Title: Uniform Wrappers: Bridging Concave to Quadratizable Functions in Online Optimization
Venue: NeurIPS 2025
Accepted: yes
Keywords: DR-submodular Optimization

Convex Optimization
Abstract: This paper presents novel contributions to the field of online optimization, particularly focusing on the adaptation of algorithms from concave optimization to more challenging classes of functions.
Key contributions include the introduction of uniform wrappers, a class of meta-algorithms that could be used for algorithmic conversions such as converting algorithms for convex optimization into those for quadratizable optimization.
Moreover, we propose a guideline that, given a base algorithm $\mathcal{A}$ for concave optimization and a uniform wrapper $\mathcal{W}$, describes how to convert a proof of the regret bound of $\mathcal{A}$ in the concave setting into a proof of the regret bound of $\mathcal{W}(\mathcal{A})$ for quadratizable setting.
Through this framework, the paper demonstrates improved regret guarantees for various classes of DR-submodular functions under zeroth-order feedback. Furthermore, the paper extends zeroth-order online algorithms to bandit feedback and offline counterparts, achieving notable improvements in regret/sample complexity compared to existing approaches.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces uniform wrappers, a meta-algorithmic framework that systematically converts online convex optimization (OCO) algorithms into algorithms for quadratizable function classes (a generalization of concave/DR-submodular functions). The core innovation lies in providing provable regret guarantees through automated transformation of both algorithms and their regret proofs. By applying this framework to zeroth-order base algorithms (e.g., ZO-FTRL), the work achieves improved regret bounds for zeroth-order feedback and bandit feedback—surpassing prior state-of-the-art results. The framework can recover or obtain superior regret for DR-submodular maximizations in different setting.\n\nSTRENGTHS AND WEAKNESSES:\nThe article's content is very dense, leading to a somewhat cluttered presentation. I believe the core contribution is that any OCO algorithm can be converted into an algorithm for quadratizable function classes via the proposed uniform wrappers, and that the regret bound proof for the original OCO algorithm can be transformed into a regret bound proof for the new algorithm. If I understand correctly, this would be a very nice result. The remaining parts essentially serve as examples demonstrating the generality of this core conclusion. Therefore, I recommend that the authors significantly enhance the introduction by adding a practical guide outlining the conversion process and proof transformation. Specifically: Assuming a researcher in a related field proposes a new OCO algorithm, how should they use the authors' framework to derive a online quadratizable optimization algorithm?\n\nI'd be happy to reevaluate if the authors could alleviate my confusions as shown in the question part.\n\nQUESTIONS:\n1. Could the authors provide a concise user guide for converting an OCO algorithm into an online quadratizable optimization algorithm, along with an explanation of how the regret bound proof is transformed? This should include specifying which technical parts of the paper should be applied and in what order. If the authors have already provided such part somewhere in the paper, it will be appreciate that the authors could remind me the location.\n\n2. Could the authors provide examples of potential applications for quadratizable function classes, or provide references that demonstrate the significance of this function class?\n\n3. For the future work, could the regret lower bound proof of some OCO setting be convert to that of the online quadratizable optimization?\n\ntypos:\n58-59: \"fully adaptive Another...\"\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 1, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-06-20"}

comments: SUMMARY:
This paper introduces a meta-algorithmic framework called "uniform wrappers" that transforms algorithms designed for concave/convex optimization into ones applicable to non-convex, quadratizable optimization problems, specifically targeting classes of DR-submodular functions. The main results are: development of a conversion guideline that adapts both algorithms and their proofs; application to zeroth-order feedback, bandit feedback, and offline optimization;

STRENGTHS AND WEAKNESSES:
Strengths: 1) the proposed uniform wrappers is a novel development. It enables systematic translation of actions and oracles while maintaining regret guarantees. 2) The detailed proof adaptation framework leads to new regret bounds under zeroth-order and bandit settings. These theoretical results either match or improve existing regret/sample complexity bounds. 3) The framework handles multiple function classes, including monotone, non-monotone, weakly DR-submodular functions. This demonstrates the framework’s generality by recovering and unifying prior SOTA results.

Weakness: 1) The paper is purely theoretical. No simulations is provided to demonstrate practical performance of the meta-algorithms (e.g., on DR-submodular test functions).

QUESTIONS:
1. For many practical functions, how should we construct a suitable wrapper W? Are there heuristics or automatable steps, especially for less structured non-convex functions?

