Title: A Novel General Framework for Sharp Lower Bounds in Succinct Stochastic Bandits
Venue: NeurIPS 2025
Accepted: yes
Keywords: Bandits; Lower bounds; Sparsity; Low rankness
Abstract: Many online learning applications adopt the stochastic bandit problem with a linear reward model, where the unknown parameter exhibits a succinct structure. We study minimax regret lower bounds which allow to know whether more efficient algorithms can be proposed. We introduce a general definition of succinctness and propose a novel framework for constructing minimax regret lower bounds based on an information-regret trade-off. When applied to entry-sparse vectors, our framework sharpens a recent lower bound by (Hao et al, NeurIPS 2020). We further apply our framework to derive novel results. To the best of our knowledge, we provide the first lower bounds for the group-sparse and low-rank matrix settings.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents an algebraic approach to defining *succinctness* and constructs a framework for the linear bandit problem with a succinct parameter, which encompasses entry-sparse vectors, low-rank matrices, and group-sparse matrices. The paper provides a regret lower bound under this framework, and the resulting lower bounds for each application are either new or improved.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n\n1. The paper identifies core and common properties of three distinct problems (entry-sparsity, low-rank, and group-sparsity) and conducts the analysis under minimal assumptions. This approach is indeed novel and general. I believe it sets a strong example of how theoretical approach can unify multiple problem classes.\n\n2. The derived lower bounds improve upon previous ones, demonstrating the strength of the analysis.\n\n3. The introduction of the $Q$ and $R$ norms in the framework provides insight into which norm should be used in the boundedness assumptions for individual problem settings.\n\n\nWeaknesses\n\n1. The final step of the proof of the main theorem, as well as the proofs of the corollaries, is somewhat difficult to follow. For instance, it took me some time to understand that the phrase “we obtain the lower bound $\\min( \\frac{ns\\varepsilon}{24}\\exp(-2\\varepsilon [ \\frac{n}{p}+ \\frac{h}{q}]), \\frac{C_0}{3}h )$” in line 237 means that the worst-case regret is lower bounded by this expression for *any* $h$, as the intermediate steps are largely omitted. In the proofs of the corollaries, I would appreciate a bit more elaboration on how the bounds for $C_{\\min}(\\mathcal{H})$, $p$, and $q$ are obtained. While I understand it is mostly tedious calculation, I believe it would be helpful for readers if the authors briefly indicated what exactly needs to be computed.\n\n2. The final step of the proof of Theorem 3.8 introduces certain conditions on $n$, for example, $9s^2p \\le C_0^2 n$ in line 240, which are not mentioned in the theorem statement. Including these assumptions explicitly would improve clarity.\n\n3. The lower bound of $\\Omega(\\sqrt{sdn})$ from [11] is mentioned in the related work section but is omitted from Table 1, which could be slightly misleading.\n\nQUESTIONS:\n1. In Corollary 4.2, it seems like it should be $Q(\\cdot) = || \\cdot ||_ {2, \\infty}$ and $R(\\cdot) = || \\cdot ||_ {2, 1}$, which correspond to the maximum and the sum of $\\ell_ 2$ norms of the rows, respectively. Could the authors confirm this, or clarify the exact definitions of the matrix norms in the paper?\n\n2. I am curious about where the term *succinct* comes from. Is it a concept newly introduced in this paper, or is it an existing one? If it is the former, I suggest noting this in the abstract, as facing the word *succinct* before its description adds slight confusion.\n\n\n**Minor comments (Typos)**\n\nI could not find the definition of $\\Delta_{\\mathcal{X}}$, which appears for the first time in line 210. I could easily guess its definition, but I believe the authors would prefer to have it explicitly defined.  \nIn line 224, should it not be $1/2$ instead of $1/4$?  \nIn line 258, should it not be $C_{\\min}(\\mathcal{A}) > 0$ instead of $\\ge 0$, as the latter is always true?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 4, "ORIGINALITY": 3, "DATE": "2025-06-29"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a novel “template” or framework to prove lower bounds on bandit problems with a so-called “succinct” structure (e.g. sparsity, defined in section 3). Following instantiations of their methods, the authors prove sharper or new lower bounds on Sparse bandits, Group-Sparse bandits and Low-Rank bandits.\n\nSTRENGTHS AND WEAKNESSES:\n### Strengths:\n\n* The technical contribution is strong: the paper provides a clear and relatively novel method to prove lower bounds\n* Results are significant and may have major impact on succinct bandits research because the lower bounds that are proved are mainly not matched by existing upper bounds\n\n### Weaknesses: \n\n* Perhaps some additional citations could be provided, in particular in section 2.2 (KL lower bound on l.112 “or other options”). For now this section gives the feeling that this general framework wasn’t known before in the bandit literature, though similar ideas are discussed in many papers and gathered nicely in Lattimore &Szepesvari (2020). In general, I would have liked to understand better what was done wrong before and that you fixed. Perhaps this is written somewhere, or understandable in between the line and I missed it but I felt like I kept asking myself “What is different from known frameworks”? \n* Comments and discussions are lacking\n* Some vocabulary in Section 3 is a bit confusing and not always easy to connect with the bandit problems. Generally Section 3 is the hardest section to follow because it tries to be as general as possible in order to instantiate results into applications afterwards. It makes sense as a presentation choice, but I find that I often prefer to anchor my thoughts into a concrete example so I found myself reading sections 4 and 3 back and forth and trying to connect the dots. I think it’s a minor comment and I will adjust my judgement according to other reviewers’ opinions. Perhaps a way to help the reader would be to use sparse linear bandits as a running example?\n\nQUESTIONS:\n* I guess your framework applies to K-armed bandits as well since it’s a sub-case of the linear bandits. What are known lower bounds for the S-sparse K-armed bandits? Is your technique also improving that? \n* Your results seems to highlight gaps between lower and upper bounds, either in the C_min dependency or in the other problem quantities. All sections 4.1, 4.2, 4.3 are purely technical and do not discuss the implications of your results. Can you comment on the gaps you opened and whether you believe the lower bounds could be attainable? or perhaps what may cause sub-optimality in the upper-bounds?\n\nLIMITATIONS:\nOverall, this is a strong theory paper that has many contributions and few limitations. Given that there is a bit more space on page 9, I would like the authors to discuss a bit the implications of their results (see questions).", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 4, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nA new model of \"succinct\" bandits is introduced that generalizes some previously studied models. Their model of succinctness relaxes the notion of orthonormal sets in a normed vector space. This allows the lower bound to apply to a variety of settings. For instance, when the 1-succinct points are 1-sparse vectors, s-succinct vectors correspond to k-sparse vectors, and so on. The authors apply their technique to a few settings establishing the first lower bounds for low rank and \"group sparse\" matrix bandits, and improving the lower bound on entry-sparse bandits.\n\nI think of this as similar in spirit to the theory of matroids for greedy algorithms, where a specific structure in the instance guarantees approximation guarantees.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\nThe authors provide a novel generalization that yields to lower bounds for bandit problems that have additional \"sparsity\" structure. This seems like a fruitful generalization, and beside the possibility of resulting in further lower bounds in other settings, it seems to extract the notion of sparsity relevant for bandit algorithms.\n\nWeakness:\nThe paper is very dense.\n1. Proving that the instances in Section 4 actually satisfy Assumption 3.7. For instance, line 273, 288 just say \"We can verify that ...\".\n2. Explaining in terms of sparsity what Assumptions 3.7 (ii) and (iii) mean. It seems like (ii) imposes a restriction on what is allowed to be denoted \"succinct\" vectors. Is $\\max \\Phi$ related to $C_min$ used later? If we just take $p=q=1$, then we already get a lower bound that differs mostly in $s$ terms, so is this specific useful in distinguishing this work from prior work?\n3. Where does the construction in line 203 of $\\Theta$ come from? Again I think that a concrete example with sparsity would make this easier to read.\n\nOverall, I think this paper could be a little longer, with some proofs explained in more detail. The key to using this framework elsewhere is understanding Assumption 3.7, and I dont think it is clear how to even verify these assumptions.\n\nQUESTIONS:\n1. A direct comparison between succinctness and sparsity would be useful in Section 3.1. It seems like succinctness itself doesnt imply much structure and is just a set that could be specified in a problem, while the notion of a support is similar to that of orthonormality/independence, and s-succinct vectors _do_ have implied some structure. I think this is confusing, but there must be some reason for not just writing s-succinct vectors as decomposing into 1-succinct vectors, but rather as decomposing into a support of 1-succinct vectors.\n2. What is the purpose of the first part of Assumption 3.7 (i)? The lower bound just needs the existence of an action space, so I dont see why you need to impose a restriction on it as an assumption (that is, why do you need to say that $C_0 < 0$? If $C_0>0$ for some action set, then maybe the lower bound is just not valid).\n3. What do these assumptions mean for the entry sparse bandit? Specifically, does Assumption 3.7 say that the ground truth $\\theta$ is $k$-succinct? If so, what is the significance of the parameter $s$? How does Theorem 3.8 improve on the entry sparse lower bound? Is it just a better lower bound instance (so there is now a proof for the improved lower bound purely in terms of sparsity)? While this is not a necessary component of the narrative, it would be helpful in appreciating the technical merit of the paper.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 1, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-03"}

