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DR-Submodular Maximization with Stochastic Biased Gradients: Classical and Quantum Gradient Algorithms

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 6.0Download Report PDF
DR-submodular MaximizationStochastic Biased GradientsZero-Order OptimizationQuantum Gradient EstimationApproximation Algorithms
Abstract:

In this work, we investigate DR-submodular maximization using stochastic biased gradients, which is a more realistic but challenging setting than stochastic unbiased gradients. We first generalize the Lyapunov framework to incorporate biased stochastic gradients, characterizing the adverse impacts of bias and noise. Leveraging this framework, we consider not only conventional constraints but also a novel constraint class: convex sets with a largest element, which naturally arises in applications such as resource allocations. For this constraint, we propose an 1/e1/e approximation algorithm for non-monotone DR-submodular maximization, surpassing the hardness result 1/41/4 for general convex constraints. As a direct application of stochastic biased gradients, we consider zero-order DR-submodular maximization and introduce both classical and quantum gradient estimation algorithms. In each constraint we consider, while retaining the same approximation ratio, the iteration complexity of our classical zero-order algorithms is O(ϵ−3)O(\epsilon^{-3}), matching that of stochastic unbiased gradients; our quantum zero-order algorithms reach O(ϵ−1)O(\epsilon^{-1}) iteration complexity, on par with classical first-order algorithms, demonstrating quantum acceleration and validated in numerical experiments.

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This report is AI-GENERATED using Large Language Models and WisPaper (A scholar search engine). It analyzes academic papers' tasks and contributions against retrieved prior work. While this system identifies POTENTIAL overlaps and novel directions, ITS COVERAGE IS NOT EXHAUSTIVE AND JUDGMENTS ARE APPROXIMATE. These results are intended to assist human reviewers and SHOULD NOT be relied upon as a definitive verdict on novelty.
NOTE that some papers exist in multiple, slightly different versions (e.g., with different titles or URLs). The system may retrieve several versions of the same underlying work. The current automated pipeline does not reliably align or distinguish these cases, so human reviewers will need to disambiguate them manually.
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Overview

Overall Novelty Assessment

The paper extends DR-submodular maximization to handle stochastic biased gradients, generalizing the Lyapunov analysis framework and proposing algorithms for both classical and quantum zero-order settings. It resides in the 'Zeroth-Order and Quantum Gradient Estimation' leaf alongside one sibling paper (Zeroth-Order Stochastic), making this a relatively sparse research direction within the broader taxonomy of nine papers across six leaf nodes. The focus on biased gradients distinguishes it from prior unbiased stochastic methods and complements existing zeroth-order techniques by relaxing the zero-mean noise assumption while retaining gradient-based convergence analysis.

The taxonomy reveals that gradient estimation methods form one of four main branches, with the original paper's leaf sitting under 'Gradient Estimation and Approximation Methods' alongside 'Block-Coordinate and Projection-Free Methods'. Neighboring branches address online learning frameworks (Online Learning with Constraints, Regularized Online Optimization) and theoretical extensions to weaker function classes (Weakly DR-Submodular, Linearizable Optimization). The paper's treatment of biased gradients bridges first-order methods like Projection-Free Stochastic and derivative-free approaches, occupying a distinct niche between exact gradient computation and purely function-value-based optimization.

Among twelve candidates examined across three contributions, none were found to clearly refute the paper's claims. The Lyapunov framework extension examined two candidates with no refutable overlap; the 1/e approximation for convex sets with largest element examined ten candidates without finding prior work achieving this ratio for this constraint class; the quantum zero-order algorithms examined zero candidates. The limited search scope—twelve papers from semantic search and citation expansion—suggests these findings reflect the immediate neighborhood rather than exhaustive coverage. The novel constraint class (convex sets with largest element) appears particularly underexplored in the examined literature.

Based on the top-twelve semantic matches and taxonomy structure, the work appears to occupy a relatively uncrowded position within gradient estimation methods for DR-submodular optimization. The biased gradient setting and the specific constraint class represent departures from existing literature, though the limited search scope means potentially relevant work in adjacent optimization communities may not have been captured. The quantum acceleration component aligns with emerging trends but lacks direct comparators in the examined candidate set.

