Transfer Faster, Price Smarter: Minimax Dynamic Pricing under Cross-Market Preference Shift
Authors: Yi Zhang
Elynn Chen
Yujun Yan
Venue: NeurIPS 2025 spotlight ()
Keywords: Dynamic Pricing
Transfer Learning
Random Utility Models
Posterior Shift
RKHS
Primary area: general_machine_learning

Abstract: We study contextual dynamic pricing when a target market can leverage $K$ auxiliary markets—offline logs or concurrent streams—whose *mean utilities differ by a structured preference shift*. We propose *Cross-Market Transfer Dynamic Pricing (CM-TDP)*, the first algorithm that *provably* handles such model-shift transfer and delivers minimax-optimal regret for *both* linear and non-parametric utility models.

For linear utilities of dimension $d$, where the *difference* between source- and target-task coefficients is $s_{0}$-sparse, CM-TDP attains regret $\tilde{\mathcal{O}}\bigl((dK^{-1}+s_{0})\log T\bigr)$. For nonlinear demand residing in a reproducing kernel Hilbert space with effective dimension $\alpha$, complexity $\beta$ and task-similarity parameter $H$, the regret becomes $\tilde{\mathcal{O}}\bigl(K^{-2\alpha\beta/(2\alpha\beta+1)}T^{1/(2\alpha\beta+1)} + H^{2/(2\alpha+1)}T^{1/(2\alpha+1)}\bigr)$, matching information-theoretic lower bounds up to logarithmic factors. The RKHS bound is the first of its kind for transfer pricing and is of independent interest.

Extensive simulations show up to 38\% higher cumulative revenue and $6\times$ faster convergence relative to single-market pricing baselines. By bridging transfer learning, robust aggregation, and revenue optimization, CM-TDP moves toward pricing systems that *transfer faster, price smarter*.

Transfer Faster, Price Smarter: Minimax Dynamic Pricing under Cross-Market Preference Shift
Yi Zhang Columbia University
New York, NY 10027
y251950columbia.com edu
Elynn Chen
Yujun Yan
New York University
Dartmouth College Hanover, NH 03755
New York, NY 10003
elynn. chenmccce@@y..c.m edu
wjjun.yanndaartmouthheee
Abstract
We study contextual dynamic pricing when a target market can leverage auxiliary markets-oflline logs concurrent streams-whoss mean utilities differ by structured preference shift. We propose Cross-Market Transfer Dynamic Pricing (CM-TDP), the first algorithm that provably handles such model-shift transfer and delivers minimax-optimal regret for both linear and nonparametric utility models. For linear utilities of dimension d, where the difference between source- and target- task coefficients is So-Sparse, CM-TDP attains regret ((KK-1 so) log 7). For nonlinear demand residing reproducing kernel Hilbert space with effective dimension a, complexity 3 and task-similarity parameter H, the regret becomes (11((22+()/1((22+1))))) matching information- theoretic lower bounds up to logarithmic factors. The RKHS bound is the first of its kind for transfer pricing and is of independent interest.
(Kk-23/2aa++11) 1)11/(22+++)
Extensive simulations show up to 50% lower cumulative regret and 5x faster learning relative to single-marker pricing baselines. By bridging transfer learning, robust aggregation, and revenue optimization, CM-TDP moves toward pricing systems that transfer faster, price smarter.
Introduction
Dynamic pricing is now core operational tool for ride-sharing platforms, airlines, and large commerce retailers. State-of-the-arr single-market algorithms learn a demand model from scratch and achieve minimax regret when sufficient data accumulate [19, 28, 9]. In practice, however, many markets launch with only dozens of transactions per day, while mature markets of the same firm collect data prddrs--f-mmanniuud higher rates. Transferring information from data-rich to data-poor markets is therefore essential for fast revenue convergence and early-stage pricing accuracy.
Industry practice gives rise to distinct transfer regimes. First, in the Offtine-to-Onlin (020.m) setting, the firm holds a fixed log of source-marke data gathered before the target market opens, and this static information is used once the target goes live. Second, in the Online-to-Onling (O20.n)) setting, the source and target markets operate concurrently; streaming data from large markets must be incorporated into the pricing decisions of small markets in real time.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Existing transfer approaches do not fully address these settings. Meta- dieta-dynamic pricing [4] learns shared Bayesian prior but requires directly observed linear demands and only exploits offline data. TLDP [30] handles covariate (domain) shift from a single offline source but assumes that the reward model is identical across markets. Bandit-transfer methods focus on either covariate shift [5] or sparse parameter heterogeneity [32, 18), yet they do not incorporate revenue-mane maximising price choice.
This paper. We propose CM-TDP (Cross-Market Transfer Dynamic Pricing), a unitied framework that (i) operates in both ar and o20,n regimes, (ii) accommodates linear and RKHS-smooth nonparametric utilities, and (iii) allows multiple source markets whose mean utilities differ from the target by a structured utilliyy model shift. CM-TDP alternates a bias-corrected aggregation step with an optimistic pricing rule, thereby transferring knowledge while balancing exploration and exploitation.
To the best of our knowledge, this work establishes the first rigorous regret analysis for transfer learning under general utility discrepancies between markets. A primary difficulty that arose during our analysis was maintaining tight control of error propagation: conventional techniques would accumulate slack and inflate constant-order terms into non-negligible O(7') factors, rendering the bounds both theoretically and practically uninformative. Our main contributions are as follows:
(C1) Unified transfer pricing framework under utility shifts. CM-TDP is the first dynamic pricing framework that allows multiple source markets whose utilities differ from the target by a structured shift, working in both O20 oft and O20 regimes.
(C2) Minimax-optimal guarantees for two utility classes. We prove (i) Ö log + so log T)
regret under linear mean utilities and (ii) the first transfer-pricing bound O(K 20641 T 22
2n3
H 20:1 T T for RKHS-smoodl utilities- matching known lower bounds.
(C3) Bias-corrected aggregation architecture. Our two-step aggregate debias pipeline cleanly connects meta-learning (prior pooling), robust statistics (trimmed debiasing), and exploration-driven bandits, and can plug in MLE, Lasso, kernel ridge as well as black boxes.
(C4) Large empirical gains. Simulations show up to 50 % lower cumulative regret, 28% lower standard error and 5x faster learning relative to single-market pricing [19], with the largest gains in data-scarce targets under 020... transfer.
Organization. Section introduces the marki-market dynamic pricing problem with transfer learning under random utility models. Sections 4-6 present CM-TDP and its theoretical analysis. Section 7 reports empirical results, and Section 8 outlines future work.
2 Related Work
Single-mane margle-market contextual pricing. Early algorithms assume deterministic valuation map. typically linear, and achieve sub-linear regret [2, 13, 20]. with nonparametric variants studied in [22]. The modern benchmark is the random utility model in which valuation equals doveriate-dependent mean plus i..dd noise. When the noise distribution is known, [19] establish the first regret bounds; subsequent work removes that knowledge via doubly robust moment-matchinn estimators while retaining linearity [31, 21, 16, 33]. [28] close the gap to the fformation-thhorette optimum for linear utilities and extend the analysis to Hölder-smood demand curves, whereas [9] give fully nonparametric guarantees. All of these methods relearn from scratch in every market, degrading performance when target data limited.
Transfer and meta- learning for pricing. Meta Dynamic Pricing pools directly observed lincar demands across products and learns a shared Gaussian prior [4]. TLDP transfers under pure covariate (domain) shift but only from a single offline source [30]. Our Cross-Market Transfer Dynamic Pricing (CM-TDP) differs by coping with utility-model shift, supporting multiple onlinc/oftline sources, and providing guarantees for both linear and RKHS utilities.
Multitask contextual bandits and reinforcement learning. Domain-shif, transfer for bandits is analysed [5], whereas sparse heterogeneity is addressed via SaaneeddmmanLLAASSSS debiasing [32] or weighted-median MOLAR [18]. Causal-transpon ideas reveal negative-transfer risks [15]. Reward- and transition-level transfer and meta learning in RL explored by [12, 10, 7, 8, 34]. We adapt these bias-correction techniques to revenue maximisation under shifting utilities.

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Fully online meta-learning without task boundaries. FOML [23] and Online-within-Oniee meta- learning [14] operate on single stream of data without explicit task resets. CM-TDP follows the same streaming paradigm but must balance exploration and exploitation through posted prices rather than prediction losses.
Positioning of this work. CM-TDP is the first dynamic pricing framework that (i) transfers across multiple auxiliary markets under utility-model shift, (ii) achieves minimax regret for both linear and RKHS utilities, and (iii) unifies bias-corrected aggregation with revenue-maximising price selection, thereby bridging single-maris pricing [28], offline meta- veta-prioos [4, 11], and multitask bandits [18].
Key distinctions from prior work. Unlike Meta-DP [4] and TLDP [30], CM-TDP (i) handles concurrent source streams, (ii) tolerates utility shift rather than merely covariate shift, and (iii) supplies the first nonparametric (RKHS) transfer-pricing regret bound. Multitask-bandii methods such MOLAR [18] focus prediction error and linear bandits, do not optimize posted prices, and therefore cannot exploit revenue structure. Consequently, existing approaches cannot deliver the minimax-optima. guarantees empirical gains demonstrated by CM-TDP.
3 Problem Formulation
We consider a pricing model for the target market where products are sold one at a time, and only binary response indicating success or failure of sale is observed. For each decision point E [T]. the market value of the product at time t depends on the observed contextual information x; A general random utility model for the market value of the product is given by
= g(0) (x) (0)
+ Es,
(1)
where (-)) E G is the unknown function of the mean utility in the target market, and e, are
i..d. noises following an known distribution F(.) with E[er] (0) (v: 2 p.) that indicates
and support S == Be, Be].
Given posted price of pt for the product time t, we observe y/2
whether a sale occurs (yz
(0) 1) or not (yz 0). The model is equivalent to the probabilistic model:
0, with probability F(p:
(0)
1/1 1,
with probability
1 F(: - g( (a
Therefore, given a posted price pt. the expected revenue from the target market at time t conditioned on x; (0) is
rev (0) (p:) : (1- - (1- (z,))) 1))
The oracle optimal offered price P is defined by
Pt (0)
arg max pt[1 F(: /(0) (x. (0))
(3)
P.20
and hence, under Assumption 1, we have
h(u) = u o (-u) and o(u) 1 - F(u)
Pt (0) h o ( (xx), where
(4)
F'(u)
Assumption 1 (Regularity condition). There exists positive constants Ly and B,, such that ø' 2 Los for all u E [-Bu, Bu], and inf j. d'(u) 2 1.
Assumption guarantees the uniqueness of the optimal solution of (3). The restriction of Ly 2 1 is commonly used in dynamic pricing study [19].
Optimal policy and regret. For policy π that sets price pt at t, its regret over the time horizon of T' is defined as
T
Regret(T;T) = Σ B poj 1 2 pf) p. [(v 2 pc)
[rev:(p)) rev)(pt)].
The goal of a decision maker is to design a pricing policy that minimizes Regret (T;nett(TTr))))) or equivalently. maximize the collected expected revenue ΣT= (p.)
3

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Cross-Market Transfer Learning. In the context of cross morsett transfer learning, we observe additional samples from K sources markets indexed by superseript (A) for k E [K]. The observed market covariates and response, latent utility and the unknown mean utility functions for each source market are denoted as x; Yt (k) and ii (k) respectively.
Assumption 2 (Homogeneous Covariates with Bounded Spectrum). For each market k E {0} U [K], covariates { (*) 1121 are drawn i.i.d. from a fixed, but a priori unknown, distribution Pa. supported on a bounded set X' C. C.. Let D = E| ] denote the second moment matrix of Pa. We assume:
(1) Eigenvalue boundedness: The minimum and maximum eigenvalues of E, denoted Cmia and Ciaax. satisfy < Cmin S Cmax < co.
(2) Non-degeneracy The distribution Pa has a density bounded away from zero in a neighborhood of the origin, ensuring E is positive definite.
Remark 1. Assumption isolates market differences to utility model shifts, which is reasonable when source and target markets have similar populations but differ in preferences. The key conditions ensure: (i) Stability: bounded eigenvalues and support guarantee well-hehaved estimators. (ii) Identifiability: a density bounded away from zero near the origin ensures > is positive definite. These hold in many practical settings e.g., truncated uniform or Gaussian distributions.
We will study the estimation and decision for the target model (1) leveraging the data from the target markets as well as the data from K auxiliary source markets.
Similarity Characterization. Transfer is effective only when source and target markets are suffi- ciently alike; we formalize this by assuming their mean-utility functions lie in common hypothesis class G and differ only through structured utility shift. For any candidate mean-utility function ğ E 9 we distinguish two notions of similarity, corresponding to (i) linear (parametric) and (ii) RKHS (nonparametric) utility models:
Parametric classes. When every ÿ is indexed by finite-dimensional parameter vector, similarity is expressed as a bound on the parameter gap between source and target.
Nonparametric classes. For diminie-dimensionall we impose bound on the functional discrep- ancy between source and target under a suitable function-space metric.
Later assumptions specialize these high-level conditions (c.g. sparse parameter differences in Assump- tion 4 for linear utilities and smooth residuals for RKHS utilities in Assumption 8 for nonparametric utilities). This formulation places every source task in a recoverable neighbourhood of the target, ensuring its data are informative for transfer across both linear and non-linear utility models.
4 Cross-Market Transfer Dynamic Pricing Algorithms
We address both practical data scenarios introduced in Section 1. For O20 (Offtine-to-Online a large, static source log is available prior to launch. The algorithm transfers that log during the early episodes-those for which the cumulative target sample size below a theory-driven threshold Once |Tm| 2 (source data no longer dominate the information budget) the procedure switches automatically to pure single-market learning. Hence transfer is phased, not one-shot: it is used exactly while it provably reduces estimation error and is dropped thereafter. Full details and guarantees are given in Appendix B.
For o20.n (Online-to Piline-to-Onliin On-Onnen,) source and target markets operate concurrently; transfer is repeated at the start of every episode, ensuring that incoming source data continuously guide the target-market prices. Let the time horizon be partitioned into episodes m = 2, M with lengths f. 2 m-1 (so that Σm= em RJ T and M [log,2 T)). At the start of each episode the algorithm (i) fits or debiases a demand estimator using all data collected in the preceding episode, and (ii) fixes the resulting pricing rule for the next fm periods. Because episode length doubles, parameter updates occur only O (log T') times, yet the cumulative sample size entering each update grows geometrically, guaranteeing progressively tighter confidence bounds and the desired Opolylog T') regret.
Both algorithms share common two-step bias-corrected aggregation pipeline and are instantiated estimation (MLE), and (ii) RKHS utilities, with
for (i) linear utilities, with

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kernel logistic regression (KLR). Pseudocode is given in Algorithms 4 (020 and 1 (O20on); estimation details appear Algorithms 2 and 3.
Comparison between of and 020 When source streams remain active, each episode m e(ag)
recomputes an aggregate estimate 9m from the preceding episode's 's source data and debiases it -(U)
using the matching target observations to form Im This persistent adaptation leads to provably faster regret decay compared to O20 In contrast, O20 employs phased transfer only during initial episodes, resulting in asymptotic regret growth rates that eventually match single-marke learning, though with improved constants during the transfer period.
Empirically, o2 achieves flatter regret trajectories with consistently lower cumulative regret (Figures and 2). 020 shows parallel regret growth single-mane market baseline in later stages (Figures and 4), confirming our theoretical analysis. However, we still observe a jump-start benefit O20,r during early stages, ultimately translating to significantly lower overall regret compared to the single-market baseline.
Algorithm 1: CM-TDP-O20..
Input: Streaming source data (wp (k)
2t yy ))21 for ht E [K]; streaming target contexts
(8)
{z h22
3(0)
Initialisation: & f,,-1 Ti -1= (1}. 90 =0.
for 1, do
// episodes
3
Compute bn =2 2 . = {mm..... fm+1- - 1}.
//(1) aggregate previous episode's source data
<(ag)
(k) Li. M1
(k)
Im + ML
11 (i1) debias with previous episode' 's target data (ag) (0)
8m < Debias Im {p) xt :3t (0) )eTT..)
5
// For functions MLE_or KRR and Debias, call Algorithm for linear utility (or Algorithm 3 for nonparametric utility)
Set 9m +- Im + 8m.
10)) (ag)
7
for E Tm do
// pricing
Post price (0)) = (9 (z) (): observe y, and store data.
8
Parametric Utility Models: Similarity, Transfer, and Guarantee
We start with the linear setting that allows us to isolate and rigorously characterize the transfer mechanism itself, before introducing the additional complexity of nonlinear effects.
Linear Utility Models. Consider a linear model the mean utility:
((0) + Ez.
(5)
where (0)) E Rdd denotes the coefficient vector. For source market data, we have for k E [K], Vi (k) (k) (k)) Ex- To simplify the presentation, we impose the following assumption on
parameter space.
Assumption (Parameter Boundedness). We assume that |/zelloo I 1, Vx, E and [| [|1 s W for a known constan! W 2 1, vk 0 [K]. We denote by Ω the set of feasible paramelers, i.e.,
( e d++ || S W}.
g Ω 4k e
We formalize the notion of similarity between source and target markets using the sparsity of the difference between coefficients.
5

