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REVIEW 2 major objections 5 minor 49 references

Continuum-armed Bandit Optimization with Batch Pairwise Comparison Oracles

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper claims that biased pairwise comparison oracles, queried in blocks of n periods, suffice for near-optimal continuum-armed bandit learning, with regret $\tilde{O}(T^{(k+d)/(2k+d)})$ for $k$-smooth functions and…

desk verdict Real contribution in the core oracle model and minimax rates; the NRM application corollary has a repairable feasibility gap that should be addressed in review. read the letter →

arxiv 2505.22361 v1 pith:JBIYVAN5 submitted 2025-05-28 cs.LG stat.ML

Xiangyu Chang , Xi Chen , Yining Wang , Zhiyi Zeng This is my paper
classification cs.LGstat.ML MSC 62L0568Q3290B05
keywords continuum-armedbanditpairwisecomparisonoraclebatchedfeedbackregretboundsHöldersmoothfunctionclassesstronglyconcaveoptimizationlocalpolynomialregressionnetworkrevenuemanagement
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

This paper tries to establish that continuum-armed bandit optimization remains nearly as easy when the only feedback is a biased pairwise comparison oracle queried in batches. Instead of observing $f(x)$, the learner chooses two actions $(x,x')$ and spends $n$ periods to get an estimate of $f(x')-f(x)$ with error of order $\sqrt{\ln(1/\delta)/n}$. The paper proves two algorithmic results: for $k$-H\"older smooth objectives, cumulative regret is $\tilde{O}(T^{(k+d)/(2k+d)})$, matching the minimax rate for the strictly easier setting with direct function observations; for strongly concave objectives, regret is $\tilde{O}(\sqrt{T})$. These rates matter because pricing and inventory problems with censored demand naturally deliver exactly this kind of difference oracle, and the paper shows the new algorithms improve existing regret bounds for joint pricing-inventory control and network revenue management.

What carries the argument

The argument is carried by two reductions. First, the domain is partitioned into $J^d$ cubes of width $1/J$, and on each cube a truncated Taylor expansion (the local polynomial feature map $\phi_j(x)$ with $\nu=\binom{k+d-1}{d}$ features) makes $f(x)-f(x_s)$ approximately linear in $\phi_j(x)-\phi_j(x_s)$ with error $O(J^{-k})$. This turns each cube into a linear bandit, solved by a batched LinUCB variant (Algorithm 1) in which the batch size $n_t$ is chosen so that $\det(\Lambda+n_t\phi_t\phi_t^\top)$ doubles at every round, and a baseline anchor $x_s$ makes the pairwise oracle usable. Second, an iterative doubling wrapper (Algorithm 2) keeps the anchor near-optimal, and a tournament successive elimination over cubes (Algorithm 3) guarantees the optimal cube is never eliminated while all surviving cubes are near-optimal. For strongly concave functions, Algorithm 4 uses finite differences with step $h_\tau$ and increasing batch sizes $\beta_\tau$ to build inexact gradients, then runs fixed-step proximal gradient ascent with epoch lengths growing geometrically, which controls the accumulation of gradient estimation error.

What would settle it

Simulate Algorithm 3 on a one-dimensional quadratic ($k=2$, $d=1$) with an oracle whose error is $\sqrt{(\gamma_1+\gamma_2\ln(1/\delta))/n}$ plus an extra bias of $c/n^{1/4}$ that decays slower than $1/\sqrt{n}$; if the empirical cumulative regret grows faster than the claimed $\tilde{O}(T^{3/5})$ rate as $T$ increases, the theorem's reliance on Definition 1 is falsified.

