Transfer Learning for Nonparametric Contextual Dynamic Pricing

Fan Wang; Feiyu Jiang; Yi Yu; Zifeng Zhao

REVIEW 2 major objections 4 minor 13 references

Transfer Learning for Nonparametric Contextual Dynamic Pricing

T0 review · 2 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read The paper proves that a transfer-learning pricing algorithm, TLDP, is minimax optimal for nonparametric contextual dynamic pricing under covariate shift, with regret rate $n_Q(n_Q+(\kappa n_P)^{(d+3)/(d+3+\gamma)})^{-1/(d+3)}$ up to…

desk verdict A solid, likely-correct first minimax result for transfer learning in nonparametric dynamic pricing, with two fixable proof gaps and a real parameter-knowledge limitation. read the letter →

arxiv 2501.18836 v1 pith:UTOZLDGE submitted 2025-01-31 cs.LG math.STstat.MEstat.TH

Fan Wang , Feiyu Jiang , Zifeng Zhao , Yi Yu This is my paper · ORCID

classification cs.LGmath.STstat.MEstat.TH MSC 62C2062L0568T05

keywords dynamicpricingtransferlearningcovariateshiftnonparametriccontextualbanditsminimaxregretLipschitzrewardadaptivepartitioningexplorationcoefficient

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper aims to prove that pre-collected data from a related market can be used to price in a new market at the statistically optimal rate, and that a specific algorithm achieves that rate. It studies nonparametric contextual dynamic pricing under covariate shift: the source and target markets share the same expected-revenue function, but covariate distributions differ, and the reward function is only assumed Lipschitz. The proposed TLDP algorithm adaptively partitions the covariate-price space, uses source data as a head start, and the authors prove a regret upper bound that matches a new minimax lower bound up to a logarithmic factor, so the rate $n_Q(n_Q + (\kappa n_P)^{(d+3)/(d+3+\gamma)})^{-1/(d+3)}$ is optimal, where $\gamma$ is a transfer exponent and $\kappa$ an exploration coefficient. A byproduct is the first minimax lower bound for the target-only Lipschitz problem, where the regret rate becomes $n_Q^{(d+2)/(d+3)}$; this settles the fundamental cost of nonparametric dynamic pricing and tells a practitioner exactly when source data help.

What carries the argument

The engine is the adaptive decomposition of the joint covariate-price space $\mathcal{Z}=[0,1]^d\times[0,1]$ into a growing family of active $\ell_\infty$-balls, managed by a UCB index $I_t(B) = C_I r(B) + \min_{B'\in\mathcal{A}_t}\{I_t^{\mathrm{pre}}(B') + C_I\|c(B)-c(B')\|_\infty\}$ with $I_t^{\mathrm{pre}}(B)=v_t(B)+C_I r(B)+\mathrm{conf}_t(B)$, where $\mathrm{conf}_t(B)$ is the source-and-target-count confidence width in (7). Two problem parameters carry the transfer: the transfer exponent $\gamma$, measuring how much the source covariate distribution covers the target's, and the exploration coefficient $\kappa$, measuring the minimal price density the source data provide; together they form the adjusted source size $(\kappa n_P)^{(d+3)/(d+3+\gamma)}$. The smallest explored radius $\tilde r$ in (12), the refinement threshold $T_B^Q$ in (10), and the source head-start counts $n_B^P(D^P)$ convert source data into fewer needed target pulls, and the matching lower bound shows no other mechanism can do better.

What would settle it

Run TLDP on the exact family of problems constructed in the proof of Theorem 2, with Bernoulli rewards and sign-vector reward functions over separated covariate grids, and compare its empirical cumulative regret with the lower-bound floor $c n_Q(n_Q + (\kappa n_P)^{(d+3)/(d+3+\gamma)})^{-1/(d+3)}$; if TLDP's regret is not within a logarithmic factor of that floor across the family, the claimed minimax optimality is false.