LIMITATIONS:
n/a
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-01

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe main contribution of the paper is providing a procedure named the uniform wrapper that converts algorithms and their regret guarantees from online concave optimization to online quadratizable optimization, which includes online DR-submodular maximization as a special case. By applying the proposed framework, the paper can recover almost all the known results in the literature for online/offline DR-submodular maximization (and indeed its generalization, online/offline up-concave optimization).\n\nSTRENGTHS AND WEAKNESSES:\nQuality. The paper presents a systematic method for solving online quadratizable optimization. It is, in general, technically sound. All the claims are well supported by theoretical analysis. \n\nClarity. Perhaps due to the paper's extensive coverage of scenarios, substantial space is devoted to presenting results, leaving limited room for presenting and explaining the approach. The description of conversion steps in Section 6 is overly simplistic, making it difficult to grasp the essence without consulting the appendix. Although Section 7 provides an example of zeroth-order FTRL and proposes a variant algorithm, the authors still fail to explain how this algorithm successfully integrates with the uniform wrapper. In a word, the main body alone provides an inadequate explanation for the algorithm's technical success.\n\nSignificance. Seeking a unified algorithmic framework for online quadratizable optimization represents a fascinating research attempt. The proposed method is, in general, successful since it either beats the SOTA or at least recovers it. However, it is worth noting that the approach only beats the SOTA in a limited number of cases. This means that the approach does not contain too many substantial insights that can break the barrier for online up-concave optimization. This paper may be regarded as an interim summary in some sense.\n\nOriginality. The paper proposes a unified framework for adapting online concave optimization algorithms to algorithms for online quadratizable optimization, improving the previous results such as [Pedramfar and Aggarwal, 2024a]. This validates its originality.\n\nQUESTIONS:\nIt is recommended to leave more room for introducing the uniform wrapper. A good manner is to describe the whole procedure in detail for a concrete scenario.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces a general technique for reducing problems involving (online) optimization of quadratizable losses to the much more well-studied problem of online optimization of concave losses (and in the course of doing so, recovering or improving many state-of-the-art results on quadratizable optimization). Quadratizability is a generalization of concavity that allows for proving approximate optimization guarantees (e.g. that some form of gradient descent will reach an alpha-approximation to the optimal); the condition that f is (upper) quadratizable can be ~written as f satisfying\n\nalpha * f(y) - f(h(x)) <= beta * (<g(f, x), y - x> - ||y-x||^2).\n\nThe key observation of this paper is that many quadratizable functions actually satisfy a slightly stronger constraint, where the functional g(f,x) is equal to $\\nabla G(f)(x)$ for some simple transformation G(f). In particular, they can construct such transformations (which they call “uniform wrappers”) for fairly broad classes of quadratizable functions, including monotone gamma-weakly concave functions with bounded curvature, monotone gamma-weakly concave functions over convex sets containing the origin, and non-monotone concave functions over general convex sets. \n\nThis transformation is fairly powerful and allows for transferring a wide variety of results from the concave optimization regime to the quadratizable optimization regime (although it does seem like there is no completely generic claim along these lines, and every application of this framework does require a little bit of custom work). One example the authors give is the generalization of a classic zeroth-order FTRL algorithm (due to Saha and Tewari) to the quadratizable setting, where it immediately gives an O(T^{2/3}) alpha-regret algorithm for the class of alpha-upper-linearizable losses.\n\nSTRENGTHS AND WEAKNESSES:\nOptimization of (online) quadratizable losses is strongly connected to several problems in online submodular optimization, an area of key importance in machine learning and algorithm design. This paper introduces what seems to me (a non-expert in this area) a fairly powerful technique for translating between this problem and (the very well-studied problem of) online convex optimization.\n\nEarlier work by Pedramfar and Aggarwal (who also introduced the class of quadratizable functions) provided a method to automatically translate between these two regimes, but their method is somewhat more lossy than the method presented in this paper (in the zeroth-order FTRL application mentioned above, the algorithm of Pedramfar and Aggarwal results in suboptimal T^{3/4} rates). The downside is that this new method seems to be somewhat less automatic (e.g., one step is to “rewrite the steps of the analysis so that the application of concavity is limited to a single line”). But the authors do successfully apply it to a fairly large class of problems, and so the generality of this method is fairly convincing.\n\nThe paper was in general well-written and easy to understand, although as someone who is not an expert in the area, it would have been helpful to have a bit more exposition explaining which of the new results in Table 1 arethe most significant improvements over the existing state-of-the-art algorithms. (Right now my impression is that the improvements largely are limited to previously not-well-studied domains, e.g. stochastic-within-a-cone queries, but there are many results and I could have missed some important ones).\n\nQUESTIONS:\nSee section above. Feel free to reply to any part of this review.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-14"}