Dynamic Pricing in the Linear Valuation Model using Shape Constraints

Daniele Bracale; Kevin Stoll; Moulinath Banerjee; Salam Turki; Yuekai Sun

REVIEW 2 major objections 4 minor 2 cited by

Dynamic Pricing in the Linear Valuation Model using Shape Constraints

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

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Pith's one-line read This paper claims that pricing with an OLS slope estimate and an isotonic fit of the noise survival function — no tuning parameters — keeps cumulative regret at $\tilde{O}(T^{\nu(\alpha)} d^{\alpha/(\alpha+2)})$ under only $\alpha$-Hölder…

desk verdict The antitonic-regression contribution is real, but the headline regret bound sits on an OLS concentration lemma whose conditions are not verified and can fail for general price ranges. read the letter →

arxiv 2502.05776 v4 pith:QLDFLFBY submitted 2025-02-09 stat.ML cs.LG

Daniele Bracale , Moulinath Banerjee , Yuekai Sun , Kevin Stoll , Salam Turki This is my paper · ORCID

classification stat.MLcs.LG MSC 62G0562G0862L0591B24

keywords dynamicpricingcontextuallinearvaluationmodelisotonicregressionantitonicshape-constrainedestimationHöldercontinuityregretbounds

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 proposes a dynamic pricing policy for the linear valuation model in which the seller only sees whether the posted price fell below the customer's latent valuation. Its central claim is that estimating the valuation slope with ordinary least squares and the unknown noise survival function with isotonic (antitonic) regression — with no bandwidth, penalty, or other tuning parameter — keeps cumulative expected regret at order $\tilde{O}(T^{\nu(\alpha)} d^{\alpha/(\alpha+2)})$ under the single assumption that the noise distribution is $\alpha$-Hölder continuous for some $\alpha \in (0,1]$. That matters because existing nonparametric and bandit-style methods either demand Lipschitz-or-smoother noise, need user-chosen parameters such as kernel bandwidths or regularization constants, or are too expensive to deploy. If the result is right, practitioners get a cheap, parameter-free pricing rule whose regret guarantee degrades gracefully as the noise distribution becomes rougher.

What carries the argument

The engine is the two-stage per-epoch estimator: ordinary least squares on $(x_t, H y_t)$ with $H = p_{\max}-p_{\min}$ during the first half of the exploration phase recovers the slope $\theta_0$, and antitonic regression — nonparametric least squares subject to the fitted curve being non-increasing, computed with the Pool Adjacent Violators Algorithm — recovers the survival function $S_0$ from binary indicators observed at uniform design points $w_t = p_t - \hat\theta_k^\top x_t$ in the second half. The object that carries the proof is the surrogate $S_\theta(u) = \mathbb{E}_x[S_0(u + (\theta-\theta_0)^\top x)]$: it is non-increasing and $\alpha$-Hölder uniformly in $\theta$, it sits within $O(\|\theta-\theta_0\|_2^\alpha)$ of $S_0$, and it is exactly the conditional sale probability that the censored data reveal when $\theta \neq \theta_0$ (Proposition 4.6). Theorem 4.8 supplies the uniform concentration inequality for the antitonic estimator around $S_\theta$, and Theorem 4.10 combines that error with the OLS error raised to the power $\alpha$, choosing per-epoch exploration lengths $d^{\alpha/(\alpha+2)}\tau_k^{\nu(\alpha)}$ so that the two errors balance.

What would settle it

Run the exploration phase of Algorithm 1 on a synthetic problem that satisfies Assumptions 4.1, 4.2, and 4.7 but has asymmetric noise with support $U=(a,b)$ and $p_{\min}$ strictly above $a$, so the intercept bias $c=a-p_{\min}+\int_a^b S_0(u)\,du$ is nonzero; if the OLS error $\|\hat\theta_k-\theta_0\|_2$ levels off at a positive constant instead of shrinking at the $\sqrt{d\log n_k/n_k}$ rate of Lemma 4.3, then the Theorem 4.10 regret bound cannot hold as stated, and the bias condition would have to be added to the assumptions.

