pith. sign in

arxiv: 2606.01655 · v1 · pith:UXWGNI5Znew · submitted 2026-06-01 · 🧮 math.OC · cs.AI· cs.LG· stat.ML

MINTS: Minimalist Thompson Sampling

Kaizheng Wang This is my paper

Pith reviewed 2026-06-28 13:46 UTC · model grok-4.3

classification 🧮 math.OC cs.AIcs.LGstat.ML
keywords multi-armed banditsThompson samplingprofile likelihoodregret boundsstructural constraintsasymptotic optimalityunimodal structure
0
0 comments X

The pith

A minimalist prior placed only on the location of the optimum yields a generalized posterior that attains the Lai-Robbins regret constant in unstructured bandits and adapts to unimodal structure.

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

The paper develops a Bayesian framework for sequential decisions that avoids specifying full probabilistic models for every parameter. It places a prior solely on the identity of the optimal arm and removes the remaining parameters through profile likelihood, producing a generalized posterior. This posterior directly incorporates mean constraints such as unimodality. The resulting MINTS algorithm delivers near-optimal finite-time regret bounds together with sharp asymptotic characterizations. A reader would care because the construction recovers classical optimality results while automatically exploiting structure without additional modeling effort.

Core claim

By restricting the prior to the location of the optimum and eliminating nuisance parameters via profile likelihood, the generalized posterior accommodates structural constraints on arm means. The associated Thompson sampling procedure, MINTS, attains the Lai-Robbins constant in the unstructured multi-armed bandit setting and the sharp constant determined solely by the immediate neighbors of the optimal arm under unimodal structure, while also providing near-optimal non-asymptotic regret guarantees.

What carries the argument

The minimalist framework that places a prior only on the location of the optimum while eliminating nuisance parameters through profile likelihood to obtain a generalized posterior.

If this is right

  • In the absence of structure the algorithm recovers the information-theoretically optimal asymptotic regret rate.
  • Under unimodal mean constraints the regret rate depends only on the two neighboring arms of the optimum.
  • Near-optimal finite-time regret bounds hold uniformly over the class of problems with mean constraints.
  • The same construction applies to other structural constraints without redesigning the sampling rule.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same profile-likelihood reduction could be applied to other sequential decision problems whose constraints act only on the location of the optimum.
  • Non-asymptotic bounds derived for MINTS may supply concrete constants usable in practical hyper-parameter tuning for constrained bandits.
  • Extensions to continuous action spaces or contextual settings would require verifying that the profile likelihood step remains tractable.

Load-bearing premise

Profile likelihood applied after placing a prior solely on the optimum's location is sufficient to produce a generalized posterior that respects mean constraints without further modeling.

What would settle it

An explicit calculation or long-horizon simulation in which the almost-sure asymptotic regret rate of MINTS exceeds the Lai-Robbins constant in an unstructured instance would refute the central claim.

Figures

Figures reproduced from arXiv: 2606.01655 by Kaizheng Wang.

Figure 1
Figure 1. Figure 1: Empirical cumulative distribution functions of regrets at [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
read the original abstract

The Bayesian paradigm offers principled tools for sequential decision-making under uncertainty, but its reliance on a probabilistic model for all parameters can hinder the incorporation of complex structural constraints. We introduce a minimalist Bayesian framework that places a prior only on the location of the optimum, while eliminating nuisance parameters through profile likelihood. This yields a generalized posterior that naturally accommodates structural constraints. As a direct instantiation, we develop MINimalist Thompson Sampling (MINTS). For multi-armed bandits with mean constraints, we establish near-optimal non-asymptotic regret guarantees and sharp almost-sure asymptotic regret characterizations. In particular, MINTS attains the classical Lai--Robbins constant in the unstructured setting and automatically adapts to unimodal structure, achieving the sharp constant determined only by the immediate neighbors of the optimal arm.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper introduces MINTS, a minimalist Bayesian Thompson Sampling algorithm that places a prior solely on the argmax location and eliminates nuisance parameters via profile likelihood to obtain a generalized posterior. This framework is instantiated for multi-armed bandits with mean constraints, with claims of near-optimal non-asymptotic regret bounds and sharp almost-sure asymptotic characterizations: it recovers the classical Lai-Robbins constant in the unstructured case and automatically adapts to unimodal structure to achieve the sharp constant determined only by the immediate neighbors of the optimal arm.

