymized link. https://docs.google.com/document/d/e/2PACX-1vRBfGzJo3ETCltTWfOi_0p4RjLbsUJo6g9z0J-Ckm2m6fL0fahJWSrrptiFwOGCyxhtHNuyHsQP0tOh/pub\n\n**Q1** Our theoretical guarantees hold for any $d$.  Like most nonparametric works, however, our approach does encounter curse of dimensionality in high-dimensional settings, which can be seen from its regret upper bound. To mitigate this, one can impose additional structures on the reward function $f(x,p)$, where $x=(x^{(1)}, ..., x^{(d)})\\in \\mathbb R^d$ is the context. For example, one can adopt a sparse additive model $f(x,p)=\\sum_{i=1}^d g_i(x^{(i)}, p)$, with sparsity $s \\ll d$ (i.e. most $g_i(\\cdot,\\cdot)$ are zero functions). In this setting, the effective dimension is $s$ rather than $d$, albeit at the potential cost of model misspecification. Following your comment, we will provide a detailed discussion on this aspect.\n\n**Q2** First, we note that the context $X$ can encode useful information such as product specifications and consumer features, thus allowing the revenue function $f(x,p)$ to remain roughly stable across different products conditioned on a rich $X=x$. For example, in our real data experiments in Section 5.2, loans differ in their terms (e.g. duration). The source and target data differ in market conditions (i.e. different US regions). Despite these discrepancies, our method still improves performance, showing its robustness. Second, we further test robustness of our method under posterior drift (i.e. when revenue functions are different between source and target). Following the setup of Configuration 1 in Scenario 1, we modify the true source revenue function as $f(x, p) = \\theta_0' + x^{\\top} \\theta' p + \\tilde{\\theta}' p^2,$ where $\\theta_0' \\sim \\mathcal{N}(\\theta_0, \\sigma^2)$, $\\theta' \\sim \\mathcal{N}(\\theta, \\sigma^2 I_d)$, and $\\tilde{\\theta}' \\sim \\mathcal{N}(\\tilde{\\theta}, \\sigma^2)$, with $\\theta_0, \\theta, \\tilde{\\theta}$ being the target revenue function parameters. We set $\\sigma = r \\cdot \\min(|\\theta_0|, \\|\\theta\\|_\\infty, |\\tilde{\\theta}|)$ and control the drift severity by varying $r$. Results (Figure 1 in the link above) show our method performs well and is robust under moderate posterior drift.\n\n**Q3** One key hyperparameter is the smallest exploration radius $\\tilde{r}$. While the regret bound remains valid for any choice of $\\tilde{r}$ as discussed in Remark 4.2, the optimality of our TLDP algorithm relies on the specific $\\tilde{r}$ given in equation (13). That choice depends on two parameters often unknown in practice: the transfer exponent $\\gamma$ and the exploration coefficient $\\kappa$.  In practice, $\\kappa$ can be estimated from the source data, while $\\gamma$ can be estimated by measuring the empirical overlap between the source and target covariate distributions using a small portion of the target data (e.g. $\\sqrt{n_Q}$ samples). In our synthetic experiments, we used the true values of $\\kappa$ and $\\gamma$ to compute $\\tilde{r}$. To assess robustness, we have conducted additional simulations under Configuration 1 in Scenario 1, using a list of mis-specified values of $(\\kappa,\\gamma)$, while keeping the true values fixed at $\\kappa = 0.6$ and $\\gamma = 1$. Results (Figure 2 in the link above) indicate that our method remains robust under moderate misspecification of these parameters.\n\n**Q4** As discussed in our response to your previous question, one limitation would be that to achieve optimal regret, our approach requires the knowledge of the unknown quantities $(\\kappa, \\gamma)$. It would be interesting to rigorously extend our method such that it is adaptive to the knowledge of $(\\kappa, \\gamma)$ and further establish theoretical guarantees for such an algorithm. In the revision, we will discuss more on this aspect of our limitation. \n\n**Q5** If the demand (thus revenue) function depends explicitly on time (e.g. seasonality, holiday, weekly effects), we can include time as an additional dimension in the context space. Our partition-based procedure then proceeds similarly and the theoretical guarantees still hold. However, if the revenue function $f(x,p)$ changes over time in a rather arbitrary fashion, we then need to rely on additional procedures such as change-point detection or moving-window estimation to counter the non-stationarity. We conjecture the regret upper bound will inevitably inflate in this case. As for inventory consideration, we conjecture one needs to reformulate the revenue maximization problem into a constrained optimization due to inventory limit. Based on deterministic approximation, Wang et al 2025 (on dynamic pricing with covariates) proposes a dual-based method for dynamic pricing under parametric models. We believe similar ideas can be used to help solve price $p_t$ in our Algorithm 1 under inventory constraint. We will comment on these aspects in the revision.", "claims_and_evidence": "The claims of the paper appear valid and are supported by the numerical evaluation.", "methods_and_evaluation_criteria": "The methods used are suitable and interesting. The regret criterion makes sense. The presentation of the concepts used is dense in some places but overall ok.", "theoretical_claims": "The theoretical claims and proofs appear solid. However, I did not check all proofs provided in the Appendix in full detail.", "experimental_designs_or_analyses": "The experimental design is reasonable. Small synthetic examples as well as real-world data based experiments are performed. Results are compared against other state-of-the-art baselines. Results are discussed and insights are inferred.", "relation_to_broader_scientific_literature": "The literature using related methodologies is sufficiently discussed.", "essential_references_not_discussed": "I do not miss essential references.", "other_strengths_and_weaknesses": "Strengths:\n\n- The paper derives analytical results (theoretical bounds).\n\n- The provided experiments are convincing.\n\nWeaknesses:\n\n- Limitations, scalability, and required parameter tuning of the approach could be better discussed.", "other_comments_or_suggestions": "I suggest to better discuss limitations, scalability, and required parameter tuning of the approach.", "questions_for_authors": "I have the following questions:\n\n(1) Does the approach scale for larger d?\n\n(2) What happens if the other product is not similar?\n\n(3) Which hyperparameters have to be tuned?\n\n(4) What are the limitations of the approach?\n\n(5) Would it be possible to extend the setup to time-dependent demand and/or inventory considerations – as typical for e.g., airline ticket market, accommodation, etc.? Please discuss.\n\n\nAfter reading the other reviews and the rebuttal, I decided to keep my score.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This work studies the contextual dynamic pricing problem with binary, unknown- and unparametric-distributional demands. Motivated by utilizing the sales history of other related products as a “warm start”, the authors consider a transfer learning framework for dynamic pricing on a covariate shift situation. Their TLDP algorithm achieves the minimax regret rate (as they also prove a matching lower bound), which also implies a two-stage phenomenon: as the source data size goes larger, the regret will switch from a cold-start rate ($n^{\frac{d+2}{d+3}}$) to a much smaller rate that benefits more from the source dataset. The authors conduct numerical experiments by comparing (and outperforming) TLDP with existing ABE and Explore-then-UCB algorithms, as well as demonstrating the influence of $n_p$, $\kappa$, and $C_r$ on the regret.