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Extended reading notes

Core claim

The central claim is that a $\gamma_1,\gamma_2$-consistent pairwise comparison oracle, one that for any pair $(x,x')$ and query block of length $n$ returns $y$ with $|y-(f(x')-f(x))|\le \sqrt{(\gamma_1+\gamma_2\ln(1/\delta))/n}$ with probability $1-\delta$, is a sufficient feedback model for near-optimal continuum-armed bandit optimization. Theorem 1 states that for $f\in\Sigma_d(k,M)$, Algorithm 3 achieves cumulative regret at most $C(M^2\nu^2+\nu^3+M\nu^2\sqrt{\gamma_1+\gamma_2\ln T})T^{(k+d)/(2k+d)}\ln^4 T$, matching the minimax lower bound for unbiased function observations up to polylog factors. Theorem 2 states that for $f\in\Sigma_d(2,M)\cap\Gamma_d(\sigma)$, Algorithm 4 achieves regret $\tilde{O}(d^2+d\sqrt{T})$, matching the $\Omega(\sqrt{T})$ lower bound. The paper further claims that when the oracle is instantiated through existing procedures for joint pricing-inventory with censored demand and for blind network revenue management, the resulting corollaries improve the best known regret bounds: smoother objectives yield the $T$-exponent $(k+d)/(2k+d)$ instead of $(d+2)/(d+4)$, and the NRM dimensional factor drops from $d^{3.5}$ to $d^{2.25}$ in the leading $\sqrt{T}$ term.

Load-bearing premise

The load-bearing premise is that the oracle's error concentrates as $\sqrt{(\gamma_1+\gamma_2\ln(1/\delta))/n}$ with known constants $\gamma_1,\gamma_2$, uniformly over every compared pair and every batch length; if real comparison procedures, especially those imported from the inventory and network-revenue applications, do not meet that condition with known constants, the stated regret bounds do not follow.

Editorial extensions

If this is right

  • For any $k$-H\"older smooth objective in $d$ dimensions, the regret exponent improves from the $k=2$ value $T^{(d+2)/(d+4)}$ to $T^{(k+d)/(2k+d)}$, so extra smoothness is automatically converted into faster learning even under the weaker comparison oracle.
  • The weaker pairwise oracle costs only logarithmic factors: its minimax rate in $T$ is the same as that of the standard continuum-armed bandit with direct noisy function values.
  • Strong concavity is enough to reach $\sqrt{T}$ regret even with biased batched comparisons, matching the lower bound for problems with multi-point unbiased gradient feedback.
  • For joint pricing and inventory with censored demand, the corollary improves the best known upper bound whenever the partially optimized reward $G(p)$ is smoother than twice differentiable, i.e., $k>2$.
  • For blind network revenue management, the corollary keeps the $\tilde{O}(\sqrt{T})$ rate while improving the polynomial dependence on the number of products from $d^{3.5}$ to $d^{2.25}$ in the leading term.

Reading between the lines

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

  • Because the bandit layer only uses the oracle's concentration rate and known constants, any setting that can produce a difference estimator with the same $1/\sqrt{n}$ error, such as dueling feedback, censored demand, or inverse-demand estimation, can reuse Algorithms 3 and 4 without redesign, provided the constants $\gamma_1,\gamma_2$ are known.
  • A testable design question the paper leaves implicit is that the imported NRM oracle is only guaranteed for feasible demand rates, while Algorithm 4's gradient query points $x_\tau\pm h_\tau e_j$ may leave the feasible polytope $A\lambda\le\gamma$; projecting the gradient queries back into the feasible region would close this gap.
  • The practical parameter heuristics in the paper (choose $k=3$ or $4$, set $M=\ln T$) suggest the algorithms may perform well even with misspecified smoothness, but the theory does not cover adaptation, so measuring regret under misspecification would be a natural stress test.
  • The tournament-elimination structure suggests a general recipe for other infinite-arm problems with batch comparison oracles: maintain a small set of candidate cells, compare them with long queries, and run batched linear bandits inside the surviving cells.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper studies continuum-armed bandit optimization under a novel pairwise-comparison oracle in which a pair (x, x') is queried for n consecutive periods and the learner receives an estimate of f(x') - f(x) with error roughly sqrt((gamma1 + gamma2 ln(1/delta))/n) with probability 1 - delta. The central contribution is two algorithms: Algorithm 3, combining local polynomial approximation, batched LinUCB, and tournament successive elimination, which attains eO(T^{(k+d)/(2k+d)}) regret for Holder functions of order k; and Algorithm 4, an inexact projected-gradient method with finite-difference gradient estimates, which attains eO(d sqrt(T)) regret for strongly concave and twice-smooth functions. Theorems 1 and 2 state these guarantees, with proofs given in the supplementary material. The paper further claims that these algorithms improve existing regret bounds for joint pricing and inventory control with censored demand and for network revenue management, via oracle constructions imported from prior work by the same authors. The numerical section reports simulations on synthetic objectives and inventory instances.