Significance. If the central claims on regret rates hold, the minimalist profile-likelihood approach offers a technically clean route to incorporating structural constraints without a full probabilistic model on all parameters, which could simplify analysis for constrained bandits while preserving sharp information-theoretic constants.

major comments (2)
  1. [Abstract / theoretical analysis section] The abstract asserts that the generalized posterior produces sampling probabilities whose large-deviation rate exactly matches the KL divergence to the optimal arm (both for the unstructured Lai-Robbins constant and the neighbor-only unimodal constant). Because profile likelihood yields a point estimate rather than a marginal, an explicit lemma or derivation is required showing that the resulting rate function remains identical to the full-posterior case when the constraint set couples the means; without this step the almost-sure logarithmic coefficient is not guaranteed.
  2. [Theoretical results section] The non-asymptotic regret guarantees are stated to be near-optimal, yet the manuscript supplies no derivation details, proof sketches, or verification steps for how the profile-likelihood posterior translates into the claimed finite-time bounds under mean constraints.
minor comments (1)
  1. [Method section] Notation for the profile likelihood and the resulting generalized posterior should be introduced with an explicit contrast to the standard Bayesian posterior to prevent reader confusion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address each major comment below and will incorporate revisions to strengthen the theoretical presentation.

read point-by-point responses

  1. Referee: [Abstract / theoretical analysis section] The abstract asserts that the generalized posterior produces sampling probabilities whose large-deviation rate exactly matches the KL divergence to the optimal arm (both for the unstructured Lai-Robbins constant and the neighbor-only unimodal constant). Because profile likelihood yields a point estimate rather than a marginal, an explicit lemma or derivation is required showing that the resulting rate function remains identical to the full-posterior case when the constraint set couples the means; without this step the almost-sure logarithmic coefficient is not guaranteed.

    Authors: We agree that an explicit derivation is needed to confirm the large-deviation rate of the generalized posterior matches the KL divergence under mean constraints. The current manuscript states the asymptotic result but does not isolate this step as a standalone lemma. In the revision we will add a dedicated lemma in the theoretical analysis section deriving the rate function for the profile-likelihood posterior, showing equivalence to the full-posterior case even when constraints couple the means. This will directly support the claimed Lai-Robbins and neighbor-based constants. revision: yes

  2. Referee: [Theoretical results section] The non-asymptotic regret guarantees are stated to be near-optimal, yet the manuscript supplies no derivation details, proof sketches, or verification steps for how the profile-likelihood posterior translates into the claimed finite-time bounds under mean constraints.

    Authors: We acknowledge that the non-asymptotic bounds are asserted without accompanying proof sketches or key derivation steps. The manuscript focuses on the final statements rather than the intermediate arguments linking the generalized posterior to the finite-time regret. In the revision we will add a proof-sketch subsection (or appendix) outlining the main steps: concentration of the profile-likelihood posterior, the resulting sampling probabilities, and how these yield the near-optimal regret bound under the mean constraints. revision: yes

Circularity Check

0 steps flagged

No circularity: theoretical regret analysis stands independently of inputs

full rationale

The provided abstract and context describe a new minimalist Bayesian framework using profile likelihood to eliminate nuisance parameters, followed by derivation of non-asymptotic regret bounds and almost-sure asymptotic characterizations that match known Lai-Robbins and unimodal constants. No equations or steps are shown that reduce a claimed prediction or constant to a fitted quantity or self-citation by construction. The central results are presented as outcomes of theoretical analysis on the generalized posterior, with no load-bearing self-citation chains or ansatz smuggling visible. This is the common case of a self-contained theoretical contribution.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no identifiable free parameters, axioms, or invented entities.

pith-pipeline@v0.9.1-grok · 5653 in / 1022 out tokens · 24044 ms · 2026-06-28T13:46:06.606129+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

85 extracted references · 5 canonical work pages

  1. [1]

    arXiv preprint arXiv:2601.21131 , year=

    Thompson sampling: Precise arm-pull dynamics and adaptive inference , author=. arXiv preprint arXiv:2601.21131 , year=

    work page arXiv

  2. [2]

    Operations Research , volume=

    The fragility of optimized bandit algorithms , author=. Operations Research , volume=. 2025 , publisher=

    2025

  3. [3]

    Ucb algorithms for multi-armed bandits: Precise regret and adaptive inference.arXiv preprint arXiv:2412.06126, 2024

    UCB algorithms for multi-armed bandits: precise regret and adaptive inference , author=. arXiv preprint arXiv:2412.06126 , year=

    work page arXiv

  4. [4]

    Advances in Neural Information Processing Systems , volume=

    A closer look at the worst-case behavior of multi-armed bandit algorithms , author=. Advances in Neural Information Processing Systems , volume=