Watch

Extended reading notes

Core claim

The paper's claim is that shape alone earns the rate: binary sale/no-sale feedback in the linear model $v_t = \theta_0^\top x_t + z_t$ identifies the survival function $S_0$ under a monotonicity constraint, and least-squares estimation under that constraint converges uniformly much as in the fully observed isotonic setting. Theorem 4.10 states that under bounded i.i.d. contexts, a compact parameter space, and $\alpha$-Hölder continuity of $S_0$, the cumulative expected regret of Algorithm 1 is $\tilde{O}(T^{\nu(\alpha)} d^{\alpha/(\alpha+2)})$, where $\nu(\alpha) = 2/(2+\alpha)$ for $\alpha \in (0,1/2)$ and $\nu(\alpha) = (2\alpha+1)/(3\alpha+1)$ for $\alpha \in [1/2,1]$; at $\alpha=1$ this becomes $\tilde{O}(T^{3/4} d^{1/3})$, matching the order in $T$ of kernel-based pricing without requiring a bandwidth or second-order smoothness. The hinge of the proof is Proposition 4.6: the function actually identified from data when the fitted slope is $\hat\theta_k$, namely $S_{\hat\theta_k}(u) = \mathbb{E}_x[S_0(u + (\hat\theta_k-\theta_0)^\top x)]$, is non-increasing and $\alpha$-Hölder and lies within $O(\|\hat\theta_k-\theta_0\|_2^\alpha)$ of $S_0$, which lets the analysis isolate the OLS error from the antitonic error, balance them within each epoch, and sum epochs under the doubling schedule.

Load-bearing premise

The load-bearing premise is that the OLS estimator in the exploration phase obeys the cited concentration bound, which is only true when the price range $[p_{\min}, p_{\max}]$ covers the full support of customer valuations for every context and the intercept-bias term $c = a - p_{\min} + \int_a^b S_0(u)\,du$ is exactly zero; the paper never states or verifies that coverage-and-bias condition, and if it fails the regression slope does not converge to $\theta_0$, so the regret bound as proven does not follow.

Editorial extensions

If this is right

Under only $\alpha$-Hölder continuity of the noise distribution, the policy's cumulative expected regret is $\tilde{O}(T^{\nu(\alpha)} d^{\alpha/(\alpha+2)})$, where $\nu(\alpha)=2/(2+\alpha)$ for $\alpha\in(0,1/2)$ and $\nu(\alpha)=(2\alpha+1)/(3\alpha+1)$ for $\alpha\in[1/2,1]$ (Theorem 4.10).
At $\alpha=1$ (Lipschitz noise) the bound becomes $\tilde{O}(T^{3/4} d^{1/3})$, the same order in $T$ as kernel-based pricing but without any bandwidth choice or second-order smoothness requirement.
The rule is tuning-parameter-free: the only inputs are the horizon, the price range, the Hölder exponent, and the first-epoch length, and the simulations show the regret is insensitive to that last choice (Remark 5.1 and Figure 3).
Computationally the method is light: the antitonic fit runs in $O(n)$ time via PAVA, so each epoch costs $O(d^{\alpha/(\alpha+2)} T^{\nu(\alpha)})$ in total (Section 3.1).
Empirically the method dominates the VAPE algorithm in its own simulation setting up to $T=8000$ and outperforms tuned UCB and VAPE baselines on a real rental dataset while staying competitive with the kernel method (Sections 5 and 6).

Reading between the lines

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

Beyond the paper: the same template — uniform-price OLS for the slope and an isotonic fit of the survival function at design points with density bounded away from zero — transfers to any censored-feedback problem with a linear index, such as auction bid/no-bid or credit accept/reject decisions, since only binary outcomes and features are used.
Beyond the paper: the rate $\nu(\alpha)$ has a kink at $\alpha=1/2$, so a matching lower bound, if one exists, would likely split into the same two regimes, and an adversarial construction near that kink would reveal whether the break is intrinsic or an artifact of the proof.
Beyond the paper: because Theorem 4.8's constant depends on the design density only through $C_2 = \inf_u f_w(u)$, the design could be re-weighted toward regions where $S_0$ is steep without changing the rate, giving a cheap step toward the sequential optimal design the paper names as future work.