Significance. The abstract oracle model is well posed and substantially weaker than standard noisy function-value feedback; obtaining near-optimal rates under it is a nontrivial contribution. The tournament-plus-LinUCB analysis and the biased-gradient proximal analysis are novel, and the claimed rates match known minimax lower bounds up to polylogarithmic factors. The paper also gives reasonably complete proofs for the abstract oracle model in the supplementary material and supports the theory with numerical experiments. Credit is due for making the oracle assumptions explicit rather than hiding bias in application-specific noise models. The main weakness is the application section: the NRM transfer in Corollary 2 is not established as written because Algorithm 4 queries points that violate the imported oracle's feasibility precondition, and the imported lemmas do not supply the constants needed to instantiate Definition 1. This gap is localized and repairable, but it currently blocks the paper's claimed operations-management improvements.

major comments (2)
  1. [Section 5.2.2 and Section 5.2.3 (Corollary 2, Algorithm 4 Lines 7-8, Lemma 6)] The NRM regret bound in Corollary 2 is not established because Algorithm 4's finite-difference queries can leave the feasible set required by Lemma 6. Lemma 6 is stated only for pairs lambda, lambda' in R^d_{++} satisfying A lambda <= gamma and A lambda' <= gamma. Algorithm 4 projects only the current iterate x_tau onto Z = {A lambda <= gamma} (Line 12), but Lines 7-8 query the oracle at x_tau + h_tau e_j and x_tau - h_tau e_j. When a resource constraint is active, at least one of these two points violates A lambda <= gamma, so the comparison oracle guarantee of Lemma 6 does not apply to those calls. Consequently the d^2.25 sqrt(T) NRM regret improvement over Miao and Wang (2021) is not justified as written. This is repairable, for example by using projected or one-sided finite differences and re-deriving the gradient error bound under the feasible projection, but it must be fixed before the application result can be claimed.
  2. [Section 5.2.2 (Lemma 6) and Section 5.1.2 (Lemma 5)] The oracle lemmas imported from prior work do not provide the explicit constants needed to instantiate Definition 1 and hence the algorithm parameters. Definition 1 requires known gamma1 and gamma2 such that the error is bounded by sqrt((gamma1 + gamma2 ln(1/delta))/n) for every n and delta. Lemma 6 states only an asymptotic bound of the form O(d^2.5 ln(dT)/sqrt(n) + d^5 ln^3(dT)/n) with unspecified constants and an unparsable probability expression. For small n, especially the first epochs of Algorithm 4 where beta_tau = 1, the 1/n term cannot be absorbed into a C/sqrt(n) bound without additional conditions. Even after the feasibility issue is resolved, the authors must specify how gamma1 and gamma2 (and hence C1 in Algorithm 4) are computed from the constants hidden in Lemma 6, or state the oracle guarantee directly in the form of Definition 1. The same concern applies to Lemma 5 and Corollary 1, although Remark 5 at least points to an external source for the constants.
minor comments (5)
  1. [Lemmas 2-6, Theorems 1-2] The phrase 'with probability 1 - eO(T^{-2})' appears repeatedly and, taken literally, is not a valid probability bound because e^{O(T^{-2})} is at least 1. This should be corrected to something like 1 - e^{-Omega(T^{-2})} or 1 - e^{-Omega(T)} throughout.
  2. [Algorithm 4, Line 4] The definition of h_tau uses 'gamma' without a subscript; from Definition 1 and the proof it should be gamma2, giving h_tau = 4 sqrt((gamma1 + 2 gamma2 ln T)/(beta_tau d)).
  3. [Algorithm 4, Lines 5-8 and Theorem 2 proof] There is an inconsistency in the epoch length: for each j the algorithm makes two oracle calls of length beta_tau, so an epoch uses 2 d beta_tau periods, while Line 5 and the proof of Theorem 2 refer to 4 d beta_tau periods. This does not change the rates but must be clarified.
  4. [Theorem 1] The leading constant is typeset as 'C 1', which is easily confused with the algorithm parameter C1 used throughout Section 3; please use a distinct symbol such as C_* or K.
  5. [EC.1.4 (proof of Lemma 4)] The proof refers to 'Eq. (EC.12)' when the displayed inequality appears to be Eq. (EC.11); please fix the cross-reference.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the core bandit theorems are derived from Definition 1 and the stated function classes, not from fitted data or from the target results; the Section 5 self-citations are premises for the application corollaries, not recycled conclusions. A separate, non-circular correctness gap exists in Corollary 2's NRM transfer.