Watch

Extended reading notes

Core claim

TLDP is a UCB-type zooming algorithm. Starting with the whole space $\mathcal{Z}=[0,1]^d\times[0,1]$, it keeps a collection of active $\ell_\infty$-balls; for each ball it forms an index from the empirical average revenue, a Lipschitz radius term, and a confidence width that shrinks with the total number of target and source samples in the ball. A ball is refined only after enough target visits, unless source data already provide enough samples, in which case the refinement threshold is zero. The proof shows that with $C_I\ge C_{\mathrm{Lip}}$ and the smallest radius chosen as in (12), the cumulative regret is bounded by $C n_Q(n_Q + (\kappa n_P)^{(d+3)/(d+3+\gamma)})^{-1/(d+3)} \log^{1/(d+3)}(n_Q + (\kappa n_P)^{(d+3)/(d+3+\gamma)})$, and Theorem 2 builds a family of problems satisfying the same assumptions for which any policy suffers at least $c n_Q(n_Q + (\kappa n_P)^{(d+3)/(d+3+\gamma)})^{-1/(d+3)}$. Setting $n_P=0$ gives the target-only rate $n_Q^{(d+2)/(d+3)}$ up to logarithms, together with the first minimax lower bound under the plain Lipschitz assumption.

Load-bearing premise

The proof assumes the algorithm receives the transfer exponent $\gamma$, the exploration coefficient $\kappa$, a Lipschitz-aware constant $C_I$, and the smallest exploration radius $\tilde r$ built from constants $c_\gamma$ and $c_Q$; none of these are estimated from data, so if a practitioner cannot supply them the stated regret guarantee does not apply.

Editorial extensions

If this is right

When the adjusted source size $(\kappa n_P)^{(d+3)/(d+3+\gamma)}$ dominates $n_Q$, the regret drops to roughly $n_Q(\kappa n_P)^{-1/(d+3+\gamma)}$ up to logarithms, so larger, better-exploring, more overlapping source datasets directly shrink target regret.
When the adjusted source size is negligible relative to $n_Q$, the regret is the target-only rate $n_Q^{(d+2)/(d+3)}$ up to logarithms, so transfer neither helps nor hurts in rate.
The target-only case settles a lower bound: any Lipschitz nonparametric contextual dynamic pricing policy must suffer at least $c n_Q^{(d+2)/(d+3)}$, and TLDP with $n_P=0$ attains this up to logarithms.
The price dimension adds one to the covariate dimension in the exponent, giving an effective dimension $d+3$; this matches substituting $d+1$ for $d$ in the closest transfer-bandit rate.
Because a ball can skip target exploration entirely when source counts are high, the algorithm front-loads the benefit of a rich source dataset rather than waiting for asymptotic gains.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The parameters $\gamma$, $\kappa$, $C_I$, and $\tilde r$ are inputs rather than estimated quantities; a natural practical test is to estimate the first two from the source and target marginals and measure how regret degrades under misspecification, which the paper does not conduct.
The lower-bound construction uses Bernoulli rewards on a grid, but the rate is driven by packing the covariate-price space; a testable extension is whether the same rate holds for sub-Gaussian noise with unknown variance, since the proof only needs a KL bound between shifted Bernoulli means.
Remark 1's scale-dependent exploration coefficient $\kappa_r$ suggests an adaptive variant that uses local coverage at each dyadic scale; simulating nonuniform source price coverage would show whether such a variant matches the paper's three-regime regret picture.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

First thing to know: this is a competent theory paper, and the main theorem is probably right. The authors extend Slivkins' zooming to a covariate-shift transfer setting with a continuous price space, get a regret upper bound matching the phase transition in Cai et al., and prove a matching minimax lower bound. The target-only lower bound (Corollary 3) is also new under a plain Lipschitz assumption. The simulations are honest and the code is public. I'd give them credit for that.

The weak spots are not fatal, but they need work. The bound in (26), which limits how many times a ball can be selected before splitting, is false for the smallest dyadic radius above r_tilde: the algorithm only splits when r >= 2 r_tilde, so balls with radius between r_tilde and 2 r_tilde can be selected more than T_B^Q times. The fix is straightforward—those pulls contribute at most C1 n_Q r_tilde to the regret, which is already in the decomposition—but as written it's a gap.