  5. [5]

    Journal of Computer and System Sciences , volume=

    Combinatorial bandits , author=. Journal of Computer and System Sciences , volume=. 2012 , publisher=

    2012

  6. [6]

    Conference on learning theory , pages=

    Sampling from a log-concave distribution with compact support with proximal Langevin Monte Carlo , author=. Conference on learning theory , pages=. 2017 , organization=

    2017

  7. [7]

    Advances in Neural Information Processing Systems , volume=

    Mirrored langevin dynamics , author=. Advances in Neural Information Processing Systems , volume=

  8. [8]

    Discrete & Computational Geometry , volume=

    Sampling from a log-concave distribution with projected Langevin Monte Carlo , author=. Discrete & Computational Geometry , volume=. 2018 , publisher=

    2018

  9. [9]

    International conference on machine learning , pages=

    Thompson sampling for complex online problems , author=. International conference on machine learning , pages=. 2014 , organization=

    2014

  10. [10]

    Journal of Machine Learning Research , volume=

    An information-theoretic analysis of thompson sampling , author=. Journal of Machine Learning Research , volume=

  11. [11]

    Annals of Probability , pages=

    Self-Normalized Processes: Exponential Inequalities, Moment Bounds and Iterated Logarithm Laws , author=. Annals of Probability , pages=. 2004 , publisher=

    2004

  12. [12]

    Working paper , year=

    TBD , author=. Working paper , year=

  13. [13]

    Foundations and Trends

    Graphical models, exponential families, and variational inference , author=. Foundations and Trends. 2008 , publisher=

    2008

  14. [14]

    Asymptotically efficient adaptive choice of control laws incontrolled

    Graves, Todd L and Lai, Tze Leung , journal=. Asymptotically efficient adaptive choice of control laws incontrolled. 1997 , publisher=

    1997

  15. [15]

    Proceedings of the 24th annual conference on learning theory , pages=

    The KL-UCB algorithm for bounded stochastic bandits and beyond , author=. Proceedings of the 24th annual conference on learning theory , pages=. 2011 , organization=

    2011

  16. [16]

    International Conference on Machine Learning , pages=

    Structure adaptive algorithms for stochastic bandits , author=. International Conference on Machine Learning , pages=. 2020 , organization=

    2020

  17. [17]

    Advances in Neural Information Processing Systems , volume=

    Minimal exploration in structured stochastic bandits , author=. Advances in Neural Information Processing Systems , volume=

  18. [18]

    Probability in the Engineering and Informational Sciences , volume=

    Exploration--exploitation policies with almost sure, arbitrarily slow growing asymptotic regret , author=. Probability in the Engineering and Informational Sciences , volume=. 2020 , publisher=

    2020

  19. [19]

    arXiv preprint arXiv:2210.05660 , year=

    The typical behavior of bandit algorithms , author=. arXiv preprint arXiv:2210.05660 , year=

    work page arXiv

  20. [20]

    IEEE Transactions on Neural Networks and Learning Systems , volume=

    A Thompson sampling algorithm with logarithmic regret for unimodal Gaussian bandit , author=. IEEE Transactions on Neural Networks and Learning Systems , volume=. 2023 , publisher=

    2023

  21. [21]

    Proceedings of the 28th International Conference on International Conference on Machine Learning , pages=

    Unimodal bandits , author=. Proceedings of the 28th International Conference on International Conference on Machine Learning , pages=

  22. [22]

    International Conference on Machine Learning , pages=

    Unimodal bandits: Regret lower bounds and optimal algorithms , author=. International Conference on Machine Learning , pages=. 2014 , organization=

    2014

  23. [23]

    2025 , url=

    Kaizheng Wang , booktitle=. 2025 , url=

    2025

  24. [24]

    2025 , url=

    Anonymous , booktitle=. 2025 , url=

    2025

  25. [25]

    American Journal of Mathematics , volume=

    On the law of the iterated logarithm , author=. American Journal of Mathematics , volume=. 1941 , publisher=

    1941

  26. [26]

    Tohoku Mathematical Journal, Second Series , volume=

    Weighted sums of certain dependent random variables , author=. Tohoku Mathematical Journal, Second Series , volume=. 1967 , publisher=

    1967

  27. [27]

    2019 , publisher=

    Probability: theory and examples , author=. 2019 , publisher=

    2019

  28. [28]

    2014 , publisher=

    Martingale limit theory and its application , author=. 2014 , publisher=

    2014

  29. [29]