full rationale

The main derivation chain (Definition 1, Lemmas 1-4, Theorems 1-2) is self-contained: the regret bounds are proved from the gamma1,gamma2-consistency of the pairwise comparison oracle and the Holder/strong-concavity assumptions, with no parameter fitted to data and no target bound inserted as an assumption. The minimax optimality claims are benchmarked against lower bounds from external works (Wang et al. 2019; Agarwal et al. 2010), which is not circular. Section 5 imports oracle constructions from prior papers co-authored by Yining Wang (Chen et al. 2023, Lemma 5; Miao and Wang 2021, Lemma 6); these are inputs to the application corollaries, not the outputs being proved, and the manuscript explicitly states that 'all algorithms and analysis in this section are not new'. Thus the self-citations are not recycled as conclusions and do not make the central claim circular. The one flagged issue is a correctness gap rather than circularity: Lemma 6 requires both query points to satisfy A lambda <= gamma and A lambda' <= gamma, whereas Algorithm 4 queries x_tau + h_tau e_j and x_tau - h_tau e_j on Lines 7-8 while only x_tau is constrained to Z by the projection on Line 12; when a resource constraint is active, one finite-difference point can leave Z, so Corollary 2's NRM transfer is not established as written. This gap is repairable and does not affect the abstract bandit theorems. Score 2 reflects the presence of minor self-citations in the application section without any circular reduction.

Assumptions & free parameters 5 free parameters · 7 assumptions · 0 invented entities

The central theoretical model is axiomatized through Definition 1 and the Holder/strong-concavity function classes; the algorithms further assume knowledge of smoothness and noise constants. The application improvements import oracle constructions from previous self-authored works, which adds to the burden but is not parameter fitting. No new physical or conceptual entities with independent empirical handles are introduced.

free parameters (5)
  • k (smoothness order) = assumed known; k=2,3,4 and k=4,5,6 in experiments
    Input to Algorithm 3 and to the regret bound; not fitted to data. A lower bound in the cited literature shows knowledge of smoothness is necessary.
  • M (Holder bound) = 1e-4 for f1; 1/sqrt(nu) for f2 in experiments
    Theoretical algorithms require M as an input; experiments choose values that satisfy the Holder assumption, not fitted to outcomes.
  • gamma1, gamma2 (oracle concentration constants) = 0.01 and 0.005 for f1; 0.01 and 0.01 for concave experiments
    Chosen by hand for numerical confidence widths; the theoretical bounds treat gamma1 and gamma2 as known oracle parameters.
  • J (number of cells per dimension) = 3 in experiments; T^(1/(2k+d)) in theory
    Discretization level set by the theoretical analysis; experiments use a small fixed value.
  • sigma, eta, alpha (strong concavity and step parameters) = sigma=1, eta=1, alpha=10 in experiments
    Algorithm 4 parameters; theory sets eta=sigma/M and alpha=1/M. The experimental values are chosen by hand, not fitted.
assumptions (7)
  • domain assumption Pairwise comparison oracle consistency (Definition 1)
    The entire analysis assumes the oracle satisfies the stated high-probability error bound for every n, every pair of actions, and every confidence level. The applications claim to construct such oracles but the construction proofs are in prior works.
  • domain assumption Holder smoothness and strong concavity of the objective
    Theorems 1 and 2 assume f in Sigma_d(k,M) and f in Sigma_d(2,M) intersect Gamma_d(sigma). These function-class assumptions are standard but are postulates about the environment.
  • domain assumption Interior maximizer (Assumption 1)
    The algorithms require a ball around the maximizer to be contained in the domain so that local comparisons and gradient steps are valid.
  • ad hoc to paper Knapsack penalty function (Assumption 2)
    The convex, Lipschitz penalty psi with zero set Z is introduced specifically to model feasibility in network revenue management and to keep the regret definition tractable.
  • standard math Taylor expansion with Lagrangian remainder for local polynomial approximation (Lemma 1)
    The local polynomial bound is proved by multivariate Taylor expansion; this is a standard mathematical tool.
  • standard math Linear bandit confidence ellipsoid machinery (Abbasi-Yadkori et al. 2011)
    The batched LinUCB analysis relies on determinant-based confidence ellipsoid bounds from the cited linear bandit literature.
  • standard math External minimax lower bounds (Wang et al. 2019; Agarwal et al. 2010)
    The optimality claims are benchmarked against existing lower bounds for direct function-value feedback, which are accepted prior results.