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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

Short version: the paper is worth a look for the antitonic-regression concentration result and the Welltower case study, but Theorem 4.10 as stated is not proven. Lemma 4.3, borrowed from Fan et al., requires E[H y | x] to be affine in x. The paper's Assumptions 4.1 and 4.2 do not imply that, and the stress-test example is a genuine counterexample.

What's actually new and good: using isotonic (antitonic) regression to estimate the survival function S0 from binary current-status data is a sensible idea, and Theorem 4.8's uniform tail bound for the antitonic estimator is a real contribution—I haven't seen that stated as an explicit probability inequality before, and the proof is self-contained. The simulation study is honest: they compare with Tullii et al. in the exact setting from that paper and show better empirical regret up to T = 8000, while acknowledging their asymptotic rate is worse. The real-data emulation with transaction prices is the kind of practical evaluation this literature often lacks.

Where the soft spots are. The main one is Lemma 4.3. In the exploration phase, p_t is uniform on [pmin, pmax] and the response is H y_t. Conditional on x, E[H y | x] = ∫_{pmin}^{pmax} S0(u − θ0^T x) du. That is affine in x only if the price interval covers the support of z for every x; then the slope is θ0 and the intercept is absorbed. Assumptions 4.1–4.2 are boundedness conditions, nothing about coverage. A concrete failure: x = (1, x2), θ0 = (0, 1), x2 ~ Unif[0,1], pmin = 0, pmax = 1, z ~ Unif(−1/2, 1/2). The conditional mean takes values 0.245, 0.5, 0.755 at x2 = 0.2, 0.5, 0.8—obviously nonlinear. So the OLS estimator does not consistently estimate θ0, and Lemma 4.3, which feeds Lemma 4.11 and Theorem 4.10, is not applicable. This is fixable: add a coverage assumption (e.g., pmin ≤ inf_x θ0^T x + inf U and pmax ≥ sup_x θ0^T x + sup U), which holds automatically when pmin/pmax are the observed min/max of valuations, as in the real-data section. But the theorem as written is overbroad.

The 'tuning-parameter-free' claim is overstated: α, U, τ1, pmin, and pmax are inputs; the method avoids kernel bandwidths and UCB constants, but it is not parameter-free. Also, Lemma A.1 has a typo in the direction of the probability inequality. These are minor compared with the Lemma 4.3 gap.

Who this is for: researchers in contextual dynamic pricing who want a low-smoothness alternative to kernel methods, and anyone needing concentration inequalities for isotonic regression with binary responses. The paper deserves a serious referee—the core idea is sound, the gap is identifiable and repairable, and the empirical work suggests the method has practical legs. I would engage with it, but not take Theorem 4.10 at face value until the OLS step is either proved under the stated assumptions or the theorem is restated with the coverage condition made explicit.

Referee Report

2 major / 4 minor

Summary. The paper proposes a tuning-parameter-free dynamic pricing algorithm in a linear valuation model with an unknown noise distribution. The algorithm estimates the linear coefficient via ordinary least squares (OLS) during a randomized-price exploration phase and estimates the noise survival function via antitonic regression on a second exploration sub-phase, then exploits the fitted model. The main theoretical contribution is Theorem 4.10, an expected regret bound of order ~O(T^{ν(α)} d^{α/(α+2)}) under α-Hölder continuity of the survival function, with ν(α) defined in Equation (2). The paper also proves a uniform concentration inequality for the antitonic regression estimator (Theorem 4.8), and reports simulations and a real-data application using Welltower data.