In the lower bound proof, Step 2 asserts that the transfer exponent holds for 3 r_tilde/4 < r <= 1 without doing the calculation. I checked the key equality P_X(B(x*_i, r_tilde)) = Q_X(B(x*_i, r_tilde)); it does hold with the delta defined in (68), so the stress-test worry about that specific equality doesn't land. But the general verification for large radii is genuinely skipped, and the paper should spell it out. The construction also needs C_Q to be allowed to grow, which is fine since the upper bound in Assumption 2 doesn't enter the regret constants.

The real practical limitation is that TLDP needs gamma, kappa, C_I, c_gamma, and c_Q as inputs. The authors acknowledge this in Section 6 and leave estimation to future work. That's a standard caveat in this literature, but it does mean the theorem is a conditional guarantee.

Bottom line: the paper is for researchers working on transfer learning in bandits and dynamic pricing. It deserves a serious referee, and I'd ask for the two proof gaps to be closed.

Referee Report

2 major / 4 minor

Summary. The paper studies transfer learning for nonparametric contextual dynamic pricing under covariate shift. The target and source domains share the same reward function, while covariate distributions differ. The authors propose TLDP, a contextual zooming algorithm with a source-data head start, and prove an upper bound on cumulative regret of order n_Q (n_Q + (κ n_P)^{(d+3)/(d+3+γ)})^{-1/(d+3)} up to logarithmic factors under a Lipschitz condition on the reward. They also state a matching minimax lower bound for the same class, including a target-only corollary, and support the results with synthetic and real-data experiments. The paper claims to be the first to provide a minimax-optimal transfer learning result for nonparametric contextual dynamic pricing.

Significance. If the technical gaps identified below are repaired, this is a significant contribution. The paper is the first to bring transfer learning into nonparametric contextual dynamic pricing, and the claimed rate interpolates naturally between the target-only regime and the transfer-dominated regime. The TLDP algorithm is clearly described and the proofs are detailed; the lower bound adapts the covariate-shift construction of Kpotufe and Martinet rather than treating transfer as an afterthought. The release of code and the inclusion of a real auto-loan experiment are additional strengths. The central mathematical claims are plausible, but two load-bearing points in the proof need correction before the results can be accepted as stated.

major comments (2)

[Appendix B.1, Eqs. (26) and (41); Step 2.2] The bound |S_Q(B)| ≤ T_Q^B is used for every active ball with r(B) > r̃, but Algorithm 1's refinement loop (Step 6) splits a ball only while r(B) ≥ 2r̃. For radii in the interval (r̃, 2r̃), a ball can accumulate more than T_Q^B target pulls without being refined, so the inequality is false as written. This affects the summation leading to (27) in Step 1.2 and the analogous bound (41) in Step 3.2. The same issue appears in Step 2.2, where the assertion that the second term in (33) vanishes because T_Q^B = 0 is only valid for balls with r(B) ≥ 2r̃. The final rate is likely preserved by treating the balls with r(B) < 2r̃ separately and charging their contribution to the n_Q r̃ term, but the proof as written does not do this.
[Appendix C.1, Step 2 (transfer exponent verification)] The verification that the constructed distributions lie in the class I(γ, c_γ, κ, C_Lip, c_Q) is incomplete. Definition 1 is checked only for r ≤ 3r̃/4; for 3r̃/4 < r ≤ 1 the proof states that the inequalities hold without showing the verification. The required inequality P_X(B_X(x,r)) ≥ c_γ r^γ Q_X(B_X(x,r)) is not immediate for x in the inner patches: the source density on the inner region is c_γ r̃^γ q_1, the source mass on the outer shells is determined globally by normalization, and the balance depends on x and r. The asserted equality P_X(B_X(x_i^*, r̃)) = Q_X(B_X(x_i^*, r̃)) is also not derived. Since this membership is exactly what makes Theorem 2 a matching lower bound for the class used in Theorem 1, a complete case analysis for all r ∈ (0,1] is required.

