Transfer Faster, Price Smarter: Minimax Dynamic Pricing under Cross-Market Preference Shift

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.

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:

• Customers in coastal cities may value fresh seafood availability, while inland customers prefer comfort foods.
• Preferences for universal factors like delivery speed, order tracking, and menu navigation tend to remain stable across markets.
• Colder regions may exhibit stronger demand for hot beverages, while warmer areas favor cold desserts.

This 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.

In 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:

• Most customer populations share broadly similar and smoothly varying sensitivity to price changes or promotions.
• 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.

These differences between populations typically form a structured, smooth discrepancy rather than arbitrary deviations, aligning naturally with our RKHS-based smoothness assumptions.

The 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.

Our 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.

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.

Our regret analysis introduces key technical innovations that distinguish it from previous single-task dynamic pricing and multi-task bandit results:

• 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.

• 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.

• 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.

We will revise the manuscript to highlight these specific theoretical contributions more explicitly in the main text and theoretical appendices.

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.

(1) Handling large market differences: Our theoretical analysis explicitly quantifies how effectively transfer learning can leverage source markets as a function of two key factors:

• Magnitude of utility differences relative to dimension ($s_0/d$ or RKHS norm $H$).
• Relative availability of source versus target data.

Specifically, 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 $H$.

This 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.

(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 $F$.

Importantly, 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.

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.

References

[1] Fan, J et al. Policy optimization using semiparametric models for dynamic pricing. JASA, 2024.

[2] Shah, V. et al. Semi-parametric dynamic contextual pricing. NeurIPS, 2019.

[3] Chen, E. et al. Dynamic Contextual Pricing with Doubly Nonparametric Random Utility Models, arXiv:2405.06866.

Thank you for the positive feedback and insightful questions. We have updated our manuscript to address the following points.

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:

• Statistical efficiency may degrade, but the algorithm remains valid.
• Our bias-corrected estimator smoothly interpolates between full pooling and target-only learning, limiting downside risk.

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).

We will add a brief discussion in the revision and highlight this as a direction for adaptive transfer tuning.

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:

• 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.
• 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 $(p,x)$.
• 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.

Our 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.

We 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.

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.

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.

Cumulative Regret

Running time (in seconds)

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.

Specifically, 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.

However, 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).

We will clarify this explanation explicitly in the revised manuscript.

Q7 We now address the question regarding CM-TDP’s computational costs across settings.

Linear Utility Model

• Online-to-Online Transfer
  • Each Episode:
    • 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.
    • Debiasing: $O(d^2 \ell_{m-1} \cdot N)$ for Lasso regression on target market data.
  • Cumulative regret over $\log T$ episodes: $\mathcal{O}(d^2 T K\cdot N)$
• Offline-to-Online Transfer
  • Each Episode:
    • 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.
    • Phase 2 (No Transfer): Standard linear MLE: $O(d^2 l_{m-1} \cdot N)$.
  • Cumulative regret: $O(d^2 (n_\mathcal{K} + T) \cdot N)$
• 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).

RKHS Utility Model

Base Complexity: KRR with $n$ samples requires $O(n^3)$ for exact matrix inversion.

• Cumulative regret
  • Online-to-Online Transfer: $O(T^3 K^3 \cdot N)$
  • Offline-to-Online Transfer: $O((n_\mathcal{K}^3 + T^3) \cdot N)$
• 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.

We will add a computational discussion in the appendix to clarify these scaling properties and practical trade-offs.

Thank you for the positive feedback and insightful questions. We have updated our manuscript to address the following points.

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.

• 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].
• 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].
• 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:
  • For linear $g(x)$, they show a regret lower bound of $\Omega(d \log T)$, consistent with classical bounds in high-dimensional pricing.
  • 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 $\kappa = \alpha\beta$.

We 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.

Sketch of Lower Bound Derivation

• Linear Utility with Sparse Shift We now sketch a lower bound argument that matches our Theorem 5 (up to logarithmic factors).

  1. Hard Instance Construction: Let the source markets all share utility $\beta^{(k)} = \beta^*$, and consider two possibilities for the target:

     • $\beta^{(0)} = \beta^*$,
     • or $\beta^{(0)} = \beta^* + \delta e_j$, where $e_j$ is the unit vector and $j \in \mathcal{S} \subset [d]$, with $|\mathcal{S}| = s_0$.