Significance. If the central regret guarantee were valid, the paper would be a meaningful contribution: it weakens the standard Lipschitz assumption on the noise distribution to α-Hölder continuity, removes tuning parameters, and introduces an antitonic-regression concentration result that appears to be new. The empirical work is careful, including validation of the theoretical rate in log-log plots, comparison against kernel, UCB, and VAPE baselines, and a real-data emulation. The antitonic concentration inequality (Theorem 4.8) and its proof via Hoeffding-type bounds are a useful technical contribution independent of the pricing application.

major comments (2)

[Section 4.2-4.3, Lemma 4.11] Lemma 4.3 is invoked from Fan et al. (2021, Lemma 4.1) without verifying its key condition that the regression function E[H y_t | x_t] is affine in x_t with slope θ0. In the exploration phase I_k of Algorithm 1, p_t is uniform on [pmin, pmax] and y_t = 1{p_t ≤ θ0ᵀx_t + z_t}, so E[H y_t | x_t] = ∫_{pmin}^{pmax} S0(p − θ0ᵀx_t) dp. This is affine in x_t only if the interval [pmin − θ0ᵀx_t, pmax − θ0ᵀx_t] contains the support U for every x_t in the covariate support, and even then the intercept is θ0,1 − pmin rather than θ0,1. Assumptions 4.1 and 4.2 do not include this support-coverage condition, and pmin is allowed to be positive. A concrete admissible instance satisfying Assumptions 4.1, 4.2, and 4.7 is x_t = (1, x2t) with x2t ~ Unif[0,1], θ0 = (0,1), pmin = 0, pmax = 1, and z_t ~ Unif(−1/2, 1/2). There E[H y_t | x2t] = 0.125 + x2t/2 + x2t^2/2 on [0, 0.5] and 1.5x2t − 0.5x2t^2 − 0.125 on [0.5, 1], which is not affine; OLS of H y_t on x_t converges to slope ≈0.8125 rather than 1, so ∥θhat_k − θ0∥ does not vanish. Consequently Lemma 4.3, and the bounds in Lemma 4.11 and Theorem 4.10 that rely on it, are not established under the stated assumptions. The authors should either add an explicit assumption that [pmin − θ0ᵀx, pmax − θ0ᵀx] ⊇ U for all x in the support and set pmin = 0 (or correct the intercept bias in the OLS response), or redesign the exploration phase so that the conditional regression function is exactly linear.
[Section 4.2-4.3, Lemma 4.11] There is a mismatch between the uniform convergence guarantee in Theorem 4.8 and its use in Lemma 4.11. Theorem 4.8 bounds sup_{u ∈ U_n} |Shat_k(u) − S_k(u)|, where U_n = {u ∈ U : [u ± δ_n] ⊂ U} is a slightly shrunk version of U. However, Lemma 4.11 defines the event S_k = {sup_{u ∈ U} |Shat_k(u) − S_k(u)| ≤ C (log n/n)^{α/(2α+1)}} and uses it to bound the pointwise error at q − θhatᵀx_t for q ∈ {p_t, p*_t}. No argument is given that these arguments lie in U_n with high probability, nor that the boundary layer U \ U_n contributes negligibly. Since the regret analysis requires control on the full domain U, the proof of Lemma 4.11 is incomplete as written. The authors should either extend Theorem 4.8 to sup over U (e.g., by handling the boundary with a separate argument) or modify the event S_k to use U_n and account for the boundary contribution in the regret bound.

minor comments (4)

[Appendix A.1, Lemma A.1] The inequality in the statement of Lemma A.1 is reversed: it should read P(Mn(θ) ≤ (D log n)^{1/2}) ≥ 1 − ((n+1)/n^D)^2, as used in the proof of Theorem 4.8.
[Section 4.3, Proof of Theorem 4.10] The sentence 'Here we prove the result for α > 1' appears in the body of the proof; from the context and the case split in the theorem, this should read 'α > 1/2'.
[Section 5 and 6] The paper claims to be 'tuning-parameter-free,' but the initial episode length τ1 is a user-chosen constant. The authors address this in Remark 5.1 with a robustness simulation, which is helpful; nevertheless the abstract and contribution list may overstate the tuning-free property.
[Section 6] In the real-data application, the support U of the noise is estimated from residuals and then provided as an input to the algorithm, whereas the theory assumes U is known. This discrepancy between the theoretical setup and the implementation should be acknowledged explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the regret bound follows from independent concentration inequalities and is not fitted or self-referential.