minor comments (4)

[Algorithm 1, line 3] The word "revelantt" should be "relevant".
[Section 5.1, after Eq. (12)] The simulations set C_r = 1/4 and C_I = 1, but Theorem 1 requires C_r^4 c_γ c_Q ≥ 8; for the natural constants in the experiments (e.g., c_Q = 2^d for uniform target covariates and c_γ up to 1), this condition is not satisfied. The experiments therefore probe a regime outside the proven theorem, and this should be acknowledged or the constants adjusted.
[Conclusion, first paragraph] The conclusion notes that γ and κ are often unknown, but the algorithm also requires the constants C_I, C_r, c_γ, and c_Q as inputs in order to run with the proven guarantee. A short discussion of which constants are critical and which can be chosen heuristically would help practitioners.
[References] The reference for Cai and Pu (2022) is listed as arXiv:0000.0000, which is a placeholder and should be replaced with the actual arXiv identifier.

Circularity Check

0 steps flagged · score 1.0 of 10

No material circularity: Theorems 1 and 2 are derived from explicit assumptions via the paper's own UCB/partitioning analysis and a standard external lower-bound tool; the known-parameter requirement and a proof-verification gap are limitations, not circular reductions.

full rationale

TLDP's regret upper bound (Theorem 1) is derived from Assumptions 1-2, Definitions 1-2, and the algorithm's own UCB/partitioning analysis in Appendix B; no regret bound is imported from prior work. The exploration radius r-tilde is chosen to balance exploration and estimation, and the resulting rate follows from the proof's three regimes rather than from the desired conclusion. Theorem 2 constructs distributions with the stated transfer exponent, exploration coefficient, Lipschitz constant, and target density, then verifies membership in the class I(gamma,c_gamma,kappa,C_Lip,c_Q) before applying Proposition 8 from Kpotufe and Martinet (2020), an external result; this is the standard minimax lower-bound construction and does not fit parameters to the target claim. The only self-citations (e.g., Zhao et al. 2024) appear in literature review and real-data covariate selection and are not load-bearing. The requirement that gamma and kappa be known is a practical implementability limitation acknowledged in Section 6, not a circular step. One proof-completeness concern exists: Appendix C.1 verifies the transfer-exponent condition for r <= 3r-tilde/4 and then asserts, without showing the calculation, that the inequalities hold for 3r-tilde/4 < r <= 1 because P_X and Q_X masses coincide on the grid balls; this is a rigor/correctness issue for Theorem 2, not a reduction of the claim to its own inputs. Overall, the derivation chain is self-contained, and no circular step can be exhibited from the paper's own equations.

Assumptions & free parameters 4 free parameters · 9 assumptions · 0 invented entities

The central claim depends on Lipschitz smoothness, covariate shift, and the two structural parameters gamma and kappa, plus several constants. No new physical entities or mediators are introduced. The main practical burden is that the algorithm needs these parameters as inputs.

free parameters (4)

Transfer exponent gamma
Quantifies source-target covariate overlap. Required as input to Algorithm 1 for the threshold, confidence bounds, and the theorem constants. Unknown in practice, as acknowledged in Section 6.
Exploration coefficient kappa
Quantifies source price coverage. Required to compute r_tilde and confidence bounds. Unknown in practice, as acknowledged in Section 6.
Index constant C_I = 1 (experiments)
Must satisfy C_I >= C_Lip. In simulations C_I is set to 1, which requires knowledge of the Lipschitz constant.
Smallest radius r_tilde = C_r [log(...)/(...)]^{1/(d+3)}
Input to Algorithm 1; its optimal value depends on n_Q, n_P, gamma, kappa, and the constants c_gamma and c_Q through the condition C_r^4 c_gamma c_Q >= 8 in Theorem 1.