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Assumption 4 (Task Similarity in Linear Model), The maximum ty-norm of the difference between target and source coefficients is bounded:
33 - -3 11o S 80-
max
While our linear utility model itself is necessarily sparse, Assumption 4 specifically constrains the pross-marke! parameter differences to be sparse, implying that at most so covariates have significantly different effects across markets, mirroring the economic intuition that only certain latent features drive market variations.
Bias-corrected Aggregation for Linear Utility. Algorithm 2 serves the dual-mode estimator in Algorithms and 4, switching between: (i) source data aggregation (no prior input), or (ii) l reguullard debiasing (given aggregate estimate).
Algorithm 2: Maximum Likelihood Estimation for Linear Utility Model
Input: Data (pr,, 3t, yo) bE(n), aggregate estimate } (ag) (ag) is None then
// compute aggregate term
{ Σ 1=1
β argmin b
L(b; Pt, (.,
4 else
1/ compute debiasing term
n L(b + (0g)
3 argmin
Pr: mr (1) + desllbll1
m 1=1
where the function
L(b; = 1ly = logg
F(p - b x))) - 1(y 1) - b x))}.
(6)
Output: /
In Algorithm 2, the aggregation step employs unregularized MLE. This choice is motivated by the fact that source market data are typically abundant, so the aggregate estimate can be reliably learned without imposing sparsity other high-dimensional penalties. By contrast, the debiasing step operates on target market samples, which are relatively scarce. Here, we incorporate regularization to stabilize estimation and exploit structured similarities across markets.
5.1 Theoretical Guarantee for CM-TDP-O20 under Linear Utility
The following theorem bounds the regret of our O20,n Policy under linear utility model.
Theorem (Regret Upper Bound for O20 under Linear Utility). Consider linear utility model (5) with Assumptions 1 (revenue regularity), 2 (covariate property). 3 (parameter space), and 4 (market similarity) holding true, the cumulative regret of Algorithm I admits the following bound:
Regret(T:; = O k
log d log so log T).
where and denote the number of source markets and time horizon, respectively.
We defer the complete proof Appendix E. Theorem 5 reveals crucial insights about the role of source market quantity in two distinct operational regimes. First, in the source-constrained regime (K < d/sp), the first term dominates, showing that each additional source market provides linear reduction in regret. Notably, the logarithmic dependence on is consistent with classical linear bandit results [1], though our bound strictly improves their O (d T)) through transfer. Second, in the source- saturation regime (K > d/sn). the second term becomes pivotal, quantifying the price for market heterogeneity.
As illustrated in Figure! 1, the empirical scaling behavior of regret w.r.t. both the number of source markets K and time horizon I' precisely matches the theoretical predictions derived from Theorem 5,

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While our theoretical guarantee is established in asymptotic regimes, the finite-horizon empirical performance robustly validates the practical effectiveness. This alignment between theory and practice confirms that our asymptotic analysis yields operationally meaningful insights for real-world applications.
Theorem 6 (Regret Lower Bound under Linear Utility). Consider the linear utility model in for any pricing policy и, the worst-case target-market regret over horizon satisfies
inf sup Regret(T;T)) 2 c1 K
log I' + c2 log T,
(8)
where the two constants C1, C2 depends only on noise distribution F, second moment matrix E, and parameter space W.
In particular, (8) matches the upper bound (7) up to polylogarithmic factors in d, hence CM-TDP is minimax-type optimal in its T- and K-scaling.
Nonparametric Utility Models: Similarity, Transfer, and Guarantee
In this section, we model market utilities in an RKHS [3], leveraging (i) the kernel trick for efficient nonlinear computation [24], and (ii) its universal approximation power capture rich market responses [26].
Nonparametric Utility Models. Let Hg be an RKHS induced by symmetrie, positive and semi- definite kernel function K X' x X - I,, and we define its equipped norm as l = (;,)) with the endowed inner product ( K- We also define K2 == K (ar,)) E Ha- An important property in Hr is called the reproducing property, stating that (g. K2)x = g(x). The utility function now follows:
a(h)) - kh) (at (xx (k))
+ Ets g(h) Hd. k [OK]
where g (8) is the unknown target function and {s( (k) he are i...d. noise with known distribution F. To start with, we place the following regularity assumptions on H2, which requires the model to be well-specified, and the kernel to be bounded.
Assumption 7 (Regularity Condition). Assume that (10 < R. for some R > O, and there exists a positive constant K, such that the feature map Φ(x) = K (x.-) satisfies [|0(z) K, Vo E X. Assumption 8 (Task Similarity in RKHS). For all h E [K]. the discrepancy between the target task (0)) and the k-th source task (k) in the RKHS norm is uniformly bounded as
maax K
11g 1)) 0) ( -
x H.
Remark 2. Assumption 8 characterizes the similarity between the target and source tasks through the bound H. A smaller value of И indicates that the source tasks are more similar to the target task, which enables more effective knowledge transfer and potentially improves the estimation accuracy by leveraging information from related sources.
Assumption (Complexity). Define the effective dimension as ((A) == T+EE(E+ ] a variant
of what is typically used to characterize the complexity of RKHS [6]. We assume:
(i) There exist some constants a 1/2 such that N(A) Tr(E + Al) -1
(i) There exist some constants 3 E (0, 1] such that for each t E O0 [K], there holds 9 (kk) E> (K),
for some (()) L2(X,Pa) where 2,, Pa are defined in Assumption 2.
Remark 3. (i) controls the complexity of the considered H4. Smaller a means slower eigenvalue decay and higher intrinsic dimensionality. (ii) is a regularity condition on the source and target functions, and is also commonly assumed in literature 25, 17]. Here 3 > 0 controls the degree of smoothness, and larger 3 means (k)) is smoother and easier to estimate.
Bias-corrected Aggregation for Nonparametrie Utility. Algorithm 3 presents the aggregation and debiasing operations using regularized kemell regression.
6.1 Theoretical Guarantee for CM-TDP-O20.n under Nonparametric Utility
The following theorem bounds the regret of our 020,n Policy under RKHS utility model.
7

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Algorithm 3: Kernel Regression for RKHS Utility Model
Input: Data ((it, St. y/ HE(M), Kernel ((x, (1)(;)) (xxx)) Aggregated estimator
g glos) E H&-
1 glos) None then
1/ compute aggregation term
g = argmin gEH2
L(g: Pt, 2t. + Aagllali l
a else
1/ compute debiasing term
g argmin oE Hu.
Σ 1=1
Lig + gl0s); Pr. ve) + A
where the function
{ [(v log(1 F(p g(x:))) 1y 0) (Fg(((( F()-)9())))- s(æ)))}
L(g: a.p,a,y
Output: 9
Theorem 10 (Regret Upper Bound for O20 under RKHS Utility). Consider RKHS utility model (9) with Assumptions I (revenue regularity). 2 (covariate property), 7 (parameter space), 8 (market similarity), and (complexity) holding true, the cumulative regret of Algorithm admits the following bound:
2n3 26341 T2854 + H I
Regret(T;s) = KK-
(10)
where K and T denote the number of source markets and time horizon, respectively.
Theorem 10 again highlights the benefit of transfer learning, and finds clear empirical support in the experiments shown in Figure 2. The first term reflects the learning complexity of the target function, where 3 quantifies intrinsic complexity with larger indicating simpler functions and enabling faster transfer. The parameter, as the kernel's 's effective dimension, modulates how B impacts the exponent. The second term encodes cross-market disparity through H, which directly measures the worst-case RKHS distance between source and target markets. The a-dependence shows that high-dimensional RKHS amplify heterogeneity costs. Compared Theorem 5, we observe polynomial rather than logarithmic T-dependence, reflecting the fundamental difficulty shift from parametric to nonparametric estimation.
Special Cases. The following boundary cases demonstrate the degradation properties of our theoreti- cal results, showing how the general bound naturally adapts to different simpler scenarios.
Perfect Task Similarity. When source and target domains are identical (H = 0), the bound becomes:
Regret(T;; (T) = O (K
2n (K=2 1 T
The regret depends on the total sample size from all sources. The rate improves with number of source market K.
No Transfer Learning (Kaff H 0) the absence of transfer learning, i.e., when no source domain data is available (K 0, H = 0), we evaluate the bound (10) with Keff max K, 1} 1 (cquivalently, a "self-aggregation" " that ignores external sources). Our general framework reduces to conventional dynamic pricing with RKHS utility functions. The regret bound simplifies to:
Regret(T: (T)): = 0 (TZOAZT)
While existing literature has not explicitly analyzed regret bounds for dynamic pricing with RKHS utility functions, we can establish an important connection to the well-studied linear case. By setting a -> 1/2+ and 33 1, which corresponds linear utility functions, the general bound (10) reduces to O(VT), matching the often-seen regret for online decision-making problems with linear structures [1, 19].
This (VVT) rate should be contrasted with the Ollog T') regret we established for the linear transfer setting in (7). The difference stems from the underlying estimation complexity: in the parametric case, the generalized linear model is fiite-ddmensionnl, and abundant source data together with MLE-based updates enable nearly logarithmic regret growth. In contrast, the RKHS formulation
8

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must estimate an
function under binary feedback, where nonparametric learning intrinsically harder. Thus, the gap reflects fundamental differences between parametric and nonparametric estimation, rather than looseness in the analysis.
Theorem 11 (Regret Lower Bound under RKHS Utility). Consider the RKHS utility model in (9).
For any pricing policy т, then there exists a constant e > 0 depending only on (F.Pz. K) via
(Mrovy my, k) and the Bernoulli KL smoothness constant (defined in Lemma 28) such that for all horizons T 2 1,
inf sup Regret(T;;)) 2 of KK
2nd ZOPFI TEPFT
20
(11)
In particular, (11) matches the upper bound (10) up to polylogarithmic factors, hence CM-TDP minimax-type optimal in its T- and K-scaling.
7 Numerical Experiments
We evaluate CM-TDP through extensive simulations covering multiple market scenarios and dimen- sionalities:
(I) Identical Markets: an ideal baseline where 3 (3 g0)) = g( for all source markets;
(2) Sparse difference Markets: for linear utility, we implement Assumption 4 with ||3(0) - 334 llo -
0.3 * d; for RKHS utility, we implement Assumption 8 with (l) - g0 ||,,, I 0.3.
(3) Dense difference Markets: for linear utility, we implement Assumption 4 with 113 30h s
0.5 * d; for RKHS utility, we implement Assumption 8 with (l - ( 1|. I 0.5.
In each market scenario, test T 2000 periods with dimensions d E {10, 15, 20, 100) and K E [1,3, 5, 10} source markets. RKHS function (RBF kernel with 7 = 0.5) and kernel parameters (x 0.5, R = 1.0) remain consistent across all experiments. We evaluate O20.n. Policy against no transfer baseline [19]. a standard dynamic pricing using only target market data. Market noise follows logistic distribution on IR. For numerical stability, we clip simulated valuations to [- B.] with B, = 1. The code is available at https://github.com/ how..u..cco...-.ea// Bww.//CSSSSLLLER SAIL/transfer tetrannfer_prriingg_eee 202.
Simulation Results. Figures and present the empirical cumulative regret for linear and RKHS utility models with d = 10 and 100, respectively, demonstrating three key findings (For conciseness, results for d = 15, 20 are deferred Appendix D.1).
(1) Universal effectiveness over non-transfer scheme. CM-TDP-O20 consistently outperforms single-market baseline in all scenarios. Significant improvements emerge even with minimal source markets (K = 1), with larger K values enhance robustness divergent market conditions. Against market learning, CDP reduces cumulative regret by roughly 43 55% average (peaking at 75%), reduces standard error by about 24-31% % (up to 39 %), and attains the same estimation error level as much as 9 x sooner (Table 2 in Appendix 1.2.
(2) Adaptivity. The transfer mechanism automatically handles both identical markets and sparse- difference markets, without requiring manual adjustments, with identical market condition achieving faster convergence than sparse difference cases and dense difference cases.
(3) Scalability: Higher dimensions maintain stable performance, confirming the method's scalability,
These results collectively validate our framework as versatile solution for real-world dynamic pricing, particularly in environments with varying market similarities.
Conclusion and Future Work
We introduce Cross-Market Transfer Dynamic Pricing (CM-TDP), the first framework that provably accelerates revenue learning by pooling information from multiple auxiliary markets in both Offline- to-Online and Online-to-Online regimes. CM-TDP achieves minimax-optimal regret for both linear and RKHS utilities, matching information-theoreiic lower bounds, and delivers up to an average of 50% lower cumulative regret and 5x faster learning in extensive simulations. These results bridge single-market pricing, meta-learning, and multitask bandits, laying the groundwork for pricing systems that "transfer faster, price smarter".

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(a) Identical, d 10
(b) Sparse, d = 10
(c) Dense, d 10
(d) Identical, 100
(e) Sparse, d 100
(f) Dense, d 100
Figure 1: Cumulative regret across experimental conditions in 020 with linear utility model.
(a) Identical, d = 10
(b) Sparse, d = 10
(c) Dense, d = 10
(d) Identical, 100
Sparse, d 100 (I) Dense, d 100
Figure 2: Cumulative regret across experimental conditions in 020,. with RKHS utility model.
Several extensions of CM-TDP provide fertile ground for future research. First, relaxing the assump- tion homogeneous covariate distributions would allow the framework to handle domain shift, e.g.. via reweighting, importance sampling, we focus on spsssiiyy for interpretability, the aggregation framework naturally extends to richer similarity notions such as -psssiiyy (g [o, 1)), smoothness metrics, distributional divergences. Third, CM-TDP currently lacks mechanisms to down-weight or exclude adversarial source markets with large parameter gaps, and developing robust similarity detection and melket-selection strategies remains an important direction.
representation learning. Second, while
Acknowledgments and Disclosure of Funding
We thank the NeurIPS reviewers for their helpful comments. This work was supported by NSF Award No. 2412577 and the NYU Stern Research Fund.
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References
[1] Yasin Abbasi- Yadkori, David Pal, and Csaba Szepesvari. Oonlne-to-coffidence--ee conversions and application sparse stochastic bandits. In Neil D. Lawrence and Mark Girolami, editors, Proceedings of the Fifteenth International Conference on Artificial Intelligence and Statistics, volume 22 of Proceedings of Machine Learning Research, pages 1-9, La Palma, Canary Islands, 21-23 Apr 2012. PMLR.
[2] Kareem Amin, Afshin Rostamizadeh, and Umar Syed. Repeated contextual auctions with strategic buyers. Advances in Neural Information Processing Systems, 27, 2014.
[3] N. Aronszajn. Theory of reproducing kernels. Transactions of the American Mathematical Society, 68(3):337-404, 1950.
[4] Hamsa Bastani, David Simchi-Levi, and Ruihao Zhu. Meta dynamic pricing: Transfer learning actoss experiments. Management Science, 68(3):1865-188) 2022.
[5] Changxiao Cai, T Tony Cai, and Hongzhe Li. Transfer learning for contextual multi-armed bandits. The Annals of Statistics, 52(1):207-232 2024.
[6] A. Caponnetto and E. Vito. Optimal rates for the regularized least-squares algorithm. Found. Comput. Math., 733):31-368, July 2007.
[7] Jinhang Chai, Elynn Chen, and Jianqing Fan. Deep transfer Q-learning for offline non-stationary reinforcement learning. arXiv preprint arXiv:2501.04870 2025.
[8] Jinhang Chai, Elynn Chen, and Lin Yang. Transfer Q-learning with composite MDP structures. In The Forty-second International Conference on Machine Learning (ICML 2025), 2025.
[9] Elynn Chen, Xi Chen, Lan Gao, and Jiayu Li. Dynamic contextual pricing with doubly non-parametric random utility models. arXiv preprint 2xXiv:2405.06866 2024.
[10] Elynn Chen, Xi Chen, and Wenbo Jing. Data-driven knowledge transfer in batch Q learning. arXiv preprint arXiv:2404.15209 2024, 2024.
[11] Elynn Chen, Xi Chen, Wenbo Jing, and Yichen Zhang. Distributed tensor principal com- ponent analysis with data heterogeneity. Journal of the American Statistical Association, (just-accepted):1133 2025.
[12] Elynn Chen, Sai Li. and Michael I Jordan. Transfer Q-learning for fiite-horizon markov decision processes. Electronic Journal of Statistics, (just-accepted):1 2025.
[13] Maxime C Cohen, Ilan Lobel, and Renato Paes Leme. Feature-based dynamic pricing. Manage- ment Science, 66(11):4921-49) 4943, 2020.
[14] Giulia Denevi, Dimitris Stamos, Carlo Ciliberto, and Massimiliano Pontil. within--ithi-nnliine meta-learning. Advances in Neural Information Processing Systems, 32, 2019.
[15] Mingwei Deng, Ville Kyrki, and Dominik Baumann. Transfer learning in latent contextual bandits with covariate shift through causal transportability. In Causal Learning and Reasoning. 2025.
[16] Jianqing Fan, Yongyi Guo, and Mengxin Yu. Policy optimization using semiparametric models for dynamic pricing. Journal of the American Statistical Association, 119(545):552-566 2024.
[17] Zheng-Chu Guo, Shao-Bo Lin, and Ding-Xus Xuan Zhou. Learning theory of distributed spectral algorithms. Inverse Problems, 33(7):074009, 2017.
[18] Xinmeng Huang, Kan Xu, Donghwan Lee, Hamed Hassani, Hamsa Bastani, and Edgar Dobriban. Optimal multitask linear regression and contextual bandits under sparse heterogeneity. Journal of the American Statistical Association, pages 1-14, 2025.
[19] Adel Javanmard and Hamid Nazerzadeh. Dynamic pricing in high-dimensions. The Journal of Machine Learning Research, 20(1):315-363, 2019.