full rationale

The paper's claimed derivation chain is not circular. The central regret bound (Theorem 4.10) is obtained by decomposing regret into exploration and exploitation terms and invoking three independent ingredients: Lemma 4.3 (OLS concentration, cited from Fan et al. 2021), Proposition 4.6 (Hölder transfer from S0 to Sθ), and Theorem 4.8 (uniform concentration of antitonic regression, proved in Appendix A via Hoeffding's inequality and a design-density lemma quoted from Mösching & Dümbgen 2020). None of these inputs is defined in terms of the target regret, and no fitted parameter is renamed as a prediction. The exploration length |Ek| = d^{α/(α+2)} τ_k^{ν(α)} is selected by optimizing the paper's own upper-bound expression, not by fitting to data; the exponents ξ* and ν* are derived by balancing the d- and T-exponents in the bound. The only self-citation, Bracale et al. (2024), appears in Remark 4.4 as an example in the optimal-design literature and is not load-bearing for the main theorem. The skeptical concern about Lemma 4.3—that Assumptions 4.1 and 4.2 do not visibly ensure E[H y_t | x_t] is affine in x_t—is a potential correctness gap in the applicability of an external lemma, not a circular reduction; the cited lemma's content is independent of the present paper's conclusions. Accordingly, no specific circular step can be quoted, and the paper should not receive a circularity penalty on that basis.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central claim rests on standard assumptions in dynamic pricing plus a cited OLS lemma whose conditions are not verified. The algorithm also requires knowledge of U and alpha, which are user-specified.

free parameters (4)

alpha (Holder exponent of S0) = 1 in experiments
Input to Algorithm 1 controlling exploration length; in simulations and real data it is fixed to 1 even when the true noise may not be Lipschitz.
U (support of noise z) = (-1/2,1/2) in simulations; estimated as (-17,12) from residuals in real data
The algorithm samples w_t uniformly from U; the theory assumes U known, while practice estimates it offline.
tau_1 (initial episode length) = 100 in simulations; 150 in real application
User-defined constant; the paper shows robustness but it is a free choice.
pmin, pmax (price bounds) = 0,5 in simulations; 139,1494.63 in real data
Assumed known; the OLS consistency depends on their values (see Lemma 4.3 concern).

assumptions (6)

domain assumption Assumption 4.1: theta0 is interior point of compact convex parameter space
Standard in pricing literature; used for concentration of OLS.
domain assumption Assumption 4.2: i.i.d. bounded contexts with positive definite covariance
Ensures OLS design well-conditioned.
domain assumption Assumption 4.7: S0 is alpha-Holder continuous
The convergence rate of the antitonic estimator and the regret bound depend on this.
domain assumption Lemma 4.3 (Fan et al.) holds for the paper's OLS design
The paper cites Fan et al. Lemma 4.1 without proof; the setup with p_t uniform and response H y_t must satisfy the lemma's conditions, which is not demonstrated.
domain assumption Support U and smoothness alpha are known to the algorithm
Algorithm 1 takes U and alpha as inputs; the theory does not address their estimation.
standard math Mösching & Dümbgen (2020) empirical process bound used in Lemma A.2
External result on uniform empirical measure concentration; used to control the density of design points.