  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.

  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.

  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.

• RKHS Utility with Operator Smoothness

  1. Function Class: Define a class $\mathcal{G} \subset \mathcal{H}\_k$, such that for each task: $g^{(0)} = g^{(k)} + (\Sigma^{(0)})^\eta \omega, \quad \|\omega\|_{\mathcal{H}_k} \leq H.$

  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 $\Omega(T^{-\frac{2\alpha\beta}{2\alpha\beta + 1}})$.

  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.

Q2 (W2) We agree that markets with utility differences may also differ in covariates and noise.

• 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].
• 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.

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.

Q4 Our bias-corrected aggregation framework is modular and can indeed accommodate alternative notions of similarity, including $l_q$ constraints where $q\in [0,1]$.

• 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.
• 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.

Thus, 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.

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.

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 $\mathring{g}^{(k)}$.

Our 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 $\mathring{g}^{(k)}$.

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.

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.

References

[1] Wang, Fan et al. Transfer Learning for Nonparametric Contextual Dynamic Pricing, arXiv:2501.18836.

[2] Chai, J, Chen, E, Fan, J. Deep Transfer Q-Learning for Offline Non-Stationary Reinforcement Learning. arXiv:2501.04870.

[3] Chen, E. et al. Dynamic Contextual Pricing with Doubly Nonparametric Random Utility Models, arXiv:2405.06866.

[4] Xu, K. & Bastani, H. Multitask Learning and Bandits via Robust Statistics, Management Science, 2025.

[5] Li, S. et al. Transfer Learning for High-Dimensional Linear Regression, JRSSB, 2022.

[6] Caponnetto, A. & De Vito, E. Optimal Rates for Regularized Least-Squares Algorithm, FoCM, 2007.

[7] Bu, J. et al. Context-Based Dynamic Pricing with Separable Demand Models, Management Science, 2025.

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.

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.

That 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.

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.

Importantly, 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.

W3: Assumption on sparsity. We appreciate the reviewer’s concern and agree it’s important to contextualize the sparsity assumption:

• 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.
• 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].
• 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.
• 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.

Q1 We now address the question regarding CM-TDP’s computational costs across settings.

Linear Utility Model

• Online-to-Online Transfer
  • Each Episode:
    • 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.
    • Debiasing: $O(d^2 \ell_{m-1} \cdot N)$ for Lasso regression on target market data.
  • Cumulative regret over $\log T$ episodes: $\mathcal{O}(d^2 T K\cdot N)$
• Offline-to-Online Transfer
  • Each Episode:
    • 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.
    • Phase 2 (No Transfer): Standard linear MLE: $O(d^2 l_{m-1} \cdot N)$.
  • Cumulative regret: $O(d^2 (n_\mathcal{K} + T) \cdot N)$
• 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).

RKHS Utility Model

Base Complexity: KRR with $n$ samples requires $O(n^3)$ for exact matrix inversion.

• Cumulative regret
  • Online-to-Online Transfer: $O(T^3 K^3 \cdot N)$
  • Offline-to-Online Transfer: $O((n_\mathcal{K}^3 + T^3) \cdot N)$
• 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.

We will add a computational discussion in the appendix to clarify these scaling properties and practical trade-offs.

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.

References

[1] Fan Wang, Feiyu Jiang, Zifeng Zhao, and Yi Yu. Transfer learning for nonparametric contextual dynamic pricing. arXiv preprint arXiv:2501.18836, 2025.

[2] Jinhang Chai, Elynn Chen, Jianqing Fan. Deep Transfer $Q$-Learning for Offline Non-Stationary Reinforcement Learning. arXiv preprint arXiv:2501.04870.

[3] Kan Xu and Hamsa Bastani. Multitask learning and bandits via robust statistics. Management Science, 2025.

[4] Elynn Chen, Xi Chen, Lan Gao, Jiayu Li. Dynamic Contextual Pricing with Doubly Non-Parametric Random Utility Models. arXiv preprint arXiv:2405.06866