Title: Instance-Dependent Regret Bounds for Nonstochastic Linear Partial Monitoring
Venue: NeurIPS 2025
Accepted: yes
Keywords: Partial monitoring

exploration-by-optimization

linear bandits

online learning
Abstract: In contrast to the classic formulation of partial monitoring, linear partial monitoring can model infinite outcome spaces, while imposing a linear structure on both the losses and the observations. This setting can be viewed as a generalization of linear bandits where loss and feedback are decoupled in a flexible manner. In this work, we address a nonstochastic (adversarial), finite-actions version of the problem through a simple instance of the exploration-by-optimization method that is amenable to efficient implementation. We derive regret bounds that depend on the game structure in a more transparent manner than previous theoretical guarantees for this paradigm. Our bounds feature instance-specific quantities that reflect the degree of alignment between observations and losses, and resemble known guarantees in the stochastic setting. Notably, they achieve the standard $\sqrt{T}$ rate in easy (locally observable) games and $T^{2/3}$ in hard (globally observable) games, where $T$ is the time horizon. We instantiate these bounds in a selection of old and new partial information settings subsumed by this model, and illustrate that the achieved dependence on the game structure can be tight in interesting cases.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies the problem of adversarial linear partial monitoring with finitely many actions. This setting differs from both stochastic linear partial monitoring and finite partial monitoring, which have been studied previously. The authors show that the minimax regret rates are $O(\\sqrt{T})$ and $O(T^{2/3})$ (up to logarithmic factors) for locally observable and globally observable games, respectively. The upper bounds are achieved by combining the \"exploration by optimization\" framework of Lattimore and Szepesvári with the linear structure of the problem, resulting in a computationally efficient algorithm. The regret bounds are also specialized to various settings, such as the full-information setting and the linear bandit, recovering existing optimal results.\n\nSTRENGTHS AND WEAKNESSES:\n# Strengths\n\nI have read the main paper and skimmed through Appendices A, C, and D. The paper is clearly written, and the related works are adequately addressed. The theoretical results I checked are rigorously supported.\n\n---\n### Novelty and Significance\n\nThe adversarial linear partial monitoring is a fundamental problem in online learning, yet it has not been studied thoroughly in the literature. This paper provides a comprehensive characterization of the minimax regret rate when the action set is finite, which is a valuable contribution.\n\nAs far as I understand, the main technical contribution of the work is a computationally efficient algorithm (Algorithm 1), that cleverly leverages the linear structure within the exploration by optimization (ExO) framework. In contrast, the general ExO framework requires solving an infinite-dimensional optimization problem, which may be intractable.\n\n---\n\n# Weaknesses\n\n### The finite-action assumption\n\nThe paper assumes that there are only finitely many actions (Line 67 and 94). However, as far as I understand, the linear structure is typically used to address large or even continuous action spaces. Therefore, I am confused about the motivation and advantages of studying the finite-action linear setting. In particular,\n\n1. What is the motivation for studying the linear partial monitoring problem with *finite* actions? Are there any important applications that can be modeled by this problem?\n2. Besides computational efficiency, are there other key differences between the finite partial monitoring problem and the finite linear partial monitoring problem?\n3. What are the main challenges in extending the current results to the infinite-action case?\n\nI will raise my score if the above concerns are resolved.\n\n---\n\nThe following comments are minor.\n\n### Clarity\n\nGenerally speaking, this paper is clearly written. However, due to the complexity of the notation, it took me some time to understand the results. Below are some comments that could help improve clarity.\n\n1. In Lemma 5, the loss estimate is defined as $\\hat{y}_t = (H^\\top-\\mathbf{1}_dx_t^\\top)Q(p_t)^\\dagger M\\_{A\\_t}\\phi_t$. From my understanding, this is an unbiased estimate of the true loss $y_t$, i.e., $\\mathbb{E}_t[\\hat{y}_t]=y_t$. The definition can be derived by using least squares similar to [1, Chapter 27]. To improve clarity, an explanation of how this estimate is constructed would be helpful.\n2. Due to the complex notation, it may be difficult for readers to specialize the results (Theorem 10 and Theorem 11) to different settings. I would suggest including derivations of some of the special cases, such as full information and linear bandits, in the appendix.\n3. It is stated in Line 343-347 that \"Overall, our bounds seem more versatile [...].\" I couldn't see the reason for this claim. Could you elaborate more on the advantages of your bounds?\n\n---\n### Typos\n1. In Line 299, under the summation in the definition of $\\hat{p}$, should it be $a\\in\\mathcal{A}^\\ast$?\n2. In Line 354, I believe it should refer to Theorem 10 rather than Theorem 11.\n3. In Line 349-351, it is mentioned that in Section 4, we choose $\\hat{p}=q$. However, this does not seem to be the case (see Line 299). I assume there are some typos here.\n\n---\n\nReferences\n1. Lattimore and Szepesvári. Bandit Algorithms. 2020.\n\nQUESTIONS:\nNo further questions.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-26"}

comments: SUMMARY:
This paper proves an instance-dependent upper bound for the regret for non-stochastic adversarial linear partial monitoring which covers possibly infinite outcome spaces. The problem-independent lower regret bound and upper regret bound is characterized by the global or local observability.