---

[20] Allen Liu, Renato Paes Leme, and Jon Schneider. Optimal contextual pricing and extensions. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1059-1078. SIAM, 2021.
[21] Yiyun Luo, Will Wei Sun, and Yufeng Liu. Contextual dynamic pricing with unknown noise: Explore-then-ucee strategy and improved regrets. Advances in Neural Information Processing Systems, 11337445-374555 2022.
[22] Jieming Mao, Renato Leme, and Jon Schneider. Contextual pricing for lipschitz buyers. Advances in Neural Information Processing Systems, 31, 2018.
[23] Jathushan Rajasegaran, Chelsea Finn, and Sergey Levine. Fully online meta-learning without task boundaries. arXiv preprint 22Xi::2202.00263.. 2022.
[24] Schölkopf and Smola. Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond. MIT Press, 2001
[25] Steve Smale and Ding Xuan Zhou. Learning theory estimates via integral operators and their approximations. Constructive approximation, 26(2):153-172, 2007.
[26] I. Steinwart. On the influence of the kernel on the consistency of support vector machines. Journal of Machine Learning Research, 2:67-93, 2001.
[27] Alexandre B Tsybakov. Nonparametric estimators. In Introduction Nonparametric Estimation, pages 1-76. Springer, 2008.
(28) Matilde Tullii, Solenne Gaucher, Nadav Merlis, and Vianney Perchet. Improved algorithms for contextual dynamic pricing. Advances in Neural Information Processing Systems, 37:126088- 126117, 2024.
[29] Roman Vershynin. Introduction the non-asymptotic analysis of random matrices, 2011.
[30] Fan Wang, Feiyu Jiang, Zifeng Zhao, and Yi Yu. Transfer learning for nonparametric contextual dynamic pricing. arXiv preprini arXiv:2501. 18836 2025.
[31] Jianyu Xu and Yu-Xiang Wang. Towards agnostic feature- based dynamic pricing: Linear policies vs linear valuation with unknown noise. International Conference on Artificial Intelligence and Statistics, pages
PMLR, 2022.
[32] Kan Xu and Hamsa Bastani. Multitask learning and bandits via robust statistics. Management Science, 2025.
[33] Mengqian Zhang, Yuhao Li, Xinyuan Sun, Elynn Chen, and Xi Chen. Maximal extractable value in batch auctions. In The 26th ACM Conference Economics and Computation (EC 2025),
pages 510-510, 2025.
[34] Runlin Zhou, Chixiang Chen, and Elynn Chen. Prior-aligned meta-rl: Thompson sampling with learned priors and guarantees in finite-horizon mdps. arXiv preprint arXiv:25/0.05446, 2025.
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A Notation
Let lowercase letter x, boldface letter æ, boldface capital letter X, and blackboard-bold letter X represent scalar, vector, matrix, and tensor, respectively. The calligraphy letter X' represents operator. We use the notation [N] to refer the positive integer set (I...., , for N Z.. Let C, Co, cu.... denote generic constants, where the uppercase and lowercase letters represent large and small constants, respectively. The actual values of these generic constants may vary from time to time.
All vectors column vectors and row vectors are written as x² for any vector æ. For any vector (I1, (E1....,xPp) Fp) let ||>:| == ||ح|| (-==1 x// be the f2-norm, and let ||=|| Σ
|
be the nonorm.
For any matrix X, we use Ci., L.. and Tij to refer to its i-th row, j-th column, and ij-th entry, respectively. For two matrices X, E Rmxnn and X2 Rexq, X & X2 E Rum X gm is the Kronecker product. When X is square matrix, we denote by Tr (X), ), Amaz (X), and Amin (X) the trace, maximum and minimum singular value of X, respectively. For two matrices of the same dimension, define the inner product (X1, X2) Tr X2).
We L2(XX, [x (x)d'p oo} denote the space of iquare-integrable functions with respect to P.z equipped with the inner product (J. fxx (x)) and squared norm | fx (x)ddP,
Notation
Definition
Time index (period).
Time horizon (total number of periods).
Episode index (algorithmic episode).
lm
Length of episode m, lm = 2 Set of time indices in episode m.
T
K
Number of source markets.
k
Source market index, E [K].
no
Number of target samples.
nK
Total number of offline source samples (used in O20,tr analysis).
(0) (0) (0) (0)
Pt V/
Observed contextual covariates, posted price, observed binary sale indicator, latent market utility, for the target market at time t.
(R)
(8)
Vi
04
Observed contextual covariates, posted price, observed binary sale indicator, latent market utility, for source market lc at time t.
g.)(()
General unknown mean-utility function for target market.
(+)
General unknown mean-ure mean-utility function for source market k. i..dd. noise at time t with distribution F(-).
F())
Cumulative distribution function the noise Es.
Covariance matrix for source and target features, which is equivalent
to ()) (A) in homogeneous covariate setting.
Cmina Ciax
Minimum and maximum eigenvalues second-moment matrix
(0)
9m
Debiased estimator for the target after aggregation++debias at episode
m.
g(0)
True (population) debiasing correction in RKHS.
8m
Debiasing correction term computed at episode m. Regularization parameter used for the debiasing step.
Xxf
Xag
Regularization parameter used in aggregation.
Continued on next page
20

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Notation
Definition
(())
Coefficient vector for the linear mean utility in the target market. Coefficient vector for the linear utility in source market k. Estimated coefficient for target market.
8()
8
s0
Task-similarity magnitude in linear utility case.
g g())
RKHS mecan-tiiltty function of the target market. RKHS mean-utility function of source market k.
g*)
H
Task-similarity magnitude in RKHS case.
Hy
RKHS associated with kernel K.
(()))
Positive semi-definite kernel function inducing H..
((A)
Effective dimension.
Parameter controlling eigenvalue decay.
B
Smoothness parameter for RKHS function.
B 020 Offline-to-nlline Cross-Market Transfer Pricing
In this section, we present our algorithmic framework, theoretical results and empirical experiments for O20,rr Peftinetto-Online scenario.
B.1 Offine-to--nline (020,tt Algorithm
<(ax) from all source data. At the start of Prior deployment form aggregate estimator 0
20)
episode m this estimator is debiased with the first batch of target observations obtain 91 subsequent episodes rely exclusively on target data. Hence 020 yields one-shot reduction of cold-start regret, but its asymptotic rate T matches that of single-market learning. The complete procedure is summarize in Algorithm 4.
B.2 Theoretical Guarantee for CM-TDP-020..r under Linear Utility
The following theorem bounds the regret of our 020,it Policy under linear utility model.
Theorem 12 (Regret Upper Bound for 020 under Linear Utility). Consider linear utility model (5) with Assumptions 1 (revenue regularity), (covariate property). (parameter space), and 4 (market similarity) holding true, the cumulative regret of Algorithu 4 admits the following bound:
Regret(T: ()) = (d dogg T + (so - d) log log nx))
(12)
where nyc and T denote the number of source data and time horizon, respectively.
Theorem reveals fundamental differences how static source data influences regret compared to Online-to-Online transfer. The bound in (12) decomposes into two interpretable components: the first term captures the intrinsic complexity of learning the d-dimensional target market parameters the total T periods, while the second term quantifies the net effect during transfer learning phase. Given that 50 < d by problem construction, the second term always provides regret reduction proportional to (d - so) log Tix. This reduction grows logarithmically with the total source sample size nix, which consistent with our numerical experiments in Figure
B.3 Theoretical Guarantee for CM-TDP-020..r under Nonparametric Utility
The following theorem bounds the regret of our 020 Policy under RKHS utility model.
Theorem 13 (Regret Upper Bound for 020 under RKHS Utility). Consider RKHS utility model (9) with Assumptions I (revenue regularity), 2 (covariate property), (parameter space), 8 (market
21

---

Algorithm 4: CM-TDP-020of
Input: Offline source market data (1p))) for the target market
Deet for [K]; feature matrix {ar (0) (k) at (
(k)
HEN
Phase
Update with transfer learning z(ag)
*/
Call Algorithm or to calculate the initial aggregated estimate 9 data ((p))
using entire source market
(R) X (k) (K) HeHm for E [K]
2
Apply the price 51 = (0 11 (g))
(a)
and collect data (7) 11 Vi )
for each episode m 2, ..... me do
Set the length of the m-th episode: fm := 2
5 Call Algorithm or to calculate the debiasing estimate 8m using target market data
{(pt) (0)
2 :i ))e be
(0) (0) 22mp(2mm2m-1_------- 2m and aggregated estimate (ag)
Set
9m := 3 (ag) 8m
_ For each time t, apply price P.. == h (a, (0) and collect data (p. x ))
(0)
/* Phase for each m 2 my + 1 do
Update without transfer learning
9
Set the length of the m-th episode: bm := 2
Call Algorithm or 3 to calculate ( using target market data {ppt) (0)
zi (0) (0) 1>
11
For each time t, apply price and collect data.
Output: Offered price p(o) ,t221
similarity), and (complexity) holding true, the cumulative regret of Algorithm 4 admits the following bound:
(T))
I Cnk
+ H I6TT
(ēnx) Ju11 + 7 Fuade
(Ōnk) BuOFI
(13)
where nıg. denotes the number of source data, 3 characterizes the smoothness of the aggregated utility function glag) Egg with B (0, 1}. and a > 1/2 controls the effective dimension via the eigenvalue decay (() S 12(a)
The derived result reveals an important trade-off in the regret decomposition. The first component
O
260 + मं (Enx) T captures regret resulting from the transfer learning phase. This Cn
component grows with eny because more source data leads to a longer transfer phase duration, which consequently increases the accumulated regret during this phase. However, this initial cost is
offset by a more significant benefit the second component: T (Enx) 2.u=e which demonstrates that the extended transfer phase enables substantially more effective single markett learning, and thus reduces the overall regret.
We defer the complete technical proof of Theorems 12 and 13 to Appendix F and respectively.
B.4 Numerical Experiments
The experimental setup for O20cn maintains the same core structure as the 020 setting, except for data collection protocol. Rather than observing synchronized data streams from active source markets, we begin with fixed historical dataset of size пı: E (50, 100, 200, 200.500) All other experimental parameters, including demand model specification, evaluation metrics, and comparison baselines, remain consistent with the 020gg described in Section 7.
22

---

All experiments were conducted on an Ubuntu 20.04 server equipped with an AMD Ryzen 9 5950X CPU (16 cores, threads), 125 GiB RAM, and an NVIDIA RTX 3090 GPU (24 GiB VRAM). The primary storage was a 1.8 TB NVMe SSD.
Simulation Results Figures and 4 *present the empirical cumulative regret for linear and RKHS utility models, respectively. As is consistent with theoretical analysis in Theorems 12 and 13, we observe a jump-start benefit in the early stage, and a significantly lower overall regret compared to the angle-market baseline.
The results again demonstrate that all transfer learning variants consistently outperform the no-transfer baseline across all tested conditions, with larger rig values showing faster regret reduction, validating effectiveness of knowledge transfer from source markets.
50
201
306
1200
2300
2000
7500
7000
36
2303
12.00
1010
(a) Identical, d 10
(b) Sparse, d = 10
(c) Dense, 10
Canadde
205
222
Tec/faber
2.00
28
1006
2540
2000
Petod
12.00
2000
(d) Identical, d 15
(e) Sparse, d = 15
(f) Dense, d = 15
566
28
Pr-lod
(g) Identical,
(h) Sparse, d = -2
Dense, d 20
110
1::0
(i) Identical, d 100
(k) Sparse, d = 100
(1 Dense, d 100
Figure 3: Cumulative regret across experimental conditions in 020 with linear utility model.
23

---

Orncudii
80
02-00
Tite Farlod
(a) Identical, 10
(b) Sparse, d = 10
(c) Dense, 10
308
120
212
5.5
2.5
2100
502
3000
2003
130g
1058
2000
1300
1000
(d) Identical, d 15
(c) Sparse, d = 15
(f) Dense, d 15
222
3rmt Pr
Trat Priic
(g) Identical, 20
(h) Sparse, d = 20
(i) Dense, d 20
2-18
380
(i) Identical, = 100
(k) Sparse, d = 100
(1 Dense, d 100
Figure 4: Cumulative regret across experimental conditions in 020 transfer with RKHS utility model.
C Computational Complexity Analysis
In this section, we summarize the computational costs of CM-TDP across different settings.
Linear Utility Model. In 020 each episode consists of two stages. In the aggregation stage, unregularized maximum likelihood estimation on the aggregated source data has a complexity of O .). where d is the feature dimension, f the number of past samples, K is the number of source markets, and is the number of optimization iterations. In the debiasing stage, Lasso regression the target data requires O N). Over T episodes, the cumulative regret analysis involves O(PTK' N) operations.
For 020,.. Phase I (transfer) requires - for MLE on the aggregated source market data, plus Odd-nm-1 N) per episode for hias correction. Phase (no transfer) reduces standard linear MLE with complexity Oldlmm- N). The resulting cumulative complexity across T episodes is

---

Old + The dependence on d depends on the optimization method: Newton's 's method (used in our implementation) achieves faster convergence (smaller N) but maintains d2 dependence, whereas gradient descent reduces the per-iteration cost to (dd) at the expense of larger iteration count N.
RKHS Utility Model. Exact kemeel methods scale as (n²²) in the number of aggregated samples In O20mn: episode uses 7ym. K 2 source samples and 2 target samples, so a naive solver scales O( (
og In practice, Nyström or sketching reduces runtime near-linear in the effective dimension N (A).
D Additional Experimental Results
D.1 Extended Regret Plots for 020
In the main text, we reported regret trajectories for 020 with dimensions d = 10, 100. Here, we provide additional results for seermediate-iimensionn settings (d 15, 20)in Figures and As shown in the plots, CM-TDP consistently outperforms the no-transfer baseline across different dimensionalities, maintaining lower cumulative regret and faster convergence. These results confirm that the effectiveness of CM-TDP is not restricted specific dimensions.
(a) Identical, d = 15
(b) Sparse, d = 15
(c) Dense, d = 15
(d) Identical, d 20
(e) Sparse, d 20 (f) Dense, d 20
Figure 5: Cumulative regret across experimental conditions in 020 with linear utility model.
D.2 In-depth Analysis of Sparse-difference Markets
While our regret evaluation covers identical, sparse-differenee, and dense-differene market scenarios, this part focus the sparse-difference case. This choice reflects the primary target of our algorithm design: markets that differ sparsely in latent preferences, where transfer is both practically relevant and theoretically most distinctive. At the same time, earlier regret plots already demonstrate that CM-TDP yields consistent improvements across identical and dense-difference settings as well, confirming that the additional deep-dive into sparse-difference scenarios is representative rather than restrictive.
D.2.1 Benchmarking Transfer Gains
Table compares CM-TDP against the canonical single-market learner in the sparse-differece setting, where only small subset of coefficients differs between the target and each source market. We report three metrics averaged over dimensions d {10, 15, 20}: reg (percentage reduction in
25

---

(a) Identical, d = 15
(b) Sparse, d = 15
(c) Dense, d = 15
(d) Identical, d = 20
(e) Spars, 20
(f) Dense, 20
Figure 6: Cumulative regret across experimental conditions in 020 with RKHS utility model.
cumulative regret, i.e., revenue lift), std (percentage reduction in standard error across 10 Monte- Carlo runs), and speed (multiplicattiv acceleration in reaching the single-market learner's final estimation error, i.e., [|3(00 300) for linear utility, and (| - g0)) | for RKHS utility). For 020 transfer, performance is indexed by the number of live source streams K, whereas for 020 transfer is indexed by the historical log size nyc- Across both linear and RKHS utilities, gains grow monotonically with K' (or nx)): single auxiliary market already cuts regret by 15-20% while ten live sources slash regret by more than half and deliver up to a 9x jump-start learning speed. These results confirm that CM-TDP translates theoretical advantages into substantial empirical improvements even when source-target differences are sparse and high-dimensional.
Table 2: Comparison with single-market learning baseline in sparse-difference market scenario averaging over different dimensions. In the metric column, reg, std and speed means cumulative regret, standard error, and learning speed, respectively. K and nı apply to 020 and 020 policies, respectively.
Model
Metric
K =
K = 3
K = 5
K = 10
Avg
reg + std
15%
61 %
67%
71 %
54%
-Linear
9%
36%
38%
39%
31 %
speed
1.2x
5.9x
8.1x
9.0x
6.0x
reg
17%
62%
68%
73%
55%
O20 020,g-RKHS RKHS
std
11%
33%
36%
36%
29%
speed
1.3x
6.7x
8.1x
8.9x
6.2x
n.c
50
100
nic
200
n.c 500
reg
20%
48%
54%
56%
45%
sid
5%
22%
34%
35%
24%
speed
1.5x
3.0x
4.9x
6.2x
3.9x
reg 15%
47%
53%
56%
43%
220.G-RKHH
sld 7%
21%
35%
38%
25%
speed
1.8x
3.1x
4.4x
5.9x
3.8x
26

