Title: Online Two-Stage Submodular Maximization
Venue: NeurIPS 2025
Accepted: yes
Keywords: online

two-stage

submodular maximization

recommender systems

data summarization
Abstract: Given a collection of monotone submodular functions, the goal of Two-Stage Submodular Maximization (2SSM) (Balkanski et al. 2016) is to restrict the ground set so an objective selected u.a.r. from the collection attains a high maximal value, on average, when optimized over the restricted ground set. We introduce the Online Two-Stage Submodular Maximization (O2SSM) problem, in which the submodular objectives are revealed in an online fashion. 
We study this problem for weighted threshold potential functions, a large and important subclass of monotone submodular functions that includes influence maximization, data summarization, and facility location, to name a few.
We design an algorithm that achieves sublinear $(1 - 1/e)^2$-regret under general matroid constraints and $(1 - 1/e)(1-e^{-k}k^k/k!)$-regret in the case of uniform matroids of rank $k$; the latter also yields a state-of-the-art bound for the (offline) 2SSM problem. 
We empirically validate the performance of our online algorithm with experiments on real datasets.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper addresses the problem of two-stage submodular maximization in an online setting. At each round $t$, the algorithm selects a subset $x_t$ of size $\\ell$ in the first stage, after which the submodular objective function $f_t$ is revealed. In the second stage, the algorithm performs submodular maximization over the selected subset. The authors propose an algorithm that leverages techniques from online convex optimization and rounding methods, and demonstrate that it achieves sublinear regret when the objective functions $f_t$ belong to the class of Weighted Threshold Potentials (WTP).\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n1. The studied problem is well-motivated and significant. In many online scenarios, the large size of the ground set can pose computational challenges for the submodular optimization algorithms. In this sense, the idea of optimization in two-stages, which means first choose a subset and then perform submodular optimization on the restricted set, offers a practical and compelling solution to this issue.\n2. The paper is well-written and easy to follow. \n3. The result that $\\tilde{F}$ is also concave and Lipschitz, and thus the OCO policies can be applied is also interesting.\n\nWeakness:\n1. The main weakness of the paper lies in the practicality of the proposed approach. While the two-stage setting is designed to address scenarios where the ground set is large and optimizing over a smaller subset is computationally efficient, the method introduced in this work is itself computationally intensive. Although the authors show that the algorithm runs in polynomial time, the overall computational overhead may limit its applicability in real-world online settings. As such, the practicality of the proposed method for large-scale problems remains questionable.\n\nQUESTIONS:\n1. The proofs in the paper appear to rely heavily on techniques from prior work. Could the authors clarify the main technical contributions and specify the key challenges they had to overcome in establishing the primary results?\n2. Given the computational overhead introduced by the first stage, could the authors elaborate on why the two-stage online submodular maximization framework is more efficient and advantageous compared to traditional online submodular maximization methods?\n\nLIMITATIONS:\nyes.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-20"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies the online version of the Two-Stage Submodular Maximization (2SSM) problem, introduced by Balkanski et al. [2016].\n2SSM is an interesting problem that aims to select a subset $S$ of core elements from a ground set $V$ (1st stage), such that the expected utility over a collection of submodular functions $\\mathcal{F}$ on the selected subset $S$ is maximized (2nd stage).\nIt can be seen as a special kind of coreset problems.\n\nThe main challenge is the objective function:\n\n$$\\max_S E_{f \\sim \\mathcal{F}} [\\max_{S' \\subseteq S} f(S)],$$\n\nwhere the maximization over $S'$ is taken *inside* the expectation.\nThat means each submodular function $f$ is allowed to take on a different subset $S'$ of $S$.\nThis renders the expectation non-submodular, and difficult to handle.\n\nBalkanski et al. [2016] proposed to optimize a continuous relaxation and do randomized rounding to get a coreset $S$.\nThis paper follows the same idea but adapts it to the online setting.\nBesides, the authors leverage standard algorithms for online convex optimization and more powerful existing rounding techniques to achieve a sublinear regret bound for a specific class of submodular functions.\n\nI think this is a nice paper by combining the ideas of previous works on 2SSM and online convex optimization.\nBut relying heavily on existing components does reduce the novelty of the paper a bit.\nIn my opinion, the main discovery is the theoretical properties of the weighted threshold class of submodular functions that enable the analysis.\nMoreover, I have a question about the rounding technique (see below).\n\nRepresentation issues:\n- Typo: line 59 \"'\"\n- It would be great if the authors could briefly explain how to comprehend the size of $(1-1/e) (1-e^{-k}k^k/k!)$ compared to $(1-1/e)^2$.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1. The authors study an interesting problem, that is relevant to multiple applications such as recommenders.\n2. The authors are the first to study the online version of 2SSM.\n3. The authors leverage existing tools to achieve a sublinear regret bound for a specific class of submodular functions.\n\nWeaknesses:\n1. Concerns about the rounding techniques and the claimed bounds.\n2. Given the heavy theoretical machinery, the algorithm is not scalable (up to thousands of elements).\n\nQUESTIONS:\nThe authors state that \"we design a two-stage randomized rounding procedure, used only in our analysis, that is coupled with the rounding performed by RAOCO.\"\nI am not sure if I understand this correctly.\nIf the advanced rounding technique is used virtually, then what rounding is actually used in the algorithm?\nDoes the given algorithm still achieve the claimed regret bound?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-27"}