assumptions (9)

domain assumption f is Lipschitz continuous on Z = [0,1]^{d+1} with constant C_Lip (Assumption 1)
Used throughout to bound estimation error and to justify the zooming index.
domain assumption Target covariate distribution Q_X satisfies c_Q r^d <= Q_X(B_X(x,r)) <= C_Q r^d (Assumption 2)
Ensures the target context space is explored sufficiently at all scales.
domain assumption Covariate shift: f(x,p) = f^P(x,p) for all (x,p) (Equation 3)
Defines the transfer setting; if reward functions differ between domains, the algorithm's guarantees collapse.
domain assumption Transfer exponent condition: P_X(B_X(x,r)) >= c_gamma r^gamma Q_X(B_X(x,r)) (Definition 1)
Lower bounds source mass near every target point; the effective source sample size depends on gamma.
domain assumption Exploration coefficient kappa > 0 (Definition 2)
Requires the source price distribution to have conditional density bounded below by kappa; without it source data may not cover the price axis.
domain assumption Rewards Y are in [0,1]
Needed for bounded martingale concentration in Lemma 5.
standard math Azuma-Hoeffding inequality
Used in Lemma 5 for martingale concentration.
standard math Varshamov-Gilbert bound
Used in the lower bound construction to build a packing of hypotheses.
standard math Kpotufe-Martinet lower bound proposition (Proposition 8)
Used as a black box for the minimax lower bound.

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Cite this review

Pith. "Pith review of Transfer Learning for Nonparametric Contextual Dynamic Pricing." pith.science (2026). https://pith.science/paper/UTOZLDGE

@misc{pith2026250118836,
  author       = {Pith},
  title        = {Pith review of: Transfer Learning for Nonparametric Contextual Dynamic Pricing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UTOZLDGE}},
  note         = {Machine review of arXiv:2501.18836}
}

read the original abstract

Dynamic pricing strategies are crucial for firms to maximize revenue by adjusting prices based on market conditions and customer characteristics. However, designing optimal pricing strategies becomes challenging when historical data are limited, as is often the case when launching new products or entering new markets. One promising approach to overcome this limitation is to leverage information from related products or markets to inform the focal pricing decisions. In this paper, we explore transfer learning for nonparametric contextual dynamic pricing under a covariate shift model, where the marginal distributions of covariates differ between source and target domains while the reward functions remain the same. We propose a novel Transfer Learning for Dynamic Pricing (TLDP) algorithm that can effectively leverage pre-collected data from a source domain to enhance pricing decisions in the target domain. The regret upper bound of TLDP is established under a simple Lipschitz condition on the reward function. To establish the optimality of TLDP, we further derive a matching minimax lower bound, which includes the target-only scenario as a special case and is presented for the first time in the literature. Extensive numerical experiments validate our approach, demonstrating its superiority over existing methods and highlighting its practical utility in real-world applications.

Figures

Figures reproduced from arXiv: 2501.18836 by the authors.

**Figure 2.** Results for Configuration 2 in Scenario 1. Panel (A) and (B): varying source data size [PITH_FULL_IMAGE:figures/full_fig_p037_2.png] view at source ↗

**Figure 3.** Results for Configuration 1 in Scenario 2. Panel (A) and (B): varying source data size [PITH_FULL_IMAGE:figures/full_fig_p038_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Results for Configuration 2 in Scenario 2. Panel (A) and (B): varying source data size [PITH_FULL_IMAGE:figures/full_fig_p038_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

13 extracted references · 10 canonical work pages

[1]

nQX t=1 log ( θf ω(i) Yt Xt, pπ t θf ω(0) Yt Xt, pπ t ) + nPX t=1 log ( θf ω(i) Y P t X P t , pP t θf ω(0) Y P t X P t , pP t )# =Eω(i)

Denote the target data and source data by DQ nQ = {Xt, pπ t , Yt}nQ t=1, DP = {X P t , pP t , YP t }nP t=1, respectively. For each ω(i) ∈ Ωm, let Θ i be the joint distribution of the random variables in DQ nQ and DP induced by Qω(i) X , µω(i) , fω(i) , i.e. under Θ i: • the target data DQ nQ follow the target covariate distribution QX with density qX , th...

work page 2024
[2]