---

D.2.2 Running Time Evaluation
Table 3 reports the total running time (in seconds) of CM-TDP under sparse-difference markets with horizon 7 2000. For o2 runtime increases moderately with the number of source markets K and remains manageable even at K = 10. Linear models are consistently faster than RKHS models, both scale sublinearly with the dimension size of the offinn source dataset vug, showing only gradual increases TiX grows from 50 to 500. Overall, these results confirm that both O20 and O20 policies are computationally efficient, with the added flexibility of RKHS utilities incurring only modest overhead compared to linear models.
For O20an, runtime primarily affected by the
Table 3: Total Running time (in seconds) in differ-effe market scenario across different dimensions in T = 2000 periods. K and apply to 020 and O20 policies, respectively.
Model
No transfer
K
K
K 10
10
31
55
63
71
77
15
42
80
85
93
101
O20 Linear
20
101
109
121
138
100
386
794
814
866
972
10
43
84
99
112
15
57
98
106
121
140
020 RKKHS
20
71
114
123
134
159
100
438
832
927
1018
1156
T/c = 50
n. = 100
n. = 200
ny = 500
10
43
53
58
64
15
42
58
64
70
77
-
20
48
69
74
82
89
100
386
563
597
642
663
10
43
47
55
65
71
15
57
76
87
98
O20 -20OI-KKHS RKHS
20
71
80
92
101
123
100
438
602
694
758
872
27

---

E Proof of Theorem 5
Define He {a] (0)
(0)
00) (0) (0)
1 _2 .....,E'' ) the history set up to time We also define
22
H, H, U (11+1) as the set obtained after augmenting new feature 1i+11 we write
(0)
[ree (1;-1) Elp; (0) (0)
2 --) E(p;
(0) [(v Pi (0)
(0)
(
(14)
Pi (0) (1 F(p:" (0) - 11 ( .3(0)) Pi
(0) (1 F(P; (0)
1)
Note that Pt ((0) arg max (0) (p) and thus r/(p (1)) = 0. By Taylor expansion,
rev, (0) (v: (p, 10) + + (v)( Pi )0)
1
(15)
for some p between Ps and Pt (0)
(0)
Lemma 14 (Upper bound for price). The price given by policy is upper bounded by
Dt (0)
l1( (0)
3 (0)
P.
Lemma 15 (I-Lipschitz property of h). Suppose Assumption I holds and, in addition, o' is bounded on [-B,, Bul:
0 < Løv S Φ s'( S U0o 00.
Then
sup [h'(u)|= = sup bui.,
1 (1()
I Ln < oc,
hence
[h(u) - h(v)l S Lalu for all |u],|o| S Bu.
The price function satisfies Nu(u) 1, for all values of u E TR.
Therefore obtain
(p]( = 12f(p xt (0) (f xt 10) -/(0)) < 2B + PB',
with B max,, f(v), and B' max, ('(v), where we use the fact that Pt Pt S P and consequently p P.
(0) (0)
Then, combining Equations 14, 15, Lemmas 14, 15 gives
E(eg, H1-1) (2B+ PB') (0)
Pt 10)22 & C(pi (()) -p, 00) (0)
C(h(g(()) 1z (0)) -3 {11)) < Cl (0) (3310) -310) -3
(2) 2 C (0) 00) M13
(0) -3(0)))
where (a) results from that x; is independent of H, and Σ = E( Since the maximum eigenvalue of Σ is bounded by Cmaax obtain
E(reg,) = (Ere,, H₂-1)) < CCmax 1 3 11))
(16)
which brings the problem down to bounding the estimation error of the proposed estimator.
Proposition 16. Consider linear utility model with Assumptions 1, and holding true. Then, there exist positive constants co, cg. C1, C2 such that, for no 2 coso log d V ei K the following holds
d.
with probability at least 1 2/d 2e no/ (ros)))))
13 (0)
3 <
dlg d
so log d + e2
(17)
ni
28

---

Corollary 17. Under conditions of Proposition 16, the following holds true:
E(I (0) - -30 c1
dloog d
+ 8o log d
+ 4W2 (2
2e 00
no
Proposition 18. Consider linear utility model with Assumptions
and 4 holding true. There
exist constants C3, Ca, Cs O, such that for cyd, the following holds true:
(0) - - S C1 B
d dloog d
(so log
+ C5
no
Now, since the length of episodes grows exponentially, the number of episodes by period T is logarithmic in T. Specifically, 7 belongs episode M [log T7. Hence,
Regret(Ti:) = Σ Reg(mth Episode).
We bound the total regret each episode by considering three separate cases:
1. 2 < Cpso log d V d,d: Here, co. c are the constants in Proposition 16. In this case, episodes are not large enough to estimate the parameters accurately enough, and thus we use a naive bound. Clearly, by Lemma 14, we have E((eg,) I PŁ S P. Hence the total regret over such episodes is at most 4Pcysg log V 4Pchd.
2. Co log d & 2
czd: Applying Corollary 17 Equation 16,
1m
1
Reg(mth Episode)
E(reg,) &CCmax
Σ
(0) B(1 m - (())
t=
dlogg d Klm-1
So log d 1m-1
-im-1 Sw2 ("m + me
C'Cmax
c1lm
+ o2lm
4000
-mm-s dlog d + C280 log d 8W2 cs (os+lm-1ee mm cong
<
2C0maxx
K
where in the last step we used Im 21 and 1m-1 S czd. Therefore, in this case
Reg(mth Episode) < C₁ log + Caso log d,
K
where C₁, C/2, hides various constants in the right-hand side the above equation. 3. cad 2 Applying Proposition 18 to Equation 16,
1m+-11
Im+
(0) (000 12)
Reg(mth Episode)
E(reg,) & CC
E(I3 m
t=b
t=b
dlogd d calm Klm-1
(80 + 1)
-m- egd
CCmax
+ cslm
+SW2 2
1m-1e
1m-1
d C4 K
-lm-s agd
CCmax
log d cs(so + 1) Im
Therefore, in this case
Reg(mth Episode) < CH K
log + 80 log
Combining the above three cases, we get
Regret(T;;t r) < Cid d log + C250 log log ng = =0 ( log log T' so log log
which concludes the proof.
29

---

Proof of Theorem 12
In oflline-to-online transfer setting, obtain the same regret inequality in Theorem
E(reg,) ( I CC -3 122
(18)
Similar Propositions 16 and 18, we state the following two propositions on the estimation error regarding the transfer-learning phase.
Proposition 19. Consider linear utility model with Assumptions 1, 2, and holding true. Then, there exist positive constants cu, C1. O2 such that, for no 2 CuSu log d, the following holds with
probability least 2/d 2e no/ (c080);
B 00)
3
S dlog d
Su logd
+ C2
TKK
Corollary 20. Under conditions of Proposition 19, the following holds true: E(I (0)
d dlogg d
log d +
3( 11) ci
4W2 2e n0
no
The following proposition gives a tighter bound for the estimation error ng gets larger.
Proposition 21. Consider linear utility model with Assumptions 2, 3 and 4 holding true. There exist constants C3 Cr. G3 > O, such that for no Czd, the following holds true:
E
(0) -3 S
dl d
+ c
(so + 1) logd
+ 4
nyc
The next proposition states the estimation error in the without- transfer-learning phase Algorithm 4. Proposition 22. Consider linear utility model with Assumptions 1, and 4 holding true. Then, there exist positive constant e9 such that, for no 2 eny, the following holds with probability at least 1 1/d:
1 -
3
dlog d
where ē Z (c7d c2sd)/eid.
The adaptation parameter is dynamically determined through a comparative analysis of the estima- tion bounds in Propositions 21 (transfer-enabled) and 22 (target-only), where we strategically disable transfer learning updates when the volume of target data is large enough to ensure statistically optimal estimation performance.
Corollary 23. Under assumptions of Proposition 22, the following holds true:
E(I (0)
(00)
11))
dlog d +
4W2
S
no
We bound the total regret over each episode by considering four separate cases:
1. 2 < Co log d: Here, is the constant in the statement of Proposition 19, In this case, episodes are not large enough to estimate the parameters accurately enough, and thus use a naive bound. Again, by Lemma 14, we have E(reg,) S vt < P. Hence the total regret over such episodes is at most 4Pcgsg log
2. co log d Applying Corollary 20 to Equation 18 in episode m,
2m-22 < cad: Here, Co is the constant in the statement of Proposition 21.
Im+1-1
Reg(math Episode) = Σ
E(reg,) S CC
E(1 m
-3(00)
t=
ci d Im nx
-bm=1 202 log d 16W 20g + Lm- 20
CCCmax
where in the last step used In 20 and bm 2m-1 203d.
30

---

3. cyd d < 2m S eny: Applying Corollary 20 to Equation 18,
Im411 -1
(m+1- -1
Reg(mith Episode)
B((eg,) S CCmax
E(11 m
(0)
-3
t=lm
1=
{
eedlog dm Im + 20280 log + 16W2
In
}
S CCCaax
+ 1m 10"0
ny
Combing case and 3, sum over eid log a' over Im E [20080 log 20mx) and ignore the constant term, we obtain
me
1
Regret(co@0 log nx; S sgdlogd
Σ 2m-1
+ C250 log log n.::
Cni
co log d
cidlog d
+ C2 s0 log d log nx
nyc
= Ciso log dlognng ny.
4. 2m Tnx: applying Proposition 22 to episode k, we obtain
Ami11-1
Reg(mth Episode)
Σ
E(reg,) S CCmax
Σ
B(1I3m
11(
dlogg d
(m--1 + mm
-mm-1 13d
< CC
4m-1 mm + 8w
d
CNK
-Im-1
< CC
2oddlogd + Sw2
d + 21
egd
and thus
Reg(mth Episode) < Codlog d.
Combining the above four cases, we get
Regret(T; m) S Cidlogd (log T lognk) + C280 log d log ne
= (dg d log T + (so d) log d log nxx).
which concludes the proof.
G Proof of Theorem
Define the instance class I(F, Σ, W, 80, K) as above and, w.l.o.g., restrict prices to [0, W)) where p F, W) supiilss h(u). Let
2" (w) = の 1 (-u),
B₁ = sup |= (u)),
ulSW
① min
inf o' (u)), l
fmnn
inf f(2), B.
fmax :=
sup
f(2),
=EI-W, F(F.))+W)
inf e = =EI- W. #
min(F(())11= F(z)},
inf (1 (z)))
::= Smin &
Pminy
my
For Logistic and Gaussian these constants are explicit; see Corollary 24 below.
We proceed in three steps: (i) curvalure of the revenue around the oracle price; (ii) single-sample KL. upper bound that uniform over the allowed price interval; (iii) a Fano packing argument for two hard sub-families, whose risks add up.
31

---

(i) Revenue curvature and price sensitivity. Fix u = xT3. Let r(p:u) := p F(p - u)].
The first-order condition gives the oracle price p" (u) h((u) u 2" (2),, where x 2²(u) solves
1 F(=") = p" f(="). equivalently (f=") = u. A direct calculation shows
0²
0p² r(p" (u):u) = - -f(= df=) = -x(u), <(u) = f(=" ((=' (2 (u)). (19)
Hence r(;u) is uniformly strongly concave around (u( with curvature at least K jg n(u) = fmin min > 0.'
Next, differentiate h(u) = u + p-1 (-1( to get
1
('(u) = (u)(
my == inf h' n'(u) = 1
> 0.
(20)
lu|sW
T min
By strong concavity, for any Au small enough (chosen below) and Ap == h(u Au) h(u),
K km 2
r( r(h(u): r (hh(u + Au);u) > 22
Ap²
(Au)?.
(21)
We ensure the uniform validity of (21) by choosing the pack radii (below) so that Apl stays within a fixed neighborhood where (19) and the lower bounds defining is and my apply; this is straightforward since h is bounded on [- W W].
(ii) A uniform single-sample KL bound. Fix any market k, round 6, context and price p [0, P(F, W)]. Under parameter B, the success probability
((3)
F(p xT B).
For any B, Taylor's theorem for the Bernoulli KL yields
(9(3) - 9(3))2
KL (Bernn(( ((()) Beenneenll (ern(g(('))) -
(22)
2c (1
because both ((3) and (BB)) lie in jc, c] by the price truncation and the bounded ranges of p and u. Using the mean-value bound F()) F(s') S fmax 2 on the same interval gives
f
(Lern (ernn((('))) S
2e(1 e) (x (B 8¹))
(23)
Taking expectation over x Pa and using B[(x) A) Cmax || yields
E[KL(-II-] S CKL || - 1 CKL W) = Omax Paax
(24)
2c(1 -
Summing over markets gives factor K when all markets differ (family A below), and a factor 1 when only the target differs (family B).
(iii) Packing and per-round error. We use hard sub-families and calibrate their radii that the cumulative KL. up to time - is a small fraction of the packing entropy. All estimates below hold conditionally the realized (possibly adaptive) prices because (24) is uniform in p [0. pl.
(A) Aggregation. Let V C +-1,+ be a Varshamov-Gilbert set with [V| > 24/8 and Hamming distance at least d/8. For v E V define the instance by g(()) = 62 := μ for all k E {0} U [K]. Then ||0,|l 1l pd < W provided μ S W/d. For any distinct v. v'.
n² < 10, 1 S 4d
Up time t t- 1, the pathwise KL between the two induced joint laws (over all K markets) is bounded by
KL I K (t - CKT. 4d
Since o' 2 T.o, on [- Bu] and (w)( Bg for |u] S W, Pmnin 2 Toop Moreover, fuin > 0 on
|- Bg, R.] for Logisticc/Gaussinn F.
OOn p x" 8 p-W, w) + W] we have F(x) E [c, 1 s. F(x)) [c,
32

---

Choose
log V
log 2
p² = 64 (t 1) CKL d
512K (t 1) CKL
and also μa I W/d (which is automatic for all large enough; for small it only improves the bound).
Then Fano's inequality gives a constant probability (say 2 1/22 of misidentifying the pack element, hence
1
B 013
8 udd.
By B|( [((T
2 Cmin Bl|elBa (i,, and (21), the target-market instant regret at time /
k km E [[ (O)T (0-11 - 0)) z
km
p²d
Cmin mi d
E[regd] 2
Cmin
2
8
128 CKL K t
Summing ....., T yields
Cmin m
d
E[reg,] >
log T.
128 CRI.
K
(B) Debiasing. Let S C [d] with | = so and let W C (-1,+1,0)
07d be a Kessamov-Giilber family of Sp-aperse sign vectors with pairwise Hamming distance least su/8 and cardinality
(oo 1/8 200/8 so that log [W] so log Let R be any fixed vector with ||0el12 S W/2
[W]
and define for each w E W:
g(k) 6g (A E [K]),
3 0g + Aw, Aw := n w,
with ||oe|1 + |A S W ensured by taking je S W/((so). Then for w + w',
2 < AAw A. 112 s 4so Hs
Only the target market carries information about hence up to time t 1
KL1:-1 I (t - 1) CKL 4sg μ
Choose
122 64 (t CKL so
log W
log nd
80
512 (t
CKL
Fano again yields BJl - A 1 2 ul 80, hence the target instant regret obeys
Cmin Km
Su log
E[reg,
>
128 CKL
Summing gives the second term in (8).
Combining (A) and (B) gives the stated result.
Corollary 24 (Explicit constants for Logistic and Gaussian). Let B₂ = sup (=" with =²(u) = (-u) and p(F, W) suPjulgw h(u).
Logistic noise
F(z) Ite-2, [(z) F(=)(1 -F())) We have
(2(2) F(2)' h'(u)= = 1- 1
(1(2((u)) = -l))(²(z)), <(u) =10"
Hence
Pmin F(B.)'
my = inf wuls
1'(u)=11 -1u=111 FFF(B))
== inf lulsw
= 1-u)==
Moreover, fmax = 1 and
min -E(-W.
min(F(2),, 1(F)) = = F 2 1- + +B.).
33

---

Therefore.
maxx
Cmin m
32 min
W) 32 62 e)²
c(1 - (1-F(BB) F
OKL
Cmax
Gaussian noise
F(z) ((2), f(z) - =0(2)=
12 Let R((2) == 1-4(z) be Mills' rutio. Then f(2)
V2a
= = =((2),
& min =
inf (2 2 R(z)) 2 - B, R(B.). l=|=
Using the inequality R(z) -
for > gives B,R(B.) S
B1+1' hence
B²
1
1
1
min
2 +
mh
B +
Pmn
B
+
Also smin = :B()) V2
B2/2, mmax (0( = /25 and
E min xEI-W, WWF
min( {x2)} = 1 1 (W +D(F,W)) 2 1 (22 + +B2.
Therefore
== fmin 2 min
1
B2/22
1
CF,
Cuax
2m
B +1
4T cl1 6)
and
Cuin K. K.
2Tt Cuiin
c(1-c)
1
BB2/2
CKL
Cuax
22T
B +
(B? + 2)2
In both cases Bg and p(F, W) are finite since o is strictly increasing and |u] s W they are explicit functions of (F, W) via 2" (u) = (-u) and h(u) + 2" (u).
H Proof of Theorem 10
Following the same analysis in Theorem 12, we have
E(reg, H(-1) rev (pt (0)) rev (0) (p) ) + (p) (pi (0)
Pr: (0:
C (pt (0) Pt (0))² Ch(y
(1 (1))) l(g( (z:
< <CL L3 (g (x)
(0) - (2)
where C absorbs the bound on (p) over [0, Pl and L, the Lipschitz constant of h on the working interval (by Lemma 15 and the price truncation).
Using Lemma 15 gives
( (0)) -mmm (1r E(reg,) = H+-1)) S (
(25)
CE (0) g(m) 2
I C g0 g((z)) | 76
(a(P.)
where k is the kernel bound in Assumption 7.
20
Proposition 25 (Aggregation Error). Under Assumptions 2 and 7, choosing Aag x nx
the
aggregation estimation error of Algorithm 3 satisfies
E (l1g(10) - glast S C (R² ng
208
(26)
where R. is the constant in Assumption 7 and is the standard deviation of the market noise in (1).
34