STRENGTHS AND WEAKNESSES:
**Strengths**

1. The proposed non-stochastic linear partial monitoring model covers a wide range of settings including linear bandits, graph bandits and ill-conditioned bandits, which contributes in many ways to the literature. The provided examples are helpful to understand the coverage of the partial monitoring model.

2. The proposed algorithm is simple yet principled to obtain the proven instance-dependent regret bound. Explanations on the terms that appear in the regret bound provides understanding of the procedure.

**Weaknesses**

1. The paper is quite focused on technical description and the novelty or significance of the results are not highlighted well. I suggest including comparison with other works especially for Lemma 4,5 and Theorem 11. Including the literal meaning of definitions of $\Lambda, H$ and $B$.

QUESTIONS:
1. Could authors explain the definition of $\Lambda, H$ and $B$?

2. Could authors provide an overview of the analysis on locally observable and globally observable cases, before going into the proof, respectively?

3. What is the novel technique developed in this paper? What are the main challenges to derive the results in locally and globally observable cases?

LIMITATIONS:
Yes.
TITLE: Official Review
CLARITY: 2
IMPACT: 3
ORIGINALITY: 2
DATE: 2025-07-03

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies the setting of adversarial linear partial monitoring. They provide an implementable algorithm based on the “Exploration-by-optimization” method that obtains optimal regret bounds for locally and globally observable games respectively. Moreover, their regret bounds scale with instance-dependent quantities comparable to the one obtained in the stochastic version of this problem.\n\nSTRENGTHS AND WEAKNESSES:\nThe paper is well written and easy to follow. The main ideas behind the algorithm are clearly explained and the mathematical derivations are well presented and shed some more lights on the results. The dependencies in the regret bounds are clear and the instantiation of the regret bound on different setting of interest shows that this method recovers optimal or almost optimal rates in a variety of settings.\n\nAs I’m not as familiar as the authors with the litterature on that topic, I found the comparison with existing work a bit hard to follow, in particular, what are the known upper and lower bounds for the regret in the stochastic and adversarial setting, are the proposed algorithms implementable.\nOne of the main properties of the proposed algorithm is that it is implementable, it would have been good to either have some experiments to illustrate that or at least a clear analysis of the computational complexity of the algorithm in one of the settings of interest.\n\nQUESTIONS:\n- It is stated that the setting studied in [30] is more general than the one of the paper because the observation is taken as arbitrary. In particular, the setting of the paper does not subsume the setting of finite partial monitoring. Could you explain why the restriction is necessary for the methods of this paper to work ?\n- The concurrent work of IDS for Linear Partial Monitoring doesn’t require a finite action space for the regret bounds to hold. Is it possible to extend the methods of this paper beyond the finite action space case ?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper considers the adversarial linear partial monitoring setting. The authors provide an efficient algorithm for this setting, based on the principle of exploration-by-optimization (EXO). The authors show that the regret bounds of the proposed algorithm are near-optimal in the globally and locally observable games. \n\nPrior works, e.g. Lattimore and Szepesvari [2020], only provided efficient and optimal algorithms for the finite partial monitoring setting (the current paper extends this to the linear setting). Existing results on linear partial monitoring by Lattimore and Gyorgy [2021] involved an inefficient version of EXO.\n\nSTRENGTHS AND WEAKNESSES:\nStrenghts:\n- The paper provides a unified algorithm for the linear partial monitoring setting, and shows that this generalizes many existing setting actively researched.\n- The paper is written in a mathematically rigorous way. \n\nWeaknesses: \n- The main concern I have with paper is with the writting. The paper is rather dense with definitions, and authors do not always provide sufficient intuition. If I were to rewrite the paper, I would perhaps include less results in the main body, but then provide more intuition and proof sketch in the main. \n- The authors also do not spend enough time highighting the deltas to prior work. Sure, the paper provides the first efficient algorithm with near-optimal regret for the linear partial monitoring setting, and this setting encopasses previous settings such as full information, linear bandit, graph feedback, etc. But are there existing efficient algorithm for these setting not based on EXO? Also, the authors should also highlight any novelties in the analysis of the algorithm, and where they were able to overcome prior hurdles. \n\nSome other comments:\n- The notation in the first paragraph is not consistent with the notation in the rest of the paper. \n- Line 20 says: \"All of these components are known to the learner,\", which suggestes L is know, but then L is unobserved. \n- Line 136 claims that (3) implies (2) for $C = \\mathbb{R}^d$. Is this true? From its definition, the set $\\mathcal{P}(\\mathbb{R}^d)$ should only contain points in $\\mathrm{ext}(\\mathrm{co}(\\mathcal{X}))$, whereas (2) invloves points for all $a,b\\in \\mathcal{A}$.\n- Does the set of neighboors $N_{\\mathcal{G}}(a)$ include the set itself?\n\nQUESTIONS:\nSee questions in the strengths and weaknesses.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-05"}