Since r(B) ≥ ˜r, it holds that P E c B ≤ exp n − 4−1CE κnP ˜rd+γ+1 o , (64) which completes the proof of (60)

As a consequence, by Chernoff’s bound, we have that P{E c B =P nP B DP < CE nP κr(B)d+γ+1 ≤P E nP B DP − nP B DP > E nP B DP /2 ≤ exp − 4−1CE κnP r(B)d+γ+1 . Since r(B) ≥ ˜r, it holds that P E c B ≤ exp n − 4−1CE κnP ˜rd+γ+1 o , (64) which completes the proof of (60). Now we consider the case where nQ < (κnP ) d+3 d+3+γ and ˜ r ≥ 8 log (κnp) d+3 d+3+γ CE ...

work page 2020
[4]

Marginal Singularity, and the Benefits of Labels in Covariate-Shift

URL https://arxiv.org/abs/1803.01833. Samory Kpotufe and Guillaume Martinet. Marginal singularity and the benefits of labels in covariate-shift. The Annals of Statistics , 49(6):3299–3323,

work page Pith review arXiv
[9]

For instance, Luo et al

consider a specific form of the expected revenue function, f (x, p) = p{1 − F (p − x⊤θ)}, where F is a nonparametric cumulative distribution function (CDF) of noise influencing customer valuations and θ ∈ Rd is an unknown parameter vector representing customer sensitivity. For instance, Luo et al. (2022) developed the ExUCB algorithm and derived a regret ...

work page 2022
[10]

Corollary 2.20 in Wainwright, 2019), it holds that P nX i=1 Xi > p 2n log(1/δ2) ≤ 2δ2

Following the Azuma–Hoeffding inequality (e.g. Corollary 2.20 in Wainwright, 2019), it holds that P nX i=1 Xi > p 2n log(1/δ2) ≤ 2δ2. (55) We consider the case T >0. Note that P ∃0 ≤ t ≤ T : nX i=1 Xi + n+tX i=n+1 Xi > U(t, n, δ1, δ2) ≤P nX i=1 Xi > p 2n log(1/δ2) + P ∃0 < t≤ T : n+tX i=1 Xi > p 2(n + t) log(1/δ1) ≤2δ2 + P ∃t ∈ [T ] : n+tX i=1 Xi > p 2(n ...

work page 2019
[13]

Panel (C) varying the transfer exponent γ (top axis) and the exploration coefficient κ (bottom axis)

Panel (A) and (B): varying source data size nP and target data size nQ, respectively. Panel (C) varying the transfer exponent γ (top axis) and the exploration coefficient κ (bottom axis). Panel (D): varying the index constant CI (top axis) and the exploration radius constant Cr (bottom axis). For Panels (B), (C) and (D), we fix nQ = 10000. 1000 2000 3000 ...

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Transfer learning for contextual multi-armed bandits

Changxiao Cai, T Tony Cai, and Hongzhe Li. Transfer learning for contextual multi-armed bandits. The Annals of Statistics , 52(1):207–232, 2024a. T Tony Cai and Hongming Pu. Transfer learning for nonparametric regression: Non-asymptotic minimax analysis and adaptive procedure. arXiv preprint arXiv:0000.0000 ,

arXiv
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Dynamic pricing with demand covariates

Sheng Qiang and Mohsen Bayati. Dynamic pricing with demand covariates. arXiv preprint arXiv:1604.07463,

arXiv

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Ningyuan Chen and Guillermo Gallego

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Steve Hanneke and Samory Kpotufe

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A detailed comparison of our re- gret bound in Theorem 1 with those from non-transfer learning studies is presented in Appendix A

Appendices All technical details of this paper can be found in the Appendices. A detailed comparison of our re- gret bound in Theorem 1 with those from non-transfer learning studies is presented in Appendix A. The proofs of Theorem 1 and Theorem 2 are included in Appendices B ...