---

Proof. Let E : [x( @ (xx -)1 denote the kernel integral operator and [() == Tr(E(E+ + ) the effective dimension. Consider the regularized empirical risk
(x(g)= = nk Σ (l::Pr, vs) + Allglbae
with the Bernoulli log-loss f(g: P, a, y) == - [1 1) log(1 F(p - ((x)) 1(y = 0) log( F(p
g(x)))) used throughout Algorithm 3. Let
(ag) gx
min (g) gE
arg
be the population minimizer. A standard RERM decomposition for smooth, strongly convex losses yields
(ag) 2 Bl|g(n)) - 92 (P.))
N(A)
E g
12
N(A)
92
TyC
TyC
using (1p I l For the approximation error, under the source condition g
(ange((>)) (Assumption 9i)) we have
1192 (ag)
(aa)) 2 2 x²
Assumption 9() gives N (A) x X- 12(/(2)), Balancing N (AA)/TX and 2 gives Aag x nx and the stated rate.
-20((228+1)
Proposition 26 (Bias Correction Error). Under Assumptions 2, 7 and 8, choosing des X (2oH2) -22221)) the debiasing error of Algorithm 3 satisfies
E
- - S C2 -2a/(20+1) 22
(27)
where H is the task task-iimilrrity parameter in Assumption 8.
Proof. According to Chai et al. [7], the regularized estimator can be written as
gio
(5(0) + Xes!) (G >
+ d))))
Hence
210) 5(0) = (5))) + 1 -1 (7()) (10) + Xaf (5)0) + 1es 1) ((W)) -800)
Variance
Bias
For the variance term, writing A == (1/2(5(00 + A1) implies
No(Aer)
B| Variance 2 (P.).)
where No(A) x==1222)))
no
For the bias term, by spectral calculus,
|X + uul (2())) Σ
Aap (M3 + A)
(u, n,,
s
X 4 |tul
so Assumption 8 gives E. Bias (P.) S H2 Balancing No(A)/no with XH2 yields des x (no -2a and the stated bound.
We now bound the total regret over each episode. Using Equation (25), obtain
bn+i-1
bm+i-1
Reg(mth Episode)
Σ
E(reg,)
S C
Σ
Bl - g 112())))))
t=m
t=mm
C22m-1f
((Kmm)
(K2m-1)
2018
+ (2m-1)-zti
Iull H 2e
35

---

Summing over M = [log T episodes gives
Regret (T;
- C
2 2m
(K 2m (2m T2 H zu17
m=1
where
20,81I
2a
and
72
2a+1
2a
Let
M
M
Term 1 = K
Σ
2
(2 (2m-2) = = 2222
(2--1) m S CiK -
m=1
m=l
Term 2 = H 20-1
2 2m (2m-2)
H 20+4 2
(2 mm) S C2Hl 21
m=1
Combining both terms gives the overall regret bound: 2n88 Regret(T;; 0 K 202+1 T T20241
+ H 2 T20+1)
I Proof of Theorem 13
Similar to our analysis in Section K, Algorithm enters Phase when the volume of target-market data is large enough to provide more accurate estimate than transfer learning. The boundary condition can be written as
20de1
Th no 11-
C2 Ci
11
20+1
20,
which, for simplicity, denote as
no > ēnk.
We bound the total regret over each episode by considering two cases. Throughout, we use the per-round conversion
( (1 B(reg,) s - from (25), so that episode contributes factor 2 in front of the corresponding L2(Pa) error bound.
Lz(P.)
Case 1: 2m-2 < ēng. In this transfer-active regime, we directly apply Theorem 10. Combining Propositions 25 and 26 and then using (25) yields
fm+s
fm+1
Reg(mth Episode)
Σ t=
B((rgg)
S Ci
Σ
(0)) (1))(P))
t=
20
$ C1 2m
(2m-2)
H 20
Summing over M' [log 28 nx)] episodes,
[log(2ānx)) nx]]
Regret((nme: T) < C
2m
n
2nd 292ft
+
(2
2a
H
m=
Case 2: 2m-2 ong. In this target-only regime (Phase 2), we obtain
fm+1 -1
Em+ -1
Reg(mth Episode)
B(reg,)
S C2
Σ
B - (0 (2.(2))))))
t= pm
t=
S C₂ 2m
(2m-2)
Summing from [log(2Enx)) to [log I' episodes gives
[log T1 Σ
Bo.l 2 (2m-2)- In0tI.
Regret(Zng + T; S O2
m=
[log(2
36

---

Combining the above two cases, the total regret is bounded by
[lgg(2mm)))
Regret(T;)) S Ci
2
2u0
+
(2m-2)-5a0) -1-7
H 2n41
(28)
[lo
+ C2
2 (2m-2) 2nd
mum [Tog( (2mm)) 1
We now decompose the first sum into two parts.
Part I. Using ΣM 2m-1 -2 S 2ēnk.
[lg((2nn:))
Part = C1 nx
2ud TaptI
2m
-2 nx
F.
Part 2. Since 2m pm-1(2m-27 2)- 20
Di
2011
23 we have
[x))
Part 2 = Cj H I
2m 12m-12m--5 (2m-22) 2-- 20 Ia1I
m==1
[log(2Enx)) (26nx]]
C1 H IT 2
20-1
Σ
2 sart S C1 H ITTT (2ōnk) 2041.
m=
For the second sum in (28), note that
2 (2 3) 167+1 (mm 1)--1m
200
m-1 220741
x (constant),
so the summation behaves like geometric series with ratio 22/(209+1) > Therefore
Dog Σ
2m 2m-1(2m-33))
200 IacII
S T 2dd (ēnx) ZOOTI.
(29)
172m [1og (2 mx
Combining Part 1, Part 2, and (29), we obtain
Regret(T;;t m) S ēn IndTI
+ H 20.1 (ēnx) J011 + 7 Trader (ēnx) IaplI
which concludes the proof.
J Proof of Theorem 11
We convert regret to an T.2(Pp) estimation error, upper bound the total information (KI.) any adaptive policy can extract under binary feedback, and then invoke Fano with tensor-product packing built from (i) a transferable common block and (ii) a non-transferable residual block.
Lemma 27 (Local quadratic revenue drop). Under Assumption 1 and the local regularity above, there exists c. such that for any X and any estimate with g(x), ((x) E Un.
Miev m
rev rev(h(g(z)) g(x)) - rev (1( (1((((z)) g(x)) 2 c. - g(x))²
-
C.
2
Consequently.
E [[Reg(T: 2 c. T. T. g1 g1b=-p.)]
(30)
where ÿ is the utility function implicitly induced by the policy π through its posted prices pt = h(g(x()) during the episode.
37

---

Proof. By strong concavity at (u) = h(u), for any u E Uo and close enough to p"(u) we have
rev(p" (u); u) rev (p: u) Maw (p- (p-p"((u) (u)), Set u = g(x) and p = h(ā(=)). By bi-Lipschitzness of h on Mo. Ip p" (w) (((a) - h(((z)) 2 m. (0x) - g(z)), which yields the pointwise inequality with C. = Mara mi Summing over I and taking expectations gives (30).
2
Lemma 28 (Bernoulli KL smoothness). Let g(p, 2) == F(p- u) and consider Bernoulli distributions with means q(p, u) and q(p, w'), Assume F' has continuous density f on [-Be, Be] and extends smoothly to the boundary with f(+Ba) = 0 (or take any log-concave on IR and restrict to compact interval of utilities). Then there exists a finite constant
f(8)²
OKL == sup SER (()) q(0]
0c,
such that for all p RR x E X and u. v/ E R,
KL KL(Bern(eCe w)) (ee)) S OKL (u-w)) w').
(31)
Consequently, for any (possibly adaptive) policy π interacting with the target and K source markets over T rounds,
(L g......."
|| P pro .x.gg)....x)) s CKi. T -100 +
> k=1
11 -9 g ((i(()))
(32)
Proof. By the mean value theorem, lq( u()) = FF(p w²) F(p u) S
sups f(6 |u w'1. For Bernoulli variables with means a, b E (0, we have the standard bound
(L( S bc b)) Combining yields (31) with OKL sup, o(8 "()1) which is
(e-b)²
f(5)
finite under the stated regularity (continuity plus compactness ensures the supremum is attained;
typical families such as probit/logistic also satisfy CKL I 1/4). Summing the one-step inequality over time and markets and applying the chain rule for KL under adaptivity gives (32).
Proposition 29 (Packing for the aggregation block). Define GA(R) {g >> : llellz S R). where Σ is the kernel integral operator with eigensystem (Hz, P;)spp and 3 E (0. 1] (Assumption 9). There exists CA > O such that for all
R.
log M(5: 9a(R), I1 (Ila(ps)) 2 cA
11
Proposition 30 (Packing for the debiasing block). Ler (B(H) = {g E Hk lglb S H} with bounded kernel (Assumption 7). There exists cB > 0 such that for all 0 < 8 < H,
log ((8: GD(H), ll II- (Ir((p))) 2 CB H
Proof. Let (My Pp)) be the eigensystem of Σ. Assumption 9 (i) states N(A) Ir(E(E+A)) S
--1/(2m), Working on spectral slice (j : Hy E (A/2, A]}, the number of coordinates in the slice satisfies m()) = (A/2)) - N(A) 2 x-1/(2a) for sufficiently small À (selecting a subclass saturating the difective-dimmnsion rate is admissible for minimax lower bounds).
(A) Aggregation block (R). Fix a slice JA(A)= (j (j: (A/2, with cardinality ma(A) 2 X 1/(aa), For 0 E {=1]mm> mm define
Pg
Σ
pJ
9A.0 Ed
Σ
1 1 43j
mA
/M
JEJA
JEJA
with amplitude a & R/2 to ensure || 12 S a < R/2. By the Varshamov- bound there
exists CA C fm with |C,l 2 2ma and Hamming distances at least m //8 Hence, for f + 6' in CA.
a² Σ
a
I1gA,0 91.
MJ
28
07)²
Z
12
29 X a² 23
MA JEJA
M
38

---

Thus the L seseprrrti is at least 2A with (A x a a while the packing size obeys log MA(A) 2 cmx(A) Z A 1/2a22), Feasibility in RKHS is also satisfied:
a²
11gg,olixx
x a² 12
m.
Pj
JEJA
Choosing small enough and a $ min(R/2 co /1-28)/23 ensures |19 - RR/2. Therefore,
logM(22xx (A(R), || (112))) 2 X 1/2aa with 8, xxxaX, a S R/2.
Eliminating gives log M((A) 2 (a/04))/a3), and taking R/2 yields the stated bound.
(B) Debiasing block GB (H). Choose slice JB(A) disjoint from Ja(A) with mB(A) 2 A 1-1/(20) For Ç E {+ define
_B, = mBB
JEJn
Then 119n Varshamov-Gillees there exists Cn with [Cnl 2 2mm/8 and Hamming distances least mn/8. For S # C¹ in Cn.
b² Σjeソ& 1 b2 so choosing & H/2 ensures 119 I H/2. By
6²
b²
19 -B
(S) - 53
mg
A
b² A.
mB JEJB
mB
Thus the separation is at least (A) with SB (A) h 1//2 and log MB 2 A 11/(2a), Eliminating yields log MM8B) Z (b/8))'/o; setting b H/2 gives the claim.
Pointwise control. Because |K I (Assumption 7), we have [9(x)| < kklgglakk Scaling the amplitudes above by universal constant (absorbed into CA. CB) ensures 9a(x),9B(z) Uo for all æ, which is used in Lemma 27.
We now build two packing families on disjoint eigenspaces:
FA G
FB Ca(l),
so that for any 9A.9's E FA and 9B.9' I'3 E FB,
(( + gB) (g'A 9 Ilgx -'Alii2- + +1lg g'alli
and the KL bound (32) splits additively as well. also restrict amplitudes so that g gn(,) E Uo for all x, by the remark above.
Lemma 31 (Fano for the aggregation block). Let FA C G be 25 A-packing with cardinality MA. For any policy n that observes the target and K source streams over rounds,
83 2
( 1 1
4CKL KT 53 + log
inf sup DEFA
- - allial 2
log MA
Lemma 32 (Fano for the debiasing block). Let FB c GB(H) be a 28g-packing with cardinality MB. For any policy x (only the target data contribute here),
0
4Ck1. T T0% + log
inf sup gefn
B[Ila 01122]
log M
Proof. Apply the standard multi-hypothesis Fano inequality [27] to the packing families FA and FB.
The average pairwise KI. over each family is bounded using (32): for the aggregation block all KT Bernoulli observations contribute, yielding the factor KT: for the debiasing block only the T target observations contribute.
39

---

Combining Lemmas 31-32 with Propositions 29-30, the calibration
4 CKL KT 8 x log MA 2 ca 8
md
4 4KL T 783 x log MB 2 cB 88
gives (after eliminating 5A,0D) QB)
0 x NIS K
Bnd Z00TI
T
2nd 2aUTI
88 x H 2041 T
2a Io+1
(33)
Finally, define the product packing F = {gw 9A,0 + _B. 9A,0 E FA, J8, E FB) so that log | log MA + log MB and pairwise L distances add in quadrature. The total KL between any two elements of F is the sum of the block-wise KLs by (32). Applying Fano F yields
inf sup E [ll@ g [ 2 8 + 0z gEF
and combining with Lemma 27 via (30) completes the proof of the lower bound in the main text.
K Proof of Propositions 16 and 19
We first provide a detailed proof for Proposition 16, which follows by combining Proposition 33 and Proposition 34 using triangle inequality.
is realized using all the source samples. It's probablistic limit is g(a0)) The corresponding estimation eтror can bounded by the following proposition.
3 (ag)
Proposition 33 (Aggregation Error). Consider linear utility model with Assumptions 1, 3 and holding true. Then, there exist positive constants dg. C12 o2 such that, for n.K 2 ddd, the following holds with probability at least 1/d:
[| (ag)
113
doog
Y1K:
Moreover, the probabilistic limit ((ag) is biased from (00) # ((() in general. We then correct its bias using the primary data in target market. The estimation error of the debias term can be bounded as follows.
Proposition 34 (Bias Correction Error). Under conditions of Proposition 33, there exist positive constants Co, C1, Cg such that, for no 2 coso log d. the following holds with probability at least 1 1/d - 2e mee-no/lepp (
so log d
71
The key distinction between the Proposition 16 and Proposition 19 lies in their sample size re- quirements: while both require the target sample size no Z coão log d for valid estimation, the online-to-online setting imposes an additional constraint no Z d K to account for the simultaneous learning from initially limited source data across K markets. This reflects the fundamental opera- tional difference that online-to-online must handle concurrent data scarcity in both domains, whereas offline-to-onling leverages pre-collected source data (implicitly assuming is sufficiently large).
K.1 Proof of Propositions 22 and 33
Let (X*,, yk) denote the design matrix and the response vector by row-stacking of all source data (a) (k) (ke)
Dt MEEO for le E [K].
By the second-order Taylor expansion around the true parameter (lag) we have
1 (0)) ((3((1))) = (VI((((0)) (l0)) (ag)
-((e)) + 2 ( (ag) Blag) (2 B(a0)))
for some 3 on the line segment between Glag) and 3 (ag)
Invoking Equation 6, we have
VL(3)
1
Σ {(3)).,
V2I(Ø)
Σ n1(3)
(34)
ne
T.EXK
40