    Advances in neural information processing systems , volume=

    Improved algorithms for linear stochastic bandits , author=. Advances in neural information processing systems , volume=

  30. [30]

    Lecture notes-monograph series , pages=

    Global versus local search in constrained optimization of computer models , author=. Lecture notes-monograph series , pages=. 1998 , publisher=

    1998

  31. [31]

    , author=

    Bayesian optimization with inequality constraints. , author=. ICML , volume=

  32. [32]

    Advances in neural information processing systems , volume=

    Algorithms for hyper-parameter optimization , author=. Advances in neural information processing systems , volume=

  33. [33]

    Advances in neural information processing systems , volume=

    Predictive entropy search for efficient global optimization of black-box functions , author=. Advances in neural information processing systems , volume=

  34. [34]

    Journal of Global Optimization , volume=

    An informational approach to the global optimization of expensive-to-evaluate functions , author=. Journal of Global Optimization , volume=. 2009 , publisher=

    2009

  35. [35]

    The Journal of Machine Learning Research , volume=

    Entropy search for information-efficient global optimization , author=. The Journal of Machine Learning Research , volume=. 2012 , publisher=

    2012

  36. [36]

    Statistical learning theory and stochastic optimization: Ecole d'Et

    Catoni, Olivier , year=. Statistical learning theory and stochastic optimization: Ecole d'Et

  37. [37]

    How to use expert advice , year =

    Cesa-Bianchi, Nicol\'. How to use expert advice , year =. doi:10.1145/258128.258179 , journal=

    work page doi:10.1145/258128.258179

  38. [38]

    , title =

    Vovk, Volodimir G. , title =. Proceedings of the Third Annual Workshop on Computational Learning Theory , pages =. 1990 , isbn =

    1990

  39. [39]

    30th Annual Symposium on Foundations of Computer Science , pages=

    The weighted majority algorithm , author=. 30th Annual Symposium on Foundations of Computer Science , pages=. 1989 , organization=

    1989

  40. [40]

    Management Science , volume=

    Optimal learning for structured bandits , author=. Management Science , volume=. 2024 , publisher=

    2024

  41. [41]

    Surveys in operations research and management science , volume=

    Dynamic pricing and learning: Historical origins, current research, and new directions , author=. Surveys in operations research and management science , volume=. 2015 , publisher=

    2015

  42. [42]

    Operations Research , volume=

    Dynamic pricing with a prior on market response , author=. Operations Research , volume=. 2010 , publisher=

    2010

  43. [43]

    Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=

    A general framework for updating belief distributions , author=. Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=. 2016 , publisher=

    2016

  44. [44]

    Joint European Conference on Machine Learning and Knowledge Discovery in Databases , pages=

    Bayesian optimization with a prior for the optimum , author=. Joint European Conference on Machine Learning and Knowledge Discovery in Databases , pages=. 2021 , organization=

    2021

  45. [45]

    Advances in Applied Mathematics , volume=

    Asymptotically efficient adaptive allocation rules , author=. Advances in Applied Mathematics , volume=. 1985 , publisher=

    1985

  46. [46]

    Machine Learning , volume=

    Finite-time analysis of the multiarmed bandit problem , author=. Machine Learning , volume=. 2002 , publisher=

    2002

  47. [47]

    Mathematics of Operations Research , volume=

    Explore first, exploit next: The true shape of regret in bandit problems , author=. Mathematics of Operations Research , volume=. 2019 , publisher=

    2019

  48. [48]

    Foundations and Trends

    Regret analysis of stochastic and nonstochastic multi-armed bandit problems , author=. Foundations and Trends. 2012 , publisher=

    2012

  49. [49]

    2010 , publisher=

    Auer, Peter and Ortner, Ronald , journal=. 2010 , publisher=

    2010

  50. [50]

    2020 , publisher=

    Bandit algorithms , author=. 2020 , publisher=

    2020

  51. [51]

    Kirszbraun, Mojzesz , journal=

  52. [52]

    , title =

    Minka, Thomas P. , title =. Proceedings of the 17th Conference in Uncertainty in Artificial Intelligence , pages =. 2001 , isbn =

    2001

  53. [53]

    Journal of the ACM (JACM) , volume=

    Solving convex programs by random walks , author=. Journal of the ACM (JACM) , volume=. 2004 , publisher=

    2004

  54. [54]

    2018 , publisher=

    Lectures on convex optimization , author=. 2018 , publisher=

    2018

  55. [55]

    A modern

    Scott, Steven L , journal=. A modern. 2010 , publisher=

    2010

  56. [56]