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Pith. "Pith review of Continuum-armed Bandit Optimization with Batch Pairwise Comparison Oracles." pith.science (2026). https://pith.science/paper/JBIYVAN5

@misc{pith2026250522361,
  author       = {Pith},
  title        = {Pith review of: Continuum-armed Bandit Optimization with Batch Pairwise Comparison Oracles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JBIYVAN5}},
  note         = {Machine review of arXiv:2505.22361}
}
abstract

This paper studies a bandit optimization problem where the goal is to maximize a function $f(x)$ over $T$ periods for some unknown strongly concave function $f$. We consider a new pairwise comparison oracle, where the decision-maker chooses a pair of actions $(x, x')$ for a consecutive number of periods and then obtains an estimate of $f(x)-f(x')$. We show that such a pairwise comparison oracle finds important applications to joint pricing and inventory replenishment problems and network revenue management. The challenge in this bandit optimization is twofold. First, the decision-maker not only needs to determine a pair of actions $(x, x')$ but also a stopping time $n$ (i.e., the number of queries based on $(x, x')$). Second, motivated by our inventory application, the estimate of the difference $f(x)-f(x')$ is biased, which is different from existing oracles in stochastic optimization literature. To address these challenges, we first introduce a discretization technique and local polynomial approximation to relate this problem to linear bandits. Then we developed a tournament successive elimination technique to localize the discretized cell and run an interactive batched version of LinUCB algorithm on cells. We establish regret bounds that are optimal up to poly-logarithmic factors. Furthermore, we apply our proposed algorithm and analytical framework to the two operations management problems and obtain results that improve state-of-the-art results in the existing literature.

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Works this paper leans on

49 extracted references · 47 canonical work pages

  1. [1]

    Abbasi-Yadkori, Yasin, D \'a vid P \'a l, Csaba Szepesv \'a ri. 2011. Improved algorithms for linear stochastic bandits. Proceedings of the Advances in Neural Information Processing Systems (NIPS)\/ . 2312--2320

    work page 2011

  2. [2]

    Agarwal, Alekh, Ofer Dekel, Lin Xiao. 2010. Optimal algorithms for online convex optimization with multi-point bandit feedback. Conference on Learning Theory (COLT)\/

    work page 2010

  3. [3]

    Agarwal, Arpit, Rohan Ghuge, Viswanath Nagarajan. 2022. An asymptotically optimal batched algorithm for the dueling bandit problem. Advances in Neural Information Processing Systems (NeurIPS)\/

    work page 2022

  4. [4]

    Agrawal, Rajeev. 1995. The continuum-armed bandit problem. SIAM Journal on Control and Optimization\/ 33 (6) 1926--1951

    work page 1995

  5. [5]

    Argarwal, Arpit, Rohan Ghuge, Viswanath Nagarajan. 2022. Batched dueling bandits. International Conference on Machine Learning (ICML)\/

    work page 2022

  6. [6]

    Besbes, Omar, Yonatan Gur, Assaf Zeevi. 2015. Non-stationary stochastic optimization. Operations research\/ 63 (5) 1227--1244

    work page 2015

  7. [7]

    Besbes, Omar, Assaf Zeevi. 2009. Dynamic pricing without knowing the demand function: Risk bounds and near-optimal algorithms. Operations Research\/ 57 (6) 1407--1420

    work page 2009

  8. [8]

    Besbes, Omar, Assaf Zeevi. 2012. Blind network revenue management. Operations research\/ 60 (6) 1537--1550

    work page 2012

Show all 49 references
  1. [9]

    Besbes, Omar, Assaf Zeevi. 2015. On the (surprising) sufficiency of linear models for dynamic pricing with demand learning. Management Science\/ 61 (4) 723--739