From the technical perspective, this work is an extension of Cai et al. (2024). Although the major setup and assumptions in this paper are the same as Cai et al. (2024), the extension to a contextual continuum-armed bandits is not that trivial. Given the overall quality and contributions, this paper is worth accepting.

However, there are still a few concerns that we kindly ask the reviewers to pay attention to:

(1) The assumption that the source and target demand functions are identical. This is not trivial nor granted even with covariates shifting being allowed. The authors should include a serious discussion with rigorous analysis to demonstrate the adaptivity (as you have conducted in your rebuttal but we expect a more comprehensive version).
(2) The $n^{\frac{d+2}{d+3}}$ regret bound, or a more general version like $n^{\frac{k+d+1}{2k+d+1}}$ for k-th smooth demands, is well-expected after Kleinberg (2004), Wang et al. (2021), and Cai and Pu (2022), as long as you are adapting the bandits setting to pricing. The authors are encouraged to consider an extension to a more general continuous family.

Reference:

Cai, Changxiao, T. Tony Cai, and Hongzhe Li. "Transfer learning for contextual multi-armed bandits." The Annals of Statistics 52.1 (2024): 207-232.
Kleinberg, Robert. "Nearly tight bounds for the continuum-armed bandit problem." Advances in Neural Information Processing Systems 17 (2004).
Wang, Yining, Boxiao Chen, and David Simchi-Levi. "Multimodal dynamic pricing." Management Science 67.10 (2021): 6136-6152.
Cai, T. Tony, and Hongming Pu. "Stochastic continuum-armed bandits with additive models: Minimax regrets and adaptive algorithm." The Annals of Statistics 50.4 (2022): 2179-2204.
Decision: Accept (poster)