2011

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [1]

nQX t=1 log ( θf ω(i) Yt Xt, pπ t θf ω(0) Yt Xt, pπ t ) + nPX t=1 log ( θf ω(i) Y P t X P t , pP t θf ω(0) Y P t X P t , pP t )# =Eω(i)

Denote the target data and source data by DQ nQ = {Xt, pπ t , Yt}nQ t=1, DP = {X P t , pP t , YP t }nP t=1, respectively. For each ω(i) ∈ Ωm, let Θ i be the joint distribution of the random variables in DQ nQ and DP induced by Qω(i) X , µω(i) , fω(i) , i.e. under Θ i: • the target data DQ nQ follow the target covariate distribution QX with density qX , th...

work page 2024

[2] [2]

Since r(B) ≥ ˜r, it holds that P E c B ≤ exp n − 4−1CE κnP ˜rd+γ+1 o , (64) which completes the proof of (60)

As a consequence, by Chernoff’s bound, we have that P{E c B =P nP B DP < CE nP κr(B)d+γ+1 ≤P E nP B DP − nP B DP > E nP B DP /2 ≤ exp − 4−1CE κnP r(B)d+γ+1 . Since r(B) ≥ ˜r, it holds that P E c B ≤ exp n − 4−1CE κnP ˜rd+γ+1 o , (64) which completes the proof of (60). Now we consider the case where nQ < (κnP ) d+3 d+3+γ and ˜ r ≥ 8 log (κnp) d+3 d+3+γ CE ...

work page 2020

[3] [4]

Marginal Singularity, and the Benefits of Labels in Covariate-Shift

URL https://arxiv.org/abs/1803.01833. Samory Kpotufe and Guillaume Martinet. Marginal singularity and the benefits of labels in covariate-shift. The Annals of Statistics , 49(6):3299–3323,

work page Pith review arXiv

[4] [9]

For instance, Luo et al

consider a specific form of the expected revenue function, f (x, p) = p{1 − F (p − x⊤θ)}, where F is a nonparametric cumulative distribution function (CDF) of noise influencing customer valuations and θ ∈ Rd is an unknown parameter vector representing customer sensitivity. For instance, Luo et al. (2022) developed the ExUCB algorithm and derived a regret ...

work page 2022

[5] [10]

Corollary 2.20 in Wainwright, 2019), it holds that P nX i=1 Xi > p 2n log(1/δ2) ≤ 2δ2

Following the Azuma–Hoeffding inequality (e.g. Corollary 2.20 in Wainwright, 2019), it holds that P nX i=1 Xi > p 2n log(1/δ2) ≤ 2δ2. (55) We consider the case T >0. Note that P ∃0 ≤ t ≤ T : nX i=1 Xi + n+tX i=n+1 Xi > U(t, n, δ1, δ2) ≤P nX i=1 Xi > p 2n log(1/δ2) + P ∃0 < t≤ T : n+tX i=1 Xi > p 2(n + t) log(1/δ1) ≤2δ2 + P ∃t ∈ [T ] : n+tX i=1 Xi > p 2(n ...

work page 2019

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Panel (C) varying the transfer exponent γ (top axis) and the exploration coefficient κ (bottom axis)

Panel (A) and (B): varying source data size nP and target data size nQ, respectively. Panel (C) varying the transfer exponent γ (top axis) and the exploration coefficient κ (bottom axis). Panel (D): varying the index constant CI (top axis) and the exploration radius constant Cr (bottom axis). For Panels (B), (C) and (D), we fix nQ = 10000. 1000 2000 3000 ...

work page 2000

[7] [2009]

Transfer learning for contextual multi-armed bandits

Changxiao Cai, T Tony Cai, and Hongzhe Li. Transfer learning for contextual multi-armed bandits. The Annals of Statistics , 52(1):207–232, 2024a. T Tony Cai and Hongming Pu. Transfer learning for nonparametric regression: Non-asymptotic minimax analysis and adaptive procedure. arXiv preprint arXiv:0000.0000 ,

arXiv

[8] [2015]

Dynamic pricing with demand covariates

Sheng Qiang and Mohsen Bayati. Dynamic pricing with demand covariates. arXiv preprint arXiv:1604.07463,

arXiv