Title: A Unified Approach to Submodular Maximization Under Noise
Venue: NeurIPS 2025
Accepted: yes
Keywords: submodular maximization

noisy submodular maximization

robustness

continuous greedy algorithm
Abstract: We consider the problem of maximizing a submodular function with access to a _noisy_ value oracle for the function instead of an exact value oracle.
Similar to prior work, we assume that the noisy oracle is persistent in that multiple calls to the oracle for a specific set always return the same value.
In this model, Hassidim and Singer (2017) design a $(1-1/e)$-approximation algorithm for monotone submodular maximization subject to a cardinality constraint, and Huang et al (2022) design a $(1-1/e)/2$-approximation algorithm for monotone submodular maximization subject to any arbitrary matroid constraint.
In this paper, we design a meta-algorithm that allows us to take any "robust" algorithm for exact submodular maximization as a black box and transform it into an algorithm for the noisy setting while retaining the approximation guarantee.
By using the meta-algorithm with the measured continuous greedy algorithm, we obtain a $(1-1/e)$-approximation (resp. $1/e$-approximation) for monotone (resp. non-monotone) submodular maximization subject to a matroid constraint under noise.
Furthermore, by using the meta-algorithm with the double greedy algorithm, we obtain a $1/2$-approximation for unconstrained (non-monotone) submodular maximization under noise.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies the problem of submodular maximization problems under a noisy oracle. In particular, it is assumed that the noise is random and unbiased, and that multiple queries of the submodular objective on the same subset is persistent. In that sense, the estimate of the same marginal gain/subset can't be made more accurate through averaging over multiple queries. To address this challenge, the authors construct a surrogate function to estimate marginal gains and propose algorithms for both monotone and non-monotone submodular maximization under this noise model.\n\nSTRENGTHS AND WEAKNESSES:\n* Strength:\n1. The paper addresses a challenging and practically relevant problem. Unlike prior work that assumes either deterministic persistent noise or unbiased noise with access to repeated queries, this work considers a more realistic and difficult setting: unbiased random noise with only one query per subset, where noise is persistent.\n\n2. The proposed method is quite interesting. By estimating the marginal gain through the construction of a surrogate function, it might be possible to proposed unified algorithms for solving various submodular maximization problems.\n3. The paper is well-written and easy to follow\n\n* Weakness:\n1. A major concern lies in the correctness of the technical proofs. While I did not verify every detail, I identified a potential issue in Lemma 3.3. Specifically, to apply the Hoeffding's inequality, the random variables $f(S\\cup H_i)$ for different $i\\in[m]$ should be independent. However, since $H_i$ are sampled without replacement, $f(S\\cup H_i)$s are no longer independent. This mistake in the proof raised concerns about whether there exists other incorrect proofs in this paper.\n\nQUESTIONS:\nPlease refer to my comments under the \"Strengths And Weaknesses\" section. If the authors are able to adequately address the concern regarding the correctness of the proof, I would be open to raising my score. More generally, I recommend that the authors carefully review the technical details to ensure the correctness of all proofs.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-19"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nA fundamental problem in optimization is maximizing a submodular set function. When having access to a value oracle (i.e. an oracle giving the value of the set function for any set *precisely*), approximation algorithms for the maximization problem subject to cardinality/matroid constraints are known. The paper under consideration studies this problem with a slight twist: the value oracle is noisy, i.e. we observe a noisy version $\\tilde{f}(S)$ instead of $f(S)$, and we still would like to find a set which (approximately) maximizes the function's value. \nThe noise under consideration is multiplicative, i.e. for any set $S$, we have $\\tilde{f} (S) = \\epsilon_S \\cdot f(S)$ for noise $\\epsilon_S$. \n\nThe paper derives a black-box reduction from the noisy to the exact setting. More precisely, it states a meta-algorithm which satisfies that \"under some assumptions\", any approximation algorithm for the exact setting can be used to obtain the same approximation guarantee for the noisy case.\nFrom a technical point of view, the meta-algorithm is based on a randomized surrogate function which allows to estimate the value of the function $f$ with a reasonable error.\n\nUsing their reduction, the authors then derive (tight) approximation guarantees for the noisy setting subject to cardinality and matroid constraints in the monotone and non-monotone regime by applying their meta-algorithm to several known approximation algorithms.\n\nSTRENGTHS AND WEAKNESSES:\n*Strengths*: \nThe general framework is powerful in a sense to deliver tight approximation guarantees in the settings under consideration. It extends previous work (where surrogate functions to approximate the value of the submodular set functions where mainly deterministic) in a suitable way by randomization. \nThis allows the authors to provide a unified framework for several problem specifications (matroids/monotone/...) which were previously handled by more tailored approaches.\nThat said, the meta-algorithm is very clean -- in my opinion an additional plus of the construction in the paper. \n\n*Weaknesses*:\nFrom a purely technical perspective, the contribution of the paper seems to work in the direction of the natural extension of previous work. Working with surrogate functions was already known before and also the choice of the randomized surrogate function in the paper seems to be not super unexpected. In addition, the distribution of the noise is limited to sub-exponential distributions which ensure desirable concentration properties. This further weakens the results of the paper as it remains open if the approach can be extended to general noise as well.