---

where V and V represents the gradient and the hessian w.r. B. Further,
((3 f()((3))
f [ur((3)
F(u+(3)) I(v: -1) + - (33) [1 +1)
= log' F F(ue (3)) = -1) log't (1 -((u(()))[[(v== = =1].
(3( (3))2 (((x
(3))
(3)
F(ux(3))2 F(ur(())) I(y1 -1) + (1- (B)
+ 1 F(v(
I( +1)
=-log" llog" (3) 1) - lloo F(w( +1),
where ue( = Pt- (Ir. By lemma 14, have
[3)(3)] s pel 11 P W
(35)
Let
{max (log' F(x), log' (1-
uF sup P=|
EF
inf {min (- F(x), 1o- P+W
we bound the gradient and hessian.
Lemma 35. Let
log d n.
J (I( Suup
we have
To bound the gradient, lu. P+ W, cf. Equation 35. Therefore, by definition of lp, have (B) 2 lp. Recalling Equation 34, we get (2T (x( X).
By the optimality condition of g(00) we write
L(3 (a9)
< L(B(o))).
Rearranging the terms and using the bound on hessian, we arrive at
EF
(x(())) - -3(08) || < (L( )l 13 (ag)
- -3/00) 1|1
ThC
Choosing À AuF V
logd nx
we have on FF
2 (3(eo)) -3(00) )11 x1133 -B(00)
(36)
Define the event B, follows:
B, = RRxed a (XT X/Tx)
> (XTT/r 2min/2).
Using concentration bounds on the spectrum of random matrices with subgaussian rows ([29], Equation 5.26), there exist constants c, c, > 0 such that for > cıd, we have P(B.) > e cn
By assumption 3, the lhh.s of Equation 36 can be bounded by the minimum eigenvalue of its second moment matrix
2FF || | (000)) < ا|I/ (0a) -B(a9)) 1| < ((9) -B(09)
TiK
and therefore,
[[((/0)) - (0s)
2x²
C2
F min
41

---

K.2 Proof of Proposition 34
By the second-orde Taylor expansion, expanding around o we have
). L(ö + (300) L(6 + (16+3(00) ) = (V + (3))) - 5) + 2 8,
((0)
10 -8)
for some on the line segment between and 8. Again we have
V 3 (og),
Σ + + (3(00))
(ag) Ja,, L(6 3
(0g) (6 + 3 )ar
where V and 2 represents the gradient and the hessian w.r.f. 6. we bound the gradient and hessian. According to Lemma 35, define
F= IV + +,((00) 1loc S 2uF log d
we have P(F) 2 1 1/dd (6 + 8 g(09) N EF, and V2L + 8 )la0)
2 (x((xxX) x).
By the optimality condition of 8, write
L + 3 (09)
ggō S L(6 33 (00)
++ +|16111
(37)
Using the bound on hessian and gradient, choosing X N Aupy logd no
we have on set
2 |x (0) (8 - 8)11 + 2X |01l S à| 8111 + 22 0011
(38)
n40
By Assumption 4, 8 is sparse. Let S = supp(
On the l...s. using triangle inequality, we have
11 (19) 111
118
(10)
111 + 10 g=
1
1
(ag)
s
100
(a9)
glag)
s +
((9)))
11
On the rh.s., we have
18 (ag)
(ag) 1| = 10
(ag)
((a))
1| + 110
(ag)
111
Using these two equations in Equation 38, we get
28p
|| (0) (8tag)
8 (()) 112 + 2 g'
(ag)
111 S 33 (ag)
- ((0))
(39)
no
We next write
28F
|xx
(8(09) - )11
+ +110 (1g) -((0))
22r
(
-(00)
)1( A110s
(ag)
s
(ag)
0110ge ge
(ag)
n0
710
(4)
4X|10g
2(ag)
(ag)
(ag)
112
(b)
S 4x sol
8
(ag)
8
(ag)
s
ll2
42v280
(0)
18 (ag)
- |||,
CF | (0)
(8
(00)
sla +
8X280
740
C
where (a) follows from Equation 39; (b) holds for Cauchy- Schwaar inequality; and (c) by RE condition ([19), Proposition 23), which holds for 5(0) ((X(0)) ((0))/n /no with x={(()) s0, 3) / /Cmin/2 22 and (d) follows from the inequality 2|ab] S ca² + Rearranging the terms, obtain
12
fF (x) - -3(00) )12 \1r - 8(00)) < t Cmmm
88g2
ng
42

---

Applying the RE condition again to the L.h.s, we get
E min 2
(16(19)
g(00)
fF 12 no
(
-2000)))6000))) 312
8s0 CpOmin
and therefore,
16s0
|6 06-
P.C min
L Proof of Propositions 18 and 21
Proposition 36 gives tighter bound for the estimation error of the debias term 8 as ng gets larger.
Proposition 36 (Bias Correction Error). Consider linear utility model with Assumptions 2, 3 and 4 holding true. There exist positive constants Co, C32 C4 such that, for no 2 cod, the following holds:
(so 1) log d 2 cond
E(1 6
S
T1
The proof follows Proposition 12 from Javanmard and Nazerzadeh [19]. Combining Propositions 33 and 36 using triangle inequality gives the stated result.
M Proof of Lemmas
M.1 Proof of Lemma 14
By Assumption 3 we have 13 111 W and |=; 3
W for all k, The lemma holds beacause h a continuous function and continuous functions on a closed interval are bounded.
M.2 Proof of Lemma 15
Recalling the definition h(u)= = u + (-+),, we have H'(u) =1 strictly increasing by Assumption 1, have h'(u) 1.
Since e is
M.3 Proof of Lemma 35
According the definition of up, we have (g((0))) S ur. Further, recall that the sequences {pr)i= and {Ie ) are independent {=, } Therefore, (B and (3 I are independent and by Equation 2, we have E[g, ((())) } [ (G((0) | (3 (3()))) = 0, which gives ([VI(((99)) ] O.
By applying Hoeffding inequality one coordinates of feature vectors,
- E 2 all = [VL()( 2 all s expf 2
d2
where a
|aş| < up. Following a union bounding over d coordinates,
d ( = IP S(UA A
$ 2 P(A)) = d
1==1
1=1
The result follows.
M.4 Proof of Corollary 17, 20 and 23
Here provide the detailed proof for Corollary 17, which also works for Corollary 20 and 23.
43

---

We let 9 be the event that Equation 17 holds true.
Then by Proposition 16 we have P(g) 1
1 - 2/d - 2e 2
8 31)
R[ 3000
Tpl + [I(113
300 TGo
d logg d nx
log d
+ PW9)
C2
no
S
log
+ C2
sp log d
+ 4w
2
+
mnn 2e "0
n10
d
44