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Pith. "Pith review of Dynamic Pricing in the Linear Valuation Model using Shape Constraints." pith.science (2026). https://pith.science/paper/QLDFLFBY

@misc{pith2026250205776,
  author       = {Pith},
  title        = {Pith review of: Dynamic Pricing in the Linear Valuation Model using Shape Constraints},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QLDFLFBY}},
  note         = {Machine review of arXiv:2502.05776}
}

abstract

We propose a shape-constrained approach to dynamic pricing for censored data in the linear valuation model eliminating the need for tuning parameters commonly required by existing methods. Previous works have addressed the challenge of unknown market noise distribution $F_0$ using strategies ranging from kernel methods to reinforcement learning algorithms, such as bandit techniques and upper confidence bounds (UCB), under the assumption that $F_0$ satisfies Lipschitz (or stronger) conditions. In contrast, our method relies on isotonic regression under the weaker assumption that $F_0$ is $\alpha$-H\"older continuous for some $\alpha \in (0,1]$, for which we derive a regret upper bound. Simulations and experiments with real-world data obtained by Welltower Inc (a major healthcare Real Estate Investment Trust) consistently demonstrate that our method attains lower empirical regret in comparison to several existing methods in the literature while offering the advantage of being tuning-parameter free.

Figures

Figures reproduced from arXiv: 2502.05776 by the authors.

**Figure 1.** Picture of a general episode Jk, k = 1, 2, . . . , K. 4 Regret Analysis Before proceeding with the regret analysis we need to discuss the convergence rates of θbk to θ0 and Sbk to S0. We present our main theorems and proofs. We defer to the Appendix for the missing proofs. 4.1 Estimation of θ0 Assumption 4.1 (Bounded parameter space). The parameter θ0 ∈ R d is an interior point of Θ and the parameter space Θ is a co… view at source ↗

**Figure 2.** This plot shows the total expected regret (blue line) with [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 3.** This plot in log2 − log2 scale shows the cumulative regret over time up to T = 4000 for different values of τ1 ∈ {31, 62, 124, 248} and with F0,1/3 for which theoretical regret rate is 0.86. For each value of τ1, we repeated the simulation 36 times and displayed the corresponding 95% confidence intervals. As we can see, the regret remains similar across different values of τ1 and the the empirical slopes are close t… view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Regret comparison in the simulation setting of [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]

**Figure 5.** Figure 5: Residuals [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]

**Figure 6.** Figure 6: Revenue comparison 7 Conclusions We introduced a novel method for estimating the market noise distribution F0 by leveraging its natural shape constraint: monotonicity. Our analysis led to an expected upper bound on the total regret of order Oe(T ν(α)d α/2+α ), where ν(…

**Figure 7.** Figure 7: This plot was generated using as true F0 the one considered in Fan et al. (2021) with density f0(z) = 6 1 4 − z 2 1{z ∈ (−1/2, 1/2)}. We repeated the simulation 36 times and the corresponding 95% confidence interval. The plot is in log2 -log2 scale to show the regret…

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Reference graph

Works this paper leans on

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Lemma A.2

17 Published in Transactions on Machine Learning Research (04/2025) Before proceeding with the technical Lemma A.2, let’s define ρn ≜ logn n , and λ(·) the Lebesgue measure, and denote byPn(·) the empirical measure of the design pointswt, that means Pn(B) ≜ 1 n#{t∈T :wt∈B} for B⊂U . Lemma A.2. Let w1,w 2,...,w n i.i.d. points with densityfw that satisfies...

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Now we prove Theorem 4.8. Proof. Letn be sufficiently large so thatUn̸=∅ and such that the eventAn,γ in Lemma A.2 occurs. Sincefw is the uniform distribution, the valueC2 defined in Lemma A.2 corresponds to1/|U|. Foru∈U n the indices r(u) ≜ min{j∈{ 1,...,m } :uj≥u−δn}, j(u) ≜ max{j∈{ 1,...,m } :uj≤u}, are well-defined, because[u−δn,u ] is a subinterval of...