    2015

  2. [10]

    Broder, Josef, Paat Rusmevichientong. 2012. Dynamic pricing under a general parametric choice model. Operations Research\/ 60 (4) 965--980

    2012

  3. [11]

    Bu, Jinzhi, David Simchi-Levi, Yunzong Xu. 2022. Online pricing with offline data: Phase transition and inverse square law. Management Science\/ 68 (12) 8568--8588

    2022

  4. [12]

    Bubeck, S \'e bastien, R \'e mi Munos, Gilles Stoltz, Csaba Szepesv \'a ri. 2011. X-armed bandits. Journal of Machine Learning Research\/ 12 (5)

    2011

  5. [13]

    Chen, Boxiao, Xiuli Chao, Cong Shi. 2021. Nonparametric learning algorithms for joint pricing and inventory control with lost sales and censored demand. Mathematics of Operations Research\/ 46 (2) 726--756

    2021

  6. [14]

    Chen, Boxiao, David Simchi-Levi, Yining Wang, Yuan Zhou. 2022. Dynamic pricing and inventory control with fixed ordering cost and incomplete demand information. Management Science\/ 68 (8) 5684--5703

    2022

  7. [15]

    Chen, Boxiao, Yining Wang, Yuan Zhou. 2023. Optimal policies for dynamic pricing and inventory control with nonparametric censored demands. Management Science\/ 70 (5) 3363--3380

    2023

  8. [16]

    Chen, Qi, Stefanus Jasin, Izak Duenyas. 2019. Nonparametric self-adjusting control for joint learning and optimization of multiproduct pricing with finite resource capacity. Mathematics of Operations Research\/ 44 (2) 601--631

    2019

  9. [17]

    Chen, Xin, David Simchi-Levi. 2004 a . Coordinating inventory control and pricing strategies with random demand and fixed ordering cost: The finite horizon case. Operations Research\/ 52 (6) 887--896

    2004

  10. [18]

    Chen, Xin, David Simchi-Levi. 2004 b . Coordinating inventory control and pricing strategies with random demand and fixed ordering cost: The infinite horizon case. Mathematics of Operations Research\/ 29 (3) 698--723

    2004

  11. [19]

    Chen, Xin, David Simchi-Levi. 2012. Pricing and inventory management, R. Philips and O. Ozer, eds.\/ . Oxford University Press, Oxford

    2012

  12. [20]

    Chen, Yiwei, Cong Shi. 2019. Network revenue management with online inverse batch gradient descent method. Available at SSRN 3331939\/

    2019

  13. [21]

    Chernih, Andrew, Ian H Sloan, Robert S Womersley. 2014. Wendland functions with increasing smoothness converge to a gaussian. Advances in Computational Mathematics\/ 40 185--200

    2014

  14. [22]

    Cheung, Wang Chi, David Simchi-Levi, He Wang. 2017. Dynamic pricing and demand learning with limited price experimentation. Operations Research\/ 65 (6) 1722--1731

    2017

  15. [23]

    den Boer, Arnoud V., Boxiao Chen, Yining Wang. 2024. Pricing and positioning of horizontally differentiated products with incomplete demand information. Operations Research\/ 72 (6) 2446--2466

    2024

  16. [24]

    Dud \' k, Miroslav, Katja Hofmann, Robert E Schapire, Aleksandrs Slivkins, Masrour Zoghi. 2015. Contextual dueling bandits. Conference on Learning Theory (COLT)\/ . PMLR, 563--587

    2015

  17. [25]

    Fan, Jianqing, Irene Gijbels. 2018. Local polynomial modelling and its applications\/ . Routledge

    2018

  18. [26]

    Ferreira, Kris Johnson, David Simchi-Levi, He Wang. 2018. Online network revenue management using thompson sampling. Operations Research\/ 66 (6) 1586--1602

    2018

  19. [27]

    Gallego, Guillermo, Garrett van Ryzin. 1994. Optimal dynamic pricing of inventories with stochastic demand over finite horizons. Management Science\/ 40 (8) 999--1020

    1994

  20. [28]

    Hoeffding, Wassily. 1963. Probability inequalities for sums of bounded random variables. Journal of the American Statistical Association\/ 58 (301) 13--30