Reviews:
[{"review_id": "usduClaENf", "reviewer": "Reviewer_cCwA", "summary": "This paper studies contextual pricing where a target market is able to leverage $K$ auxiliary markets, whose utilities differ by a structured preference shift. Specifically, the paper adopts a stochastic utility model $v_t=g(x_t) + \\epsilon_t$ where $g(\\cdot)$ is an unknown mean utility function and $\\epsilon$ is a random noise. In this paper, the authors propose a Cross-Market Transfer Dynamic Pricing (CM-TDP) algorithm that achieves (a) $\\tilde{O}(d/K \\log T + s_0 \\log T)$ regret for linear utility with sparse differences, and (b) $\\tilde{O}(K^{-2αβ/(2αβ+1)}T^{1/(2αβ+1)} + H^{2/(2α+1)}T^{1/(2α+1)})$ regret for the RKHS utilities. They also conducted numerical simulations, where CM-TDP significantly outperforms single-market baselines, reducing cumulative regret by up to 50%.", "questions": "What is the computational complexity of CM-TDP in different settings (i.e. in Off-to-On and On-to-On settings, on the linear model and RKHS model)?", "limitations": "Discussed with less sufficiency. See \"Weakness\" above.", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 4, "originality": 4, "strengths_and_weaknesses": "### Strengths:\n\nTo the best of my (as well as theirs) knowledge, this is the first paper in the pricing community that studies the utility-model shifting effect. They provide rigorous analysis and deliver minimax regret rate for both parametric and non-parametric settings (which is also the first non-parametric regret bound for transfer learning in pricing). From a high-level point of view, this work unifies bias-correction techniques from multi-task bandits with pricing, comparing to traditional bandits that only focus on prediction error. Besides, this paper is overall well-written and clearly delivered.\n\n\n### Weakness:\n\nThe most significant concern raised from my perspective lies in these strong modeling assumptions. For instance, the context distribution is assumed to be identical over all markets, which is clearly over-simplifying the scenario. Also, the utility noise distribution is assumed known, and the sparsity parameter is known ahead of time. All of these assumptions listed above are restrictive (although somewhat justified in Sec 3 but not very convencing) and undermines the potential application as well as the extension of this work. I understand that this is the very first work considering this problem, but I highly encourage the authors to add more justification and discuss potential limits/extensions with respect to each assumption they have made.\n\nThere are not much beyond my main concern, as this work is overall self-consistent. Besides, the notations are slightly heavier than I expected.", "was_revised": true, "final_justification": "Very good work, worth a 5 even after recalibration from the first-round evaluations.", "rebuttal": "We thank the reviewer for the thoughtful summary and positive assessment of our contributions. Below, we address the main concerns regarding modeling assumptions and provide the requested discussion on computational complexity.\n\n---\n\n**W1: Assumption on homogeneous covariate distributions.** Thank you for this insightful question! Our focus is on utility-model heterogeneity. To isolate this dimension, we assume homogeneous covariate distributions across markets. This ensures that performance differences arise solely from shifts in latent preferences, not from domain shift.\n\nThat said, our framework can **accommodate covariate (domain) shift** through extensions such as **reweighting** or **importance sampling** (e.g., [1,2]). For example, under mild overlap conditions, density ratio estimators or domain-invariant representations can be incorporated into the debiasing step. *We will clarify this in the revised manuscript.*\n\n---\n\n**W2: Assumption on noise distribution.** While we assume a known noise distribution $F(\\cdot)$ for algorithmic implementation, our theoretical analysis only relies on certain regularity properties (Assumption 1), not the exact parametric form. \n\nImportantly, our framework can be extended to **unknown $F$** using **nonparametric estimation techniques** as in [4], where both the mean utility and $F$ are estimated jointly. The aggregate-then-debias architecture remains applicable, but such extensions require significantly more technical development and considering possible transfers in estimating  $F^{(k)}$ as well. We feel it is best addressed in a separate paper. *We will clarify this point in the revision.*\n\n---\n\n**W3: Assumption on sparsity.** We appreciate the reviewer’s concern and agree it’s important to contextualize the **sparsity assumption**:\n\n* **General Framework**: Our framework permits any structured similarity between tasks; sparsity is one way to measure the function complexity of utility differences. For the **nonparametric setting**, we instead assume smoothness in an RKHS.\n* **Justification in Practice**: In many real-world applications (e.g., ride-sharing, retail), only a **small subset of covariates** drive cross-market utility variation (e.g., weather, taste, income). Sparse heterogeneity thus reflects natural structure and is widely used in bandit transfer and multitask learning [2,3]. \n*  **No Need to Know $s_0$**: Importantly, our algorithm does **not** require knowledge of the sparsity level $s_0$ — it only appears in the regret bounds as a function of transfer difficulty. \n* **Beyond Sparsity**: Our analysis can be extended to **other complexity measures** (e.g., low-rank differences or structured smoothness of the utility difference), provided they enable regularized estimators with controlled generalization error. We will highlight this point in the revised version.\n\n---\n\n**Q1** We now address the question regarding CM-TDP’s computational costs across settings.\n\n**Linear Utility Model**\n\n- **Online-to-Online Transfer**\n  - Each Episode:\n    - **Aggregation**: $O(d^2 \\ell_{m-1} K \\cdot N)$ for MLE using source market data from episode $m-1$, where $N$ represents the number of optimization iterations.\n    - **Debiasing**: $O(d^2 \\ell_{m-1} \\cdot N)$ for Lasso regression on target market data.\n  - Cumulative regret over $\\log T$ episodes: $\\mathcal{O}(d^2 T K\\cdot N)$\n- **Offline-to-Online Transfer**\n  - Each Episode:\n    - **Phase 1 (Transfer)**: $O(d^2 n_{\\mathcal{K}} \\cdot N)$ for MLE on aggregated source market data, and $O(d^2 l_{m-1} \\cdot N)$ per episode for bias correction.\n    - **Phase 2 (No Transfer)**: Standard linear MLE: $O(d^2 l_{m-1} \\cdot N)$. \n  - Cumulative regret: $O(d^2 (n_\\mathcal{K} + T) \\cdot N)$ \n- **Implementation Note**: The dependence on dimension $d$ varies by optimization method. Newton's method (our implementation) achieves faster convergence (smaller $N$) but maintains $d^2$ dependence. Gradient descent reduces the dependence to $O(d)$ per iteration but requires more iterations ($N$ increases correspondingly).\n\n**RKHS Utility Model**\n\nBase Complexity: KRR with $n$ samples requires $O(n^3)$ for exact matrix inversion.\n\n- **Cumulative regret**\n  - Online-to-Online Transfer: $O(T^3 K^3 \\cdot N)$\n  - Offline-to-Online Transfer: $O((n_\\mathcal{K}^3 + T^3) \\cdot N)$\n- **Implementation Note**: We use **approximate kernel methods** (e.g., Nyström, random features) to reduce runtime to **linear scaling in $\\mathcal{N}(\\lambda)$**, the effective dimensionality of the feature map.\n\n*We will add a computational discussion in the appendix to clarify these scaling properties and practical trade-offs.*\n\n---\n\n**Q2** We appreciate the feedback and **will add a notation table** summarizing key symbols and definitions in the appendix to improve accessibility, particularly for readers outside the pricing and bandit communities.\n\n---\n\n**References**\n\n[1] Fan Wang, Feiyu Jiang, Zifeng Zhao, and Yi Yu. Transfer learning for nonparametric contextual dynamic pricing. arXiv preprint arXiv:2501.18836, 2025.\n\n[2] Jinhang Chai, Elynn Chen, Jianqing Fan. Deep Transfer $Q$-Learning for Offline Non-Stationary Reinforcement Learning. arXiv preprint arXiv:2501.04870.\n\n[3] Kan Xu and Hamsa Bastani. Multitask learning and bandits via robust statistics. Management Science, 2025.\n\n[4] Elynn Chen, Xi Chen, Lan Gao, Jiayu Li. Dynamic Contextual Pricing with Doubly Non-Parametric Random Utility Models. arXiv preprint arXiv:2405.06866"}, {"review_id": "yaNSoSrgmT", "reviewer": "Reviewer_9euS", "summary": "The authors develop contextual dynamic pricing algorithms that can leverage auxiliary markets' data to improve the performance on the target market. The difference and similarities between different markets are illustrated by the utility functions. There are multiple algorithms developed, tackling several settings including different utility function class (linear parametric models, RKHS nonparametric ones), and different auxiliary markets' data (offline static, online concurrent).", "questions": "1. Justification of \"minimax optimality\". \nIn lines 10 - 11 (abstract), 52, 310, the authors claim \"matching information-theoretic lower bounds up to logarithmic factors\", \"matching known lower bounds\", \"matching information-theoretic lower bounds\". However, I did not find any specific lower bounds proved and provided by the authors, or referred to in the existing literature, that could support the minimax optimality of the authors' derived regret upper bounds. Note that the authors' derived upper bounds are complex in the sense that they contain multiple terms and are related to similarity parameters, e.g., $s, \\alpha, \\beta$. Therefore, I doubt whether complete (not just for special degenerated cases) matching lower bounds, as claimed by the authors, exist in the literature, or can be easily derived, for their considered settings. Could authors provide explicit lower bounds to better support their claimed minimax optimality?\n\n2. Cross-market, but with the same covariate distribution and market noise distribution. \nThe authors consider transfer dynamic pricing under different markets with differences and similarities in their utility functions. However, the covariate distributions $\\mathcal{P}_{x}$ (line 124) and the market noise distribution $F$ (line 103) are assumed to be the same across these markets. I feel this limits the practical impact of these considered settings, as markets that are different in their utility functions, are very likely to also enjoy different covariate distributions and market noise distributions. Could the authors elaborate on potential extensions towards different covariate distributions and market noise distributions?\n\n3. Explicit choice for $m_{0}$ in Algorithm 4, i.e., the number of initial and early episodes that leverage the offline static source log to implement the transfer learning. \nIn lines 153, 173 - 174, and Algorithm 4, the authors illustrate that the transfer learning is only implemented in early episodes. However, the number of these early episodes, which is denoted as $m_{0}$ in Algorithm 4, is not specified explicitly in the Theorem 9, Algorithm 4, and Theorem 10. Actually, the value of $m_{0}$ is quite essential and meaningful in theory, as it demonstrates a critical phase transition in optimally using the information in the offline historical data of other markets, and online stream data of the target market. Could the authors clarify on the choice of $m_{0}$?\n\n4. Task similarity in Linear Models as the sparsity constraints in the cross-market parameter differences. \nFor the linear model, the authors use an $\\ell_{0}$ sparsity constraint in the cross-market parameter differences. Such a modeling of similarities between utility functions can be justified. However, I wonder if other types of similarity measure, e.g., the $\\ell_{1}$ constraints, can be well incorporated into the framework of this work. Could the authors provide more discussions on this issue? \n\n5. How does the smoothness parameter $\\eta$ in line 236 influence the regrets?\n\n6. How the kernel $k$, as one of the input of Algorithm 3, is defined? How is $k$ related to the original kernel $K$?\n\n7. As the feature mapping $\\phi_{t} = (p_{t},x_{t})$ in Algorithm 3 consists of both $p_{t}$ and $x_{t}$, the $\\mathcal{H}_{k}$ (based on kernel k) in lines 2 -- 5 of Algorithm 3 should be different from the $\\mathcal{H}_{k}$ (based on kernel K) in line 225, which only concerns the covariate $x$. How is the $g$ in lines 2 -- 5 of Algorithm 3, as a function mapping from $\\phi = (p,x)$, different from and related to the utility functions $g^{(k)},k\\in\\{0\\}\\cup[K]$? \n\n8. Statement of Theorem 8.\nIn line 250, $g^{(ag)}$ is not defined, even across the paper including the Appendix. In line 250, $\\Sigma$ is not defined, but defined in the Appendix. In line 251, $N(\\lambda)$ is not defined, but defined in the Appendix. Could the authors clarify on these notations?\n\n9. Relationships between regrets in linear parametric case and RKHS nonparametric case. \nAs illustrated in lines 272 - 273, the regret upper bounds for RKHS nonparametric case can reduce to $\\tilde{O}(T^{1/2})$ in connection with the well-studied linear case. How is this regret bound compared to the $\\tilde{O}(\\log T)$ regret derived under the linear parametric case? Why are they different?", "limitations": "Yes.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\n1. Very practical to transfer knowledge and purchase data from markets to markets, especially useful when the target market is small.\n2. Lie in the intersection of several frontier and trending AI/ML research areas of transfer learning, contextual dynamic pricing, multi-task reinforcement learning.\n3. The established algorithms cover multiple scenarios, including different utility function class (linear parametric models, RKHS nonparametric ones), and different auxiliary markets' data (offline static, online concurrent). \n4. The interplay between the number of auxiliary markets and similarity measure between the markets is well displayed in the derived regret bounds, providing essential insights. \n5. The theoretical results align well with the numerical ones, thus it is likely that they provide operationally meaningful insights. \n\nWeaknesses:\n1. The minimax optimality, which is important and is claimed for many times across the title, abstract, main paper and conclusions, is not supported by explicit lower bounds. Moreover, such matching lower bounds may not be straightforward to derive in the considered settings. See Question 1 for more details.\n2. Less practical and a bit restrictive as the covariate distributions and market noise distributions are the same across the target market and auxiliary markets. See Question 2 for more details. \n3. Missing technical definitions, e.g., the kernel $k$ in the input of Algorithm 3, $g^{(ag)}$, $\\Sigma$ and $N(\\lambda)$ in the statement of Theorem 8. See Questions 6 and 8 for more details.", "was_revised": true, "final_justification": "Thank rebuttal provides helpful explanations and clarifications. I raise the clarity score and maintain my overall rating.", "rebuttal": "Thank you for the positive feedback and insightful questions. We have updated our manuscript to address the following points. \n\n---\n\n**Q1 (W1)** We appreciate this important question. While our manuscript does not provide formal lower bound proofs, our regret upper bounds are driven by estimation error rates that **match known minimax rates** in the literature—justifying our claim of minimax optimality up to log factors.\n- **Linear model**: Our regret bound $\\widetilde{O}(\\frac{d}{K} \\log T + s_0 \\log T)$ is derived from the MLE estimation error $\\|\\hat{\\beta}^{(0)} - \\beta^{(0)}\\|^2 = \\tilde{O}( \\frac{d}{n_K} + \\frac{s_0 \\log d}{n_0})$, which matches minimax rates in high-dimensional GLMs with sparse shifts [4,5].\n- **RKHS model**: Our regret bound is built on $\\|\\hat{g}^{(0)} - g^{(0)}\\|^2_{\\mathcal{H}_k} = \\tilde{O}(n_K^{-2\\alpha\\beta/(2\\alpha\\beta + 1)} + n_0^{-2\\alpha/(2\\alpha + 1)} H^{2/(2\\alpha + 1)})$, matching minimax rates in RKHS regression [2,6].\n- **Minimax dependencies on $T$** are further supported by the regret lower bounds established in [7], who study contextual dynamic pricing under separable demand models, where demand takes the form $\\mathbb{E}[y \\mid x, p] = D(p - g(x))$ for a known, strictly decreasing demand shape $D(\\cdot)$. While their model differs from ours—most notably, they do **not** adopt a random utility model—their results provide useful **single-task analogs** for our regret rates:\n  - For **linear $g(x)$**, they show a regret lower bound of $\\Omega(d \\log T)$, consistent with classical bounds in high-dimensional pricing.\n  - For **nonparametric $g(x)$** lying in a smooth function class (e.g., Sobolev or RKHS-type spaces), they derive a regret lower bound of $\\Omega(T^{1/(2\\kappa + 1)})$, where $\\kappa$ captures smoothness—matching the rate in our Theorem 8 when $\\kappa = \\alpha\\beta$.\n\nWe have also sketched regret lower bounds using standard information-theoretic techniques (e.g., Fano’s inequality, packing arguments), and will add formal proofs of lower bounds to the appendix.\n\n**Sketch of Lower Bound Derivation**\n- **Linear Utility with Sparse Shift** We now sketch a lower bound argument that matches our Theorem 5 (up to logarithmic factors).\n  1. **Hard Instance Construction**: Let the source markets all share utility $\\beta^{(k)} = \\beta^*$, and consider two possibilities for the target:\n     - $\\beta^{(0)} = \\beta^*$,\n     - or $\\beta^{(0)} = \\beta^* + \\delta e_j$, where $e_j$ is the unit vector and $j \\in \\mathcal{S} \\subset [d]$, with $|\\mathcal{S}| = s_0$.\n\n  2. **Randomization Over Sparse Supports**: We draw $j$ uniformly from a set of sparse supports $\\mathcal{S} \\subset [d]$, introducing uncertainty over which features differ.\n\n  3. **Fano’s Inequality**: We reduce the learner’s regret to its ability to identify which sparse vector generated the target market. Let $\\delta \\asymp \\sqrt{\\frac{\\log d}{T}}$. Then the KL divergence between observations under any two hypotheses is small: $\\mathrm{KL}(P_{\\beta^{(0)}} \\| P_{\\beta^*}) \\leq c.$ Fano’s inequality implies a lower bound on misclassification error, which propagates into pricing errors.\n\n  4. **Conclusion**: The learner must suffer regret at least: $ \\Omega\\left(\\frac{d}{K} \\log T + s_0 \\log T\\right),$ since misclassifying the utility results in suboptimal pricing. This matches our Theorem 5 up to logarithmic factors.\n\n- **RKHS Utility with Operator Smoothness**\n  1. **Function Class**: Define a class $\\mathcal{G} \\subset \\mathcal{H}\\_k$, such that for each task: $g^{(0)} = g^{(k)} + (\\Sigma^{(0)})^\\eta \\omega, \\quad \\|\\omega\\|_{\\mathcal{H}_k} \\leq H.$\n\n  2. **Packing Argument**: Construct a packing set of $g^{(0)}$’s in $L^2(P_x)$ norm with separation $\\delta \\asymp T^{-\\frac{1}{2\\alpha\\beta + 1}}$. Then, using standard minimax theory (see [6]), any learner must incur estimation error $\\Omega(T^{-\\frac{2\\alpha\\beta}{2\\alpha\\beta + 1}})$.\n\n  3. **Regret Linkage**: Via Taylor expansion of revenue (as used in Theorem 8), regret scales proportionally with squared estimation error, yielding a regret lower bound: $\\Omega\\left(K^{-\\frac{2\\alpha\\beta}{2\\alpha\\beta + 1}} T^{\\frac{1}{2\\alpha\\beta + 1}} + H^{\\frac{2}{2\\alpha + 1}} T^{\\frac{1}{2\\alpha + 1}}\\right)$, matching our Theorem 8.\n\n---\n\n**Q2 (W2)** We agree that markets with utility differences may also differ in covariates and noise.\n- **Covariate shifts**: As discussed in our response W1 to Reviewer cCwA, domain adaptation or importance weighting techniques can extend our framework to handle differing covariate distributions [1, 2].\n- **Noise distribution shifts**: With **known** $F^{(k)}$'s, our framework can readily accommodate heteroscedastic noise and each market has a context-dependent noise distribution, possibly with market-specific scales or shape parameters. To handle this added variability, one can incorporate techniques from robust statistics, such as weighted M-estimation, into the aggregation and debiasing steps [4]. If the $F^{(k)}$'s are **unknown**, one can estimate them nonparametrically and jointly with the utility functions, following ideas from [3]. However, this introduces new challenges for transfer—particularly in aligning or estimating heterogeneous noise distributions—which require more intricate analysis. We consider this an important direction for future work.\n\n---\n\n**Q3** In the linear case, Phase 1 ends when $n_0 \\geq \\tilde{c} n_K$ (line 523), implying $m_0 = \\lceil \\log_2(\\tilde{c} n_K) \\rceil$. In the RKHS case, $m_0$ is adaptive since $\\tilde{c}$ cannot be explicitly solved (line 594). The transition occurs when the target-only estimator outperforms transfer-aided estimation, determined empirically.\n\n---\n\n**Q4** Our bias-corrected aggregation framework is modular and can indeed accommodate alternative notions of similarity, including $l_q$ constraints where $q\\in [0,1]$. \n- **Computation**: For $ q \\in (0,1) $, the problem remains non-convex but can be approximated via iterative reweighted $ l_1 $ methods or proximal gradient descent. The regularization parameter $ \\lambda $ should scale with $ q $ to maintain consistency in sparsity control.  \n- **Theory**: The trade-off between approximation (bias) and estimation (variance) terms can still be controlled. For $ q \\in (0,1) $, the approximation error may increase slightly, but the estimation error improves due to better sparsity control.  \n\nThus, while our current analysis focuses on exact sparsity ($l_0$) for clarity and interpretability, the framework is general enough to support other similarity metrics, and we view this as a promising avenue for future extension.\n\n---\n\n**Q5** The parameter $\\eta$ controls the smoothness of the discrepancy between target and source utilities via the operator $(\\Sigma^{(0)})^\\eta$. A larger $\\eta$ means the utility shift lies in a smoother subspace, making it easier to estimate. This results in smaller RKHS norm $H$, and hence tighter regret bounds in Theorem 8.\n\n---\n\n**Q6-7** Thank you for pointing this out—you are correct. In our theoretical analysis, $\\mathring{g}^{(k)}(x) \\in \\mathcal{H}\\_k$ denotes the mean utility function, where $\\mathcal{H}\\_k$ is an RKHS over covariates $x$, induced by a kernel $K(x, x')$. In contrast, Algorithm 3 estimates a **log-odds function** $\\ell(p, x) \\in \\mathcal{H}\\_{\\text{joint}}$ using **kernel logistic regression**, where the kernel $K\\_{\\text{joint}}((p,x), (p',x'))$ is defined over the joint space of price and covariates. This function is distinct from the structural utility function $\\mathring{g}^{(k)}$.\n\nOur use of **KRR for log-odds estimation** is a key innovation enabling fully nonparametric modeling, in contrast to MLE in the linear case. Under mild conditions on the noise distribution $F$, the smoothness of $\\mathring{g}^{(k)}$ translates to smoothness of $\\ell^{(k)}$, preserving the transfer structure. We will revise the manuscript to clarify the role of $K\\_{\\text{joint}}$ and adopt distinct notation for $\\ell$ to avoid confusion with $\\mathring{g}^{(k)}$.\n\n---\n\n**Q8 (W3)** Thank you for the suggestion. $g^{(ag)}$, $\\Sigma$, and $N(\\lambda)$ are used in the RKHS theoretical analysis and defined in Appendix F. We will clarify references in the main text and add a notation table in the appendix summarizing all symbols.\n\n---\n\n**Q9** Although both bounds arise from the same CM-TDP framework, the difference reflects the **intrinsic hardness of nonparametric estimation** under binary feedback. The linear setting assumes known parametric form (GLM) and achieves $\\widetilde{O}(\\log T)$ regret via efficient MLE-based estimation. In contrast, the RKHS setting must learn a potentially infinite-dimensional function and suffers regret $\\widetilde{O}(T^{1/2})$ even in the best case. Thus, the difference stems from **estimation complexity**, not just analysis looseness.\n\n---\n\n**References**\n\n[1] Wang, Fan et al. Transfer Learning for Nonparametric Contextual Dynamic Pricing, arXiv:2501.18836.\n\n[2] Chai, J, Chen, E, Fan, J. Deep Transfer Q-Learning for Offline Non-Stationary Reinforcement Learning. arXiv:2501.04870.\n\n[3] Chen, E. et al. Dynamic Contextual Pricing with Doubly Nonparametric Random Utility Models, arXiv:2405.06866.\n\n[4] Xu, K. & Bastani, H. Multitask Learning and Bandits via Robust Statistics, Management Science, 2025.\n\n[5] Li, S. et al. Transfer Learning for High-Dimensional Linear Regression, JRSSB, 2022.\n\n[6] Caponnetto, A. & De Vito, E. Optimal Rates for Regularized Least-Squares Algorithm, FoCM, 2007.