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20 Published in Transactions on Machine Learning Research (04/2025) A.4 Proof of Lemma 4.11 Let nk =|Ik| and~nk =|~Ik| and t∈E′ k

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Analyzing C onEk: By Proposition 4.6 we have|Sk(u)−S0(u)|I(Ek) ≲∥ˆθk−θ0∥α 2≤Rα nk

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Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

[1] [1]

Lemma A.2

17 Published in Transactions on Machine Learning Research (04/2025) Before proceeding with the technical Lemma A.2, let’s define ρn ≜ logn n , and λ(·) the Lebesgue measure, and denote byPn(·) the empirical measure of the design pointswt, that means Pn(B) ≜ 1 n#{t∈T :wt∈B} for B⊂U . Lemma A.2. Let w1,w 2,...,w n i.i.d. points with densityfw that satisfies...

work page 2025

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Multi-task dynamic pricing in credit market with contextual information

Adel Javanmard, Jingwei Ji, and Renyuan Xu. Multi-task dynamic pricing in credit market with contextual information. arXiv preprint arXiv:2410.14839,

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Towards agnostic feature-based dynamic pricing: Linear policies vs linear valuation with unknown noise

15 Published in Transactions on Machine Learning Research (04/2025) Jianyu Xu and Yu-Xiang Wang. Towards agnostic feature-based dynamic pricing: Linear policies vs linear valuation with unknown noise. InInternational Conference on Artificial Intelligence and Statistics, pp. 9643–9662. PMLR,

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Now we prove Theorem 4.8. Proof. Letn be sufficiently large so thatUn̸=∅ and such that the eventAn,γ in Lemma A.2 occurs. Sincefw is the uniform distribution, the valueC2 defined in Lemma A.2 corresponds to1/|U|. Foru∈U n the indices r(u) ≜ min{j∈{ 1,...,m } :uj≥u−δn}, j(u) ≜ max{j∈{ 1,...,m } :uj≤u}, are well-defined, because[u−δn,u ] is a subinterval of...

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20 Published in Transactions on Machine Learning Research (04/2025) A.4 Proof of Lemma 4.11 Let nk =|Ik| and~nk =|~Ik| and t∈E′ k

+ 1⌉, which concludes the proof. 20 Published in Transactions on Machine Learning Research (04/2025) A.4 Proof of Lemma 4.11 Let nk =|Ik| and~nk =|~Ik| and t∈E′ k. Define the eventEk ={∥ˆθk−θ0∥≤ Rnk} where we recall Rnk∝ √ d lognk nk , as defined in Lemma 4.3, and Rk,t(q) =|ˆSk(q−ˆθ⊤ kxt)−S0 ( q−θ⊤ 0xt ) |, q ∈{pt,p∗ t}. Recall that ˆθk = OLS{(xt,yt)}t∈Ik...

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Analyzing C onEk: By Proposition 4.6 we have|Sk(u)−S0(u)|I(Ek) ≲∥ˆθk−θ0∥α 2≤Rα nk

Analyzing B onEk: By Proposition 4.6,Sk is α-Hölder, then E[BI(Ek)] ≲∥ˆθk−θ0∥α 2≤Rα nk. Analyzing C onEk: By Proposition 4.6 we have|Sk(u)−S0(u)|I(Ek) ≲∥ˆθk−θ0∥α 2≤Rα nk. Combining the termsRk,t(q)I(Ec k) and Rk,t(q)I(Ek) from Equation (9):we get sup q Rk,t(q) ≲ (log~nk ~nk )α/(2α+1) + (d lognk nk )α/2 . 21 Published in Transactions on Machine Learning Re...

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Learning the Distribution Map in Reverse Causal Performative Prediction

14 Published in Transactions on Machine Learning Research (04/2025) Daniele Bracale, Subha Maity, Moulinath Banerjee, and Yuekai Sun. Learning the distribution map in reverse causal performative prediction.arXiv preprint arXiv:2405.15172,

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Improved algorithms for contextual dynamic pricing

Matilde Tullii, Solenne Gaucher, Nadav Merlis, and Vianney Perchet. Improved algorithms for contextual dynamic pricing. arXiv preprint arXiv:2406.11316,

arXiv