comments: SUMMARY:
This paper proposes a general framework for deriving regret lower bounds in linear bandit problems where the loss vectors exhibit some form of structural simplicity. Concrete examples of such structure include sparse vectors, low-rank matrices, and group-sparse matrices. The framework aims to encompass these cases in a unified manner. In particular, for the sparse linear bandit setting, the paper derives an improved lower bound compared to existing results.

STRENGTHS AND WEAKNESSES:
Strengths:

The problem is formulated with technical precision, and the theoretical results are well structured.

The presentation emphasizes generality and extensibility, suggesting that the framework could potentially be applied to other problem settings.

Weaknesses:

The intuitive explanation of the model definitions and technical results is somewhat limited. For example, in Section 3.1, while several notations and definitions are introduced, they are mostly listed without accompanying interpretations or illustrative examples. Providing such context would improve readability.

It would be helpful to include a more detailed discussion of how tight the proposed lower bounds are in general. For instance, regarding Theorem 3.8, can the authors provide concrete examples where the bound becomes tight (or provably loose)? Would it be possible to state sufficient conditions under which the lower bound is tight?

QUESTIONS:
Regarding Theorem 3.8, can the authors provide concrete examples where the bound becomes tight (or provably loose)? Would it be possible to state sufficient conditions under which the lower bound is tight?

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