    1963

  21. [29]

    Huh, Woonghee Tim, Ganesh Janakiraman. 2008. (s, s) optimality in joint inventory-pricing control: An alternate approach. Operations Research\/ 56 (3) 783--790

    2008

  22. [30]

    Huh, Woonghee Tim, Paat Rusmevichientong. 2009. A nonparametric asymptotic analysis of inventory planning with censored demand. Mathematics of Operations Research\/ 34 (1) 103--123

    2009

  23. [31]

    Katehakis, Michael N, Jian Yang, Tingting Zhou. 2020. Dynamic inventory and price controls involving unknown demand on discrete nonperishable items. Operations Research\/ 68 (5) 1335--1355

    2020

  24. [32]

    Kumagai, Wataru. 2017. Regret analysis for continuous dueling bandit. Advances in Neural Information Processing Systems\/ 30

    2017

  25. [33]

    Locatelli, Andrea, Alexandra Carpentier. 2018. Adaptivity to smoothness in x-armed bandits. Conference on Learning Theory\/ . PMLR, 1463--1492

    2018

  26. [34]

    Miao, Sentao, Yining Wang. 2021. Network revenue management with nonparametric demand learning: sqrt \ T \ -regret and polynomial dimension dependency. Available at SSRN 3948140\/

    2021

  27. [35]

    Miao, Sentao, Yining Wang, Jiawei Zhang. 2021. A general framework for resource constrained revenue management with demand learning and large action space. available at SSRN 3841273\/

    2021

  28. [36]

    Nambiar, Mila, David Simchi-Levi, He Wang. 2019. Dynamic learning and pricing with model misspecification. Management Science\/ 65 (11) 4980--5000

    2019

  29. [37]

    Petruzzi, Nicholas C., Maqbool Dada. 1999. Pricing and the newsvendor problem: A review with extensions. Operations research\/ 47 (2) 183--194

    1999

  30. [38]

    Saha, Aadirupa, Pierre Gaillard. 2022. Versatile dueling bandits: Best-of-both world analyses for learning from relative preferences. International Conference on Machine Learning (ICML)\/ . PMLR, 19011--19026

    2022

  31. [39]

    Saha, Aadirupa, Tomer Koren, Yishay Mansour. 2021. Adversarial dueling bandits. International Conference on Machine Learning (ICML)\/ . PMLR, 9235--9244

    2021

  32. [40]

    Sobel, Matthew J. 1981. Myopic solutions of M arkov decision processes and stochastic games. Operations Research\/ 29 (5) 995--1009

    1981

  33. [41]

    Sui, Yanan, Masrour Zoghi, Katja Hofmann, Yisong Yue. 2018. Advancements in dueling bandits. International Joint Conference on Artificial Intelligence (IJCAI)\/ . 5502--5510

    2018

  34. [42]

    Urvoy, Tanguy, Fabrice Clerot, Raphael F \'e raud, Sami Naamane. 2013. Generic exploration and k-armed voting bandits. International Conference on Machine Learning (ICML)\/ . PMLR, 91--99

    2013

  35. [43]

    Wang, Yining, Sivaraman Balakrishnan, Aarti Singh. 2019. Optimization of smooth functions with noisy observations: Local minimax rates. IEEE Transactions on Information Theory\/ 65 (11) 7350--7366

    2019

  36. [44]

    Wang, Yining, Boxiao Chen, David Simchi-Levi. 2021. Multi-modal dynamic pricing. Management Science\/ 67 (10) 6136--6152

    2021

  37. [45]

    Wang, Zizhuo, Shiming Deng, Yinyu Ye. 2014. Close the gaps: A learning-while-doing algorithm for single-product revenue management problems. Operations Research\/ 62 (2) 318--331

    2014

  38. [46]

    Yuan, Hao, Qi Luo, Cong Shi. 2021. Marrying stochastic gradient descent with bandits: Learning algorithms for inventory systems with fixed costs. Management Science\/ 67 (10) 6089--6115

    2021

  39. [47]

    Yue, Yisong, Josef Broder, Robert Kleinberg, Thorsten Joachims. 2012. The k-armed dueling bandits problem. Journal of Computer and System Sciences\/ 78 (5) 1538--1556

    2012

  40. [48]

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