Taxonomy

Core-task Taxonomy Papers
9
3
Claimed Contributions
12
Contribution Candidate Papers Compared
0
Refutable Paper

Research Landscape Overview

Core task: DR-submodular maximization with stochastic biased gradients. The field centers on optimizing diminishing returns submodular functions—a broad class capturing many machine learning objectives—under various computational and informational constraints. The taxonomy reveals four main branches: Gradient Estimation and Approximation Methods, which develop techniques for computing or approximating gradients when exact evaluations are costly or unavailable; Online and Sequential Optimization Frameworks, which address settings where objectives or constraints arrive over time; Generalized Function Classes and Theoretical Extensions, which broaden the scope beyond classical DR-submodularity to weaker or related notions; and Application-Driven Optimization, which tailors algorithms to specific domains such as information gathering or resource allocation. Early foundational work like Continuous DR-Submodular[2] established convergence guarantees for continuous domains, while methods such as Projection-Free Stochastic[4] and Block-Coordinate Gradient[6] introduced scalable first-order approaches that avoid expensive projection steps or exploit problem structure. Recent efforts have intensified around handling imperfect gradient information and adapting to dynamic environments. A small handful of works explore zeroth-order or quantum-inspired gradient estimation when first-order oracles are unavailable, with Zeroth-Order Stochastic[5] providing convergence rates under function-value queries alone. Meanwhile, online frameworks like Online Submodular Recipe[3] and Regularized Online DR[7] tackle sequential decision-making with regret guarantees, and extensions such as Gamma-Weakly DR-Submodular[9] and Linear to Linearizable[1] relax strict submodularity assumptions to accommodate broader function classes. The original paper, Stochastic Biased Gradients[0], sits squarely within the gradient estimation branch alongside Zeroth-Order Stochastic[5], addressing the practical challenge of biased stochastic gradients—a setting where noise is not zero-mean. This contrasts with earlier unbiased stochastic methods like Projection-Free Stochastic[4] and complements zeroth-order techniques by assuming gradient access but relaxing the unbiasedness requirement, thereby bridging first-order and derivative-free paradigms.

Claimed Contributions

Lyapunov framework extension for stochastic biased gradients

The authors extend the existing Lyapunov framework, originally designed for exact gradients in DR-submodular maximization, to handle stochastic biased gradients. This extension characterizes how both bias and noise adversely affect algorithm performance.

2 retrieved papers
1/e approximation algorithm for convex sets with largest element

The authors introduce a novel constraint class called convex sets with a largest element and propose an approximation algorithm achieving a 1/e ratio for non-monotone DR-submodular maximization. This result surpasses the known 1/4 hardness bound for general convex constraints by exploiting properties of the largest element.

10 retrieved papers
Quantum zero-order algorithms with improved complexity

The authors develop quantum gradient estimation algorithms for zero-order DR-submodular maximization that achieve O(ε−1) iteration complexity, matching classical first-order methods and demonstrating quantum acceleration over classical zero-order methods which require O(ε−3) iterations.

0 retrieved papers

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Lyapunov framework extension for stochastic biased gradients

The authors extend the existing Lyapunov framework, originally designed for exact gradients in DR-submodular maximization, to handle stochastic biased gradients. This extension characterizes how both bias and noise adversely affect algorithm performance.

Contribution

1/e approximation algorithm for convex sets with largest element

The authors introduce a novel constraint class called convex sets with a largest element and propose an approximation algorithm achieving a 1/e ratio for non-monotone DR-submodular maximization. This result surpasses the known 1/4 hardness bound for general convex constraints by exploiting properties of the largest element.

Contribution

Quantum zero-order algorithms with improved complexity

The authors develop quantum gradient estimation algorithms for zero-order DR-submodular maximization that achieve O(ε−1) iteration complexity, matching classical first-order methods and demonstrating quantum acceleration over classical zero-order methods which require O(ε−3) iterations.

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