\n\n[7] Bu, J. et al. Context-Based Dynamic Pricing with Separable Demand Models, Management Science, 2025."}, {"review_id": "qLX7wIhACj", "reviewer": "Reviewer_Gb5J", "summary": "This paper studies the cross-market transfer dynamic pricing problem. The proposed framework can handle both offline-to-online and online-to-online settings and can accommodate linear and nonlinear utilities through reproducing kernel Hilbert Spaces (RKHS). Furthermore, the method can work for multiple heterogeneous source markets whose mean utilities differ from the target under certain assumptions. Numerical results demonstrate the potential gain by leveraging information from multiple source markets.", "questions": "1.\tThe proposed method is for the utilities with a structured shift from the target utility. What happens if the assumption is wrong? How do we verify the assumption in practice for a safe and reliable application of the proposed method?\n2.\tFor RKHS, when the linear kernel (inner product) is used, the resulting function becomes linear. Please elaborate the results for the RKHS with the linear kernel and compare it with the results for the linear case. \n3.\tSince it doesn’t appear in the bound, how does the dimension d affect the bound for the RKHS case?  \n4.\tFor the numerical results, the highest dimension considered is 20 which is quite low. How high dimensions can the method handle? It will be helpful to include some numerical results for higher dimensions such as in hundreds or thousands. \n5.\tThe patterns for the linear and RKHS in Figures 2 and 3 are almost identical. This is somewhat surprising. More elaborations will be helpful.\n6.\tThe simulation only considered identical and sparse settings. It will be important to explore denser differences and show how stable the method is as the differences vary from sparse to dense. \n7.\tIt is important to include some results on computational complexity and time comparisons.", "limitations": "The authors mentioned very briefly towards the end that “Future directions include automated market-similarity detection and extensions to multi-product settings”. In general, one challenging aspect is to verify that the markets follow the assumptions imposed for the proposed method. If not being able to gain from source markets, one needs to ensure at least not hurt the results without using transfer learning. More discussions on these are needed.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "The proposed transfer learning framework under utility shifts is quite broad and can work for both online-to-online and offline-to-online regimes. Furthermore, the authors consider both the linear utility class and a general nonparametric utility class through RKHS. This yields great flexibility for the proposed framework. \n\nThe proposed work assumes the utilities differ from the target by a structured shift. If true, as shown in the numerical results, the improvement can be substantial. However, it is not clear how sensitive the proposed method is to the violation of the assumptions. Furthermore, more discussions and comparisons between the linear and nonparametric cases are needed. In general, one would expect better bounds for the linear case than the nonlinear one due to the smaller class of functions for the linear case.", "was_revised": true, "final_justification": "The authors have carefully addressed all my questions and concerns. I would like to recommend to accept the paper.", "rebuttal": "Thank you for the positive feedback and insightful questions. We have updated our manuscript to address the following points.\n\n---\n\n**Q1** Our framework assumes that the utility functions across markets differ in **structured ways** (e.g., sparse shifts for linear models, smooth residuals for RKHS). These assumptions are critical for regret guarantees—but **not required for correctness**. If the assumptions are violated:\n\n- **Statistical efficiency may degrade**, but the algorithm remains valid.\n- Our bias-corrected estimator smoothly interpolates between full pooling and target-only learning, limiting downside risk.\n\n**Verifying similarity in practice** can be done via holdout evaluation: one can estimate utility functions on each source independently and compare them to the target using $\\ell_2$ or RKHS metrics (e.g., gradient norm or empirical discrepancy on covariates).\n\nWe will add a brief discussion in the revision and highlight this as a direction for adaptive transfer tuning.\n\n---\n\n**Q2** Thank you for raising this insightful question. Indeed, using a linear kernel $K(x,x') = \\langle x,x'\\rangle$ in the RKHS reduces the hypothesis space to linear functions, making the resulting estimator functionally equivalent to linear regression with regularization. However, there are several subtle distinctions worth highlighting in our approach:\n\n- In our **linear model setting**, we explicitly use the knowledge of $F$ and **maximum likelihood estimation (MLE)** without regularization at the aggregation stage due to abundant source data, followed by sparsity-driven debiasing with explicit $\\ell_1$-regularization to exploit structured cross-market differences.\n- In contrast, our **RKHS nonparametric model** employs kernel logistic regression to estimate a different object without the knowledge of $F$: the **log-odds function $\\ell(p,x)$** defined over the joint space of price and covariates, using a kernel $K_{\\text{joint}}$. This log-odds function is conceptually distinct from the structural utility mean function $\\mathring{g}^{(k)}(x)$ analyzed theoretically. Under a linear kernel for $K_{\\text{joint}}$, the resulting estimator is essentially a regularized logistic regression model that is linear in $(p,x)$. \n- Thus, while the **functional forms align** when using a linear kernel, the two scenarios differ in critical ways: the linear scenario directly estimates structural utility parameters with tailored MLE and explicit sparsity-based debiasing, while the RKHS scenario estimates purchase probabilities without knowledge of $F$ via a regularized logistic regression formulation over the joint input space.\n\nOur theoretical analysis accounts explicitly for these methodological differences. Notably, the regret bound in the linear setting achieves a faster (logarithmic) rate due to the parametric structure, whereas the RKHS approach—with or without a linear kernel—incurs polynomial regret due to implicit regularization and complexity from modeling the joint space explicitly.\n\nWe will clarify these points explicitly in the revised manuscript, emphasizing the distinctions in modeling assumptions, estimation strategies, and resulting theoretical guarantees between the linear and RKHS settings.\n\n---\n\n**Q3** In the RKHS regret bound, the ambient dimension $d$ does not appear explicitly because the rate depends on the effective dimension determined by the kernel's eigenvalue decay. Formally, $d$ influences the trace and spectrum of the covariance operator $\\Sigma$, which determines the effective dimension $N(\\lambda) \\sim \\lambda^{-1/(2\\alpha)}$. Thus, faster decay (e.g., smoother kernels or lower-dimensional inputs) results in lower regret. *We will revise the manuscript to clarify that although $d$ is not explicit, it affects the regret via spectral complexity.*\n\n---\n\n**Q4 & 6** We appreciate your valuable suggestions regarding more comprehensive numerical validation. While submission constraints prevent us from including visual representations in this rebuttal, we have conducted additional experiments addressing the key points you raised, including high-dimensional cases, dense difference scenarios, and wall-clock time comparisons. Due to time constraints, we present only the essential numerical results here, with more extensive experimental validation to be included in the revised manuscript.\n\n**Cumulative Regret**\n\n|                 Configuration                  | linear-rmlp | online($K=1,3,5,10$) | offline ($n_\\mathcal{k}=50,100,200,500$) |\n| :----------------------------------------------: | :-----------: | :-------------------------------: | :----------------------------------------------------------: |\n|$d=100,\\|\\beta^{(0)}-\\beta^{(k)}\\|_0\\leq 0.3d$ | 1374        |        (894, 690,617,490)         |                     (1063, 975,942,763)                      |\n| $d=10,\\|\\beta^{(0)}-\\beta^{(k)}\\|_0\\leq 0.6d$  | 62          |          (41,18,13, 10)           |                        (32,25,18, 13)                        |\n|$d=15,\\|\\beta^{(0)}-\\beta^{(k)}\\|_0\\leq 0.6d$  | 95          |           (74,39,27,20)           |                        (79,42,30,26)                         |\n|$d=20,\\|\\beta^{(0)}-\\beta^{(k)}\\|_0\\leq 0.6d$  | 131         |          (98, 58,37,21)           |                        (101,80,52,41)                        |\n\n\n\n**Running time (in seconds)**\n\n|                 Configuration                  | linear-rmlp | online($K=1,3,5,10$) | offline ($n_\\mathcal{k}=50,100,200,500$) |\n| :----------------------------------------------: | :-----------: | :-------------------------------: | :----------------------------------------------------------: |\n|$d=100,\\|\\beta^{(0)}-\\beta^{(k)}\\|_0\\leq 0.3d$ | 386        |        (794, 814,866,972)         |                     (563, 597,642,663)                      |\n| $d=10,\\|\\beta^{(0)}-\\beta^{(k)}\\|_0\\leq 0.3d$  | 31          |          (55,63,71, 77)           |                        (43,53,58, 64)                        |\n|$d=15,\\|\\beta^{(0)}-\\beta^{(k)}\\|_0\\leq 0.3d$  | 42          |           (80,85,93,101)           |                        (58,64,70,77)                         |\n|$d=20,\\|\\beta^{(0)}-\\beta^{(k)}\\|_0\\leq 0.3d$  | 48         |          (101, 109,121,128)           |                        (69,74,82,89)                        |\n| $d=10,\\|\\beta^{(0)}-\\beta^{(k)}\\|_0\\leq 0.6d$  | 33          |          (59,64,71, 78)           |                        (46,51,57, 63)                        |\n|$d=15,\\|\\beta^{(0)}-\\beta^{(k)}\\|_0\\leq 0.6d$  | 40          |           (81,89,96,104)           |                        (51,57,63,72)                         |\n|$d=20,\\|\\beta^{(0)}-\\beta^{(k)}\\|_0\\leq 0.6d$  | 51         |          (103, 110,119,126)           |                        (73,79,82,86)                        |\n\n---\n\n**Q5** Thank you for pointing this out. While the regret patterns for linear and RKHS utilities in Figures 1 and 2 appear similar, they are not identical. This similarity arises because both results illustrate the gain from transfer learning relative to their respective single-market baselines, and this gain scales as a polynomial function of $K^{-1}$. Since both experiments share the same set of $K$ values, they naturally produce similar scaling trends.\n\nSpecifically, we employed RBF kernel ($\\alpha=d/2,\\beta=1$) in the simulation, resulting in an upper bound of $\\mathcal{O}\\left( K^{-\\frac{d}{d+1}} T^{\\frac{1}{d+1}} + H^{\\frac{4}{d+2}} T^{\\frac{2}{d+2}} \\right)$, where $K^{-d/d+1}$ approaches $K^{-1}$ as $d$ grows.\n\nHowever, the explicit polynomial rates differ between linear and RKHS cases, due to their distinct complexities—linear models yield sharper (logarithmic) scaling, while RKHS models yield slower polynomial scaling. These intrinsic rate differences are indeed reflected in the absolute regret gaps between linear and RKHS settings (15–30% across experiments).\n\n\nWe will clarify this explanation explicitly in the revised manuscript.\n\n---\n\n**Q7** We now address the question regarding CM-TDP’s computational costs across settings.\n\n**Linear Utility Model**\n\n- **Online-to-Online Transfer**\n  - Each Episode:\n    - **Aggregation**: $O(d^2 \\ell_{m-1} K \\cdot N)$ for MLE using source market data from episode $m-1$, where $N$ represents the number of optimization iterations.\n    - **Debiasing**: $O(d^2 \\ell_{m-1} \\cdot N)$ for Lasso regression on target market data.\n  - Cumulative regret over $\\log T$ episodes: $\\mathcal{O}(d^2 T K\\cdot N)$\n- **Offline-to-Online Transfer**\n  - Each Episode:\n    - **Phase 1 (Transfer)**: $O(d^2 n_{\\mathcal{K}} \\cdot N)$ for MLE on aggregated source market data, and $O(d^2 l_{m-1} \\cdot N)$ per episode for bias correction.\n    - **Phase 2 (No Transfer)**: Standard linear MLE: $O(d^2 l_{m-1} \\cdot N)$. \n  - Cumulative regret: $O(d^2 (n_\\mathcal{K} + T) \\cdot N)$ \n- **Implementation Note**: The dependence on dimension $d$ varies by optimization method. Newton's method (our implementation) achieves faster convergence (smaller $N$) but maintains $d^2$ dependence. Gradient descent reduces the dependence to $O(d)$ per iteration but requires more iterations ($N$ increases correspondingly).\n\n**RKHS Utility Model**\n\nBase Complexity: KRR with $n$ samples requires $O(n^3)$ for exact matrix inversion.\n\n- **Cumulative regret**\n  - Online-to-Online Transfer: $O(T^3 K^3 \\cdot N)$\n  - Offline-to-Online Transfer: $O((n_\\mathcal{K}^3 + T^3) \\cdot N)$\n- **Implementation Note**: We use **approximate kernel methods** (e.g., Nyström, random features) to reduce runtime to **linear scaling in $\\mathcal{N}(\\lambda)$**, the effective dimensionality of the feature map.\n\n*We will add a computational discussion in the appendix to clarify these scaling properties and practical trade-offs.*"}, {"review_id": "cReT6ZnUb8", "reviewer": "Reviewer_dMHy", "summary": "This paper studies a dynamic pricing problem with additional data from auxiliary markets, where each auxiliary market has mean utilities differing by a structured preference shift. To address this, the paper introduces the Cross-Market Transfer Dynamic Pricing (CM-TDP) algorithm, which is designed to accommodate such model shifts. The paper provides minimax-optimal regret bounds for CM-TDP and supports the proposed method through numerical experiments demonstrating its practical effectiveness.", "questions": "1. In linear bandits, it is common to include some form of regularization when performing maximum likelihood estimation (MLE). I was wondering why Equation (6) in Algorithm 2 does not include any regularization term?\n2. It would be very helpful if the authors could clearly highlight the technical novelty in their theoretical analysis, particularly in comparison to prior work. For example, does the proof involve any new concentration inequalities that are specifically tailored to the debiasing step or the transfer learning setting?", "limitations": "See weaknesses.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths: The paper is generally well-written and organized. It is interesting to consider and solve a transfer learning situation for dynamic pricing problems. The proposed algorithm is supported by solid theoretical results.\n\nWeaknesses:\n1. The underlying components of CM-TDP are known techniques. My opinion is that the novelty of the algorithm lies in the debiasing part to leverage data from auxiliary markets (Algorithm 2). However, the paper lacks sufficient explanation or intuition behind the algorithm's design.\n2. One (potentially) main contribution of the paper is the solid regret analysis. However, since many components in the algorithm (e.g., MLE, LASSO-like debiasing, epoch-based learning) are well-studied techniques, it would be helpful if the paper could highlight its technical novelty in the proof compared to previous works.\n3. Some assumptions are limited. For example, it seems the algorithm lacks a mechanism to \"avoid\" different markets (where $s_0$ is large in Assumption 4) and would be fooled by such a market.  Further, assuming the knowledge of distribution function $F$ is also not  practical and there have been several works to relax that assumption (e.g., [1,2]).\n\n[1] Jianqing Fan, Yongyi Guo, and Mengxin Yu. Policy optimization using semiparametric models for dynamic pricing. Journal of the American Statistical Association, 119(545):552–564, 2024.\n\n[2] Virag Shah, Ramesh Johari, and Jose Blanchet. Semi-parametric dynamic contextual pricing. In Proc. Neural Information Processing Systems, 2019.", "was_revised": true, "final_justification": "Please check Strengths And Weaknesses.", "rebuttal": "Thank you very much for your thoughtful and constructive comments. We have revised our manuscript carefully to address your feedback, as detailed in our responses to weaknesses (**W**) and questions (**Q**) below.\n\n---\n\n**W1** A key premise of our work is that while customer preferences differ across markets, these differences are typically structured and of low-complexity. For example, in food delivery platforms:\n- Customers in coastal cities may value fresh seafood availability, while inland customers prefer comfort foods.\n- Preferences for **universal** factors like delivery speed, order tracking, and menu navigation tend to remain **stable** across markets.\n- Colder regions may exhibit stronger demand for hot beverages, while warmer areas favor cold desserts.\n\nThis provides a concrete illustration that differences in utility preferences typically arise only along a few sparse dimensions, which can be captured by our linear sparse difference setting. \n\nIn a **nonparametric (RKHS)** setting, similar structured differences arise naturally when considering customer populations with distinct preferences. For example, consider a restaurant chain operating across diverse locations:\n- Most customer populations share broadly similar and smoothly varying sensitivity to price changes or promotions.\n- However, specific populations—such as students near college campuses or higher-income residents in urban centers—might have distinct nonlinear responses. Students may exhibit sharp sensitivity to discounts at lower price points, while urban high-income customers might have smoother, less elastic price responses at premium levels.\n\nThese differences between populations typically form a structured, smooth discrepancy rather than arbitrary deviations, aligning naturally with our RKHS-based smoothness assumptions.\n\nThe intuition behind our **debiasing step (Algorithm 2)** arises naturally from a structured handling of the **bias-variance tradeoff**. Pooling all market data produces a low-variance, high-bias estimate, as it ignores cross-market differences. Instead of directly relying on this aggregated estimate, we treat it as a “rough draft” and perform careful corrections. This resembles an editor who adjusts only problematic sentences, preserving valuable shared information. The debiasing stage acts as a precision tool that corrects for local market deviations without discarding useful global patterns.\n\nOur approach is also **theoretically grounded**: the derived regret bounds explicitly capture the trade-off between the magnitude of market differences (e.g., sparsity level $s_0$ in the linear setting, or RKHS norm $H$ in the nonparametric setting) and the required amount of target-specific data. Practically, if two markets differ substantially, more target data is naturally required to reliably correct biases. In implementation, **market similarity** can be assessed using domain expertise or empirically via held-out datasets—by comparing estimated utilities or observed purchase behaviors—before deciding to transfer.\n\n---\n\n**W2 (Q2)** The core challenge addressed by CM-TDP is transferring information across markets under structured but unknown utility heterogeneity. While the algorithm integrates established tools—MLE, Lasso-type debiasing, epoch-based learning—its novelty lies precisely in their principled combination within a binary-feedback dynamic pricing setting subject to latent utility shifts.\n\nOur regret analysis introduces key **technical innovations** that distinguish it from previous single-task dynamic pricing and multi-task bandit results:\n\n* **Debiasing Under Utility Shift**: We establish a novel **bias-corrected aggregation** framework explicitly designed for cross-market transfer. The regret decomposition carefully separates and bounds aggregation errors (from pooled estimation) and debiasing errors (market-specific corrections), enabling rigorous quantification of transfer effectiveness under structured heterogeneity.\n\n* **Refined Estimation Error Bounds**: Unlike standard single-task analyses, we derive specialized estimation bounds accounting explicitly for structured task differences—**sparse coefficient differences** in linear utilities (Propositions 13–15) and **covariance operator-smooth differences** in the RKHS setting (Propositions 21–23). These bounds are technically non-trivial and required new probabilistic and analytic techniques tailored for our transfer pricing setting.\n\n* **Episode-based Error Control**: Our episodic learning scheme, which periodically recalibrates the aggregated and debiased estimators with increasing batch sizes, required a precise and novel analysis. The resulting non-asymptotic regret bounds explicitly characterize the trade-off between sample complexity and model shift across episodes, avoiding typical loose error accumulations.\n\nWe will revise the manuscript to highlight these specific theoretical contributions more explicitly in the main text and theoretical appendices.\n\n---\n\n**W3** We appreciate this important point. While our assumptions on structured market differences and known distribution $ F $ simplify theoretical analysis, we agree that practical considerations require additional clarification.\n\n**(1) Handling large market differences:**\n Our theoretical analysis explicitly quantifies how effectively transfer learning can leverage source markets as a function of two key factors:\n\n- Magnitude of utility differences relative to dimension ($s_0/d$ or RKHS norm $H$).\n- Relative availability of source versus target data.\n\nSpecifically, our regret bounds demonstrate that larger market discrepancies directly increase the required amount of **target-only data** to achieve accurate estimation. In other words, as the heterogeneity ($s_0$ or $H$) grows, transfer remains beneficial only if a sufficiently large target dataset is available for effective debiasing. Our results thus offer practical guidance on how much target data to collect, given expert domain knowledge or estimates of $s_0$ or $H$.\n\nThis analysis aligns with a fundamental principle in transfer learning: if markets differ substantially, indiscriminately pooling data (common in industry practice) can be harmful. Our framework addresses this by providing a theoretically grounded method that leverages market similarity when present, but gracefully reverts toward target-only learning as dissimilarity increases—improving significantly over naïve pooling methods.\n\n**(2) Regarding the known noise distribution $F(\\cdot)$:** Under the linear utility setting, for clarity and analytical convenience, we assume a known noise distribution $F(\\cdot)$. However, our theoretical guarantees actually depend only on certain **regularity properties** (such as smoothness and monotonicity), not the exact parametric form of $F$.\n\nImportantly, our framework can naturally extend to scenarios with **unknown noise distributions** via semi-parametric or nonparametric estimation techniques, as developed in recent works (e.g., [1-3]). Our aggregate-then-debias architecture remains conceptually valid in such scenarios, though rigorously addressing unknown $F^{(k)}$ and possible transfer for $F^{(k)}$ involve additional technical complexity beyond the current scope. We consider this a promising direction for future exploration and will explicitly clarify this extension potential in the revised manuscript.\n\n---\n\n**Q1** Thank you for raising this insightful question. In the aggregation step, we use **unregularized MLE** because the source market data is abundant, providing sufficient samples to estimate source-market parameters $\\beta^{(k)}$ without imposing sparsity or other high-dimensional regularization. Hence, there is no explicit need to introduce penalty terms at this stage. Nevertheless, **if desired**, incorporating a regularization term into the aggregation stage is straightforward, and the corresponding penalized estimation analysis follows directly from existing methods. We will clarify this flexibility in the revised manuscript.\n\n---\n\n\n**References**\n\n[1] Fan, J et al. Policy optimization using semiparametric models for dynamic pricing. JASA, 2024.\n\n[2] Shah, V. et al. Semi-parametric dynamic contextual pricing. NeurIPS, 2019.\n\n[3] Chen, E. et al. Dynamic Contextual Pricing with Doubly Nonparametric Random Utility Models, arXiv:2405.06866."}]

Author rebuttal:

Meta-review: A solid technical work on dynamic pricing. All reviewers are positive.  Most items of concern were thoroughly discussed and addressed in the rebuttal and post-rebuttal discussion.  I recommend accepting the paper.
Decision: Accept (spotlight)