comments: SUMMARY:
This paper focuses on the two stage submodular maximization problem in different settings and subject to cardinality and matroid constraint.

The first result in this paper follows the general steps of Balkanski et. al.   and slightly improves the approximation ratio in the offline setting. The improvement is mainly for small values of k.

The authors also consider the online setting and provide a 0.4 approximate algorithm for the general matroid constraint setting but only for a specific class of functions (WTP). In this online setting the functions are revealed in an online fashion and not the elements.

STRENGTHS AND WEAKNESSES:
The main strength of the paper is overcoming the challenge of computing the function F. This function is not submodular and also it cannot be computed in polytime. The techniques used here are interesting but seem to be limited to this specific problem. 

The online setting considered in this work is not that interesting and the motivations provided for it are not strong enough. This combined with the marginal improvements in the offline setting raises concern about the importance and strength of this paper. This is also not for general submodular functions and only for WTP. Most of the previous works cited in this paper (see Table 1) are for general submodular function and in more practical settings.

The experimental section requires significant changes. Currently it is not possible to conclude that the proposed algorithms can be used in practice. The main reason is that the datasets considered are tiny.

QUESTIONS:
1- Can the proposed algorithms be applied on much larger datasets?
2- The motivation behind the online setting is not strong in my opinion, can you elaborate on its importance and applicability?

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 3
IMPACT: 1
ORIGINALITY: 2
DATE: 2025-07-02

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe (stochastic) submodular maximization problem aims to maximize an objective subject to some constraint defined over a ground set, where the objective is picked u.a.r. from a collection of monotone submodular functions. The two-stage submodular maximization problem (2SSM) aims to find a subset of size at most $\\ell$ from the ground set such that when restricted to this subset, the previous submodular maximization problem attains a high maximal value in expectation. This paper studies the online version of 2SSM. For weighted threshold potential functions, an important subclass of monotone submodular functions, it provides an online algorithm that achieves an $O(\\sqrt{T})$ $(1-1/e)^2$-regret under a matroid constraint and the ratio is slightly improved for uniform matroid.\n\nSTRENGTHS AND WEAKNESSES:\nQuality. The paper is technically sound. The main result is supported by theoretical analysis. Besides, the performance of the proposed algorithm is validated by experiments on both real and synthetic datasets.\n\nClarity. The paper is organized well. The algorithm involves several components. The paper thoroughly explains each of them and lists the corresponding lemmas used to prove the final theorem.\n\nSignificance. The paper is the first to consider the online version of the well-studied 2SSM problem and attains sublinear regret for the problem. Though built on submodular maximization, the problem is not a standard online submodular maximization (OIM) problem. Therefore, the result extends previous results on OIM to a non-submodular setting. This could be of interest to researchers. On the other hand, the approach is somewhat limited since it works only for a subclass of submodular functions. Besides, it looks difficult to extend the algorithm to general submodular functions since they do not admit computable concave extensions.\n\nOriginality. The paper exploits an algorithmic framework developed for online submodular maximization in a non-standard submodular maximization problem. Although the overall algorithmic framework derives from prior work, this paper contains sufficient technical contributions, as the successful application of the framework requires key insights and technical lemmas.\n\nQUESTIONS:\nThe (offline) 2SSM admits a better approximation ratio for coverage functions, a subclass of weighted threshold potential functions. Can one obtain the same ratio for online 2SSM by applying the approach of the paper?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 5, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-07-03"}