\n\nQUESTIONS:\n* Based on one of the weaknesses mentioned above: What is the intuition behind the sub-exponential noise? What are the limitations of the approach towards general noise?\n\n* The robustness assumption in Definition 3.1. seems restrictive at first glance. Is this really the case (even though several approx. algorithms fulfill this definition)? Could this be extended to less restrictive definitions as well?\n\n* Is your approach also usable for, say, knapsack constraints or similar?\n\nLIMITATIONS:\nNo negative societal impact expected.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-22"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nSubmodular maximization under noise is the problem of maximizing submodular function $f \\colon 2^N \\to \\mathbb{R}_{\\ge 0}$ with access to only a noisy oracle that returns $\\tilde{f}(S) = \\xi_S f(S)$ for each $S \\subseteq N$, where $\\xi_S$ is a stochastic and persistent sub-exponential noise with mean $1$. This paper proposes a unified framework for obtaining an algorithm with the same approximation ratio guarantee from any \"robust\" existing algorithm. Using this framework, the present paper proposes a $(1-1/e)$-approximation algorithm for the monotone and matroid case, $1/e$-approximation for the non-monotone and matroid case, and $1/2$-approximation for the non-monotone and unconstrained case.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- The theoretical results are very good. It resolves the open question of whether it is possible to obtain the same approximation ratio even under the noisy setting under several constraints.\n- The idea behind the proposed algorithm is interesting. As existing studies on this topic, the proposed algorithm maximizes a surrogate function instead of maximizing the noisy objective function itself. The main idea of this paper is to estimate the function value $f(S)$ by sampling values $f(S \\cup H')$ for random $H' \\subseteq H$ with $|H'| = t$. By using the smoothing lemma (Lemma 3.5), maximizing the surrogate function leads to a good approximation to maximizing the original function. This idea is very clean and can be applied to many other settings of submodular maximization. I have not read all the proofs in the appendices and do not understand why this simple approach overcomes the barrier that the existing papers had, but the proofs of Lemma 3.5 and Appendix A seem to use submodularity in a good way.\n- The paper is clearly written.\n\n---\nWeaknesses:\n- The definitions of submodularity in the beginning part of this paper are wrong. In the Introduction section, \"$i \\in T \\setminus S$\" should be replaced by \"$i \\in N \\setminus T$\". In the beginning of Section 2, \"any element in $x \\in N$\" should be replaced by \"any element in $x \\in N \\setminus B$. I believe that these are just typos, but if these wrong definitions are used anywhere in the proof, it would be problematic.\n\nQUESTIONS:\nNo question.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-27"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies submodular function maximization under a persistent noisy value oracle model, where repeated queries to the same set yield a fixed noisy value. Building upon prior work [11, 14], the authors propose a meta-algorithm that systematically converts any robust noiseless submodular maximization algorithm into one that works under persistent noise, while preserving approximation guarantees. By instantiating this framework with measured continuous greedy and double greedy algorithms, the paper recovers or improves state-of-the-art approximation ratios across cardinality, matroid, and unconstrained settings for both monotone and non-monotone functions.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1. Introduce a meta-algorithm that robustly extends noiseless submodular maximization algorithms to the persistent noise setting, achieving near-optimal guarantees: a tight (1 -\\frac{1}{e})-approximation for monotone submodular maximization under matroid constraints, a 1/2-approximation for unconstrained non-monotone maximization (tight even under noise), and a 1/e-approximation for non-monotone maximization under matroid constraints. This includes the first theoretical guarantees for noisy non-monotone submodular maximization.\n2. Employ a novel randomized surrogate function technique that overcomes the limitations of prior deterministic smoothing approaches, allowing a unified treatment of various constraints while inheriting strong guarantees from continuous greedy and double greedy algorithms.\n3. Significantly broaden the applicability of noise-aware submodular maximization by generalizing previous algorithms that were limited to monotone functions and specific constraints, providing tight approximation guarantees under realistic persistent noise models.\n\nWeakness:\n1. Although the paper is relatively complete in theoretical analysis, the core method is mainly based on the robustness analysis of existing continuous greedy and double greedy algorithms, lacking essential breakthroughs in existing work. In particular, the surrogate function design and sampling mechanism proposed in the framework have obvious inheritance relationships with previous works such as [11][14]. It is recommended that the author clearly explain the substantial innovation compared with these works.\n2. The results shown in Table 1 of the paper have improved previous work in some scenarios. In the non-monotonic case, although it fills the theoretical gap of unconstrained and pseudo-matrix constraints, it is difficult to judge its practical value without comparing it with the empirical performance of heuristic methods.\n3. Theorem 4.2 requires that the rank constraint satisfies $r \\geq \\Omega(\\varepsilon^{-1} \\log^2 n)$. Taking the typical parameters n=10^4 and ε=0.1 as an example, this means that $ r\\geq 500$ is required to ensure that the theoretical result holds. However, many practical applications usually involve low-rank constraints (such as r=10). Does the theoretical guarantee of the framework completely fail? \n4. Although it is a theoretical paper, since it claims to propose a \"widely reusable unified algorithm framework\", its actual effects should be demonstrated through small-scale experiments or simulations to demonstrate its \"practical applicability\".\n\nQUESTIONS:\nSee the questions in weakness section.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 1, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-30"}