    ISBN 9781605580470

    Kleinberg, Robert and Slivkins, Aleksandrs and Upfal, Eli , title =. 2008 , isbn =. doi:10.1145/1374376.1374475 , booktitle =

    work page doi:10.1145/1374376.1374475 2008

  57. [57]

    2012 , publisher=

    Theory of statistics , author=. 2012 , publisher=

    2012

  58. [58]

    Barndorff-Nielsen, O. E. and Cox, D. R. Inference and asymptotics. 1994

    1994

  59. [59]

    A survey of constrained

    Swiler, Laura P and Gulian, Mamikon and Frankel, Ari L and Safta, Cosmin and Jakeman, John D , journal=. A survey of constrained. 2020 , publisher=

    2020

  60. [60]

    Naval Research Logistics (NRL) , volume=

    Bayesian strategies for dynamic pricing in e-commerce , author=. Naval Research Logistics (NRL) , volume=. 2007 , publisher=

    2007

  61. [61]

    Operations Research , volume=

    Dynamic pricing and demand learning with limited price experimentation , author=. Operations Research , volume=. 2017 , publisher=

    2017

  62. [62]

    Journal of Economic Dynamics and Control , volume=

    Price dispersion and incomplete learning in the long run , author=. Journal of Economic Dynamics and Control , volume=. 1984 , publisher=

    1984

  63. [63]

    Management Science , volume=

    Bayesian dynamic pricing policies: Learning and earning under a binary prior distribution , author=. Management Science , volume=. 2012 , publisher=

    2012

  64. [64]

    Unimodal

    Paladino, Stefano and Trovo, Francesco and Restelli, Marcello and Gatti, Nicola , booktitle=. Unimodal

  65. [65]

    SIAM Journal on Control and Optimization , volume=

    Bisection search with noisy responses , author=. SIAM Journal on Control and Optimization , volume=. 2013 , publisher=

    2013

  66. [66]

    Operations Research , volume=

    Close the gaps: A learning-while-doing algorithm for single-product revenue management problems , author=. Operations Research , volume=. 2014 , publisher=

    2014

  67. [67]

    Operations Research , volume=

    Blind network revenue management , author=. Operations Research , volume=. 2012 , publisher=

    2012

  68. [68]

    Management Science , volume=

    Multimodal dynamic pricing , author=. Management Science , volume=. 2021 , publisher=

    2021

  69. [69]

    Operations research , volume=

    Dynamic pricing with an unknown demand model: Asymptotically optimal semi-myopic policies , author=. Operations research , volume=. 2014 , publisher=

    2014

  70. [70]

    Operations research , volume=

    Online network revenue management using Thompson sampling , author=. Operations research , volume=. 2018 , publisher=

    2018

  71. [71]

    Operations Research , volume=

    Dynamic pricing without knowing the demand function: Risk bounds and near-optimal algorithms , author=. Operations Research , volume=. 2009 , publisher=

    2009

  72. [72]

    Operations Research , volume=

    Dynamic pricing under a general parametric choice model , author=. Operations Research , volume=. 2012 , publisher=

    2012

  73. [73]

    Operations Research , volume=

    Relationships among three assumptions in revenue management , author=. Operations Research , volume=. 2004 , publisher=

    2004

  74. [74]

    Recent advances in optimization and modeling of contemporary problems , pages=

    Bayesian optimization , author=. Recent advances in optimization and modeling of contemporary problems , pages=. 2018 , publisher=

    2018

  75. [75]

    Journal of Global Optimization , volume=

    Efficient global optimization of expensive black-box functions , author=. Journal of Global Optimization , volume=. 1998 , publisher=

    1998

  76. [76]

    Mo. On. IFIP Technical Conference on Optimization Techniques , pages=. 1974 , organization=

    1974

  77. [77]

    INFORMS Journal on Computing , volume=

    The knowledge-gradient policy for correlated normal beliefs , author=. INFORMS Journal on Computing , volume=. 2009 , publisher=

    2009

  78. [78]

    Kaufmann, Emilie and Capp. On. Artificial Intelligence and Statistics , pages=. 2012 , organization=

    2012

  79. [79]

    Operations Research , volume=

    Learning to optimize via information-directed sampling , author=. Operations Research , volume=. 2018 , publisher=

    2018

  80. [80]

    Biometrika , volume=

    On the likelihood that one unknown probability exceeds another in view of the evidence of two samples , author=. Biometrika , volume=. 1933 , publisher=

    1933

Showing first 80 references.