# Lazier Than Lazy Greedy

arXiv: https://arxiv.org/abs/1409.7938 · HTML: https://mlsanity.com/paper/1409.7938/

## Key facts

- Cited by 99 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Sep 2014.

## Built on by (most cited first, 30 of 99)

- [Batch Active Learning at Scale](https://mlsanity.com/paper/2107.14263/) (2021)
- [SIMILAR: Submodular Information Measures Based Active Learning In Realistic Scenarios](https://mlsanity.com/paper/2107.00717/) (2021)
- [Data Diversity Matters for Robust Instruction Tuning](https://mlsanity.com/paper/2311.14736/) (2023)
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- [Adaptive Second Order Coresets for Data-efficient Machine Learning](https://mlsanity.com/paper/2207.13887/) (2022)
- [PRISM: A Rich Class of Parameterized Submodular Information Measures for Guided Subset Selection](https://mlsanity.com/paper/2103.00128/) (2021)
- [Identifying Spurious Biases Early in Training through the Lens of Simplicity Bias](https://mlsanity.com/paper/2305.18761/) (2023)
- [Not All Poisons are Created Equal: Robust Training against Data Poisoning](https://mlsanity.com/paper/2210.09671/) (2022)
- [TALISMAN: Targeted Active Learning for Object Detection with Rare Classes and Slices using Submodular Mutual Information](https://mlsanity.com/paper/2112.00166/) (2021)
- [Data-Efficient Structured Pruning via Submodular Optimization](https://mlsanity.com/paper/2203.04940/) (2022)
- [An Experimental Design Framework for Label-Efficient Supervised Finetuning of Large Language Models](https://mlsanity.com/paper/2401.06692/) (2024)
- [Data-Efficient Contrastive Self-supervised Learning: Most Beneficial Examples for Supervised Learning Contribute the Least](https://mlsanity.com/paper/2302.09195/) (2023)
- [Combinatorial Stochastic-Greedy Bandit](https://mlsanity.com/paper/2312.08057/) (2023)
- [ToMA: Token Merge with Attention for Diffusion Models](https://mlsanity.com/paper/2509.10918/) (2025)
- [Toward Highly Efficient and Private Submodular Maximization via Matrix-Based Acceleration](https://mlsanity.com/paper/2305.08367/) (2023)
- [Federated Combinatorial Multi-Agent Multi-Armed Bandits](https://mlsanity.com/paper/2405.05950/) (2024)
- [Greedy-Based Feature Selection for Efficient LiDAR SLAM](https://mlsanity.com/paper/2103.13090/) (2021)
- [Optimal Sampling Gaps for Adaptive Submodular Maximization](https://mlsanity.com/paper/2104.01750/) (2021)
- [Using Partial Monotonicity in Submodular Maximization](https://mlsanity.com/paper/2202.03051/) (2022)
- [CLINICAL: Targeted Active Learning for Imbalanced Medical Image Classification](https://mlsanity.com/paper/2210.01520/) (2022)
- [MILO: Model-Agnostic Subset Selection Framework for Efficient Model Training and Tuning](https://mlsanity.com/paper/2301.13287/) (2023)
- [Adaptive Regularized Submodular Maximization](https://mlsanity.com/paper/2103.00384/) (2021)
- [Submodlib: A Submodular Optimization Library](https://mlsanity.com/paper/2202.10680/) (2022)
- [Accelerating Batch Active Learning Using Continual Learning Techniques](https://mlsanity.com/paper/2305.06408/) (2023)
- [Improving Robustness to Model Inversion Attacks via Sparse Coding Architectures](https://mlsanity.com/paper/2403.14772/) (2024)
- [AdaGReS:Adaptive Greedy Context Selection via Redundancy-Aware Scoring for Token-Budgeted RAG](https://mlsanity.com/paper/2512.25052/) (2025)
- [A Unified Framework for Task-Driven Data Quality Management](https://mlsanity.com/paper/2106.05484/) (2021)
- [Deep Active Alignment of Knowledge Graph Entities and Schemata](https://mlsanity.com/paper/2304.04389/) (2023)
- [Cluster-aware Semi-supervised Learning: Relational Knowledge Distillation Provably Learns Clustering](https://mlsanity.com/paper/2307.11030/) (2023)
- [Balancing the Budget: Feature Selection and Tracking for Multi-Camera Visual-Inertial Odometry](https://mlsanity.com/paper/2109.05975/) (2021)

## Abstract (arXiv)

Is it possible to maximize a monotone submodular function faster than the widely used lazy greedy algorithm (also known as accelerated greedy), both in theory and practice? In this paper, we develop the first linear-time algorithm for maximizing a general monotone submodular function subject to a cardinality constraint. We show that our randomized algorithm, STOCHASTIC-GREEDY, can achieve a (1-1/e-varepsilon) approximation guarantee, in expectation, to the optimum solution in time linear in the size of the data and independent of the cardinality constraint. We empirically demonstrate the effectiveness of our algorithm on submodular functions arising in data summarization, including training large-scale kernel methods, exemplar-based clustering, and sensor placement. We observe that STOCHASTIC-GREEDY practically achieves the same utility value as lazy greedy but runs much faster. More surprisingly, we observe that in many practical scenarios STOCHASTIC-GREEDY does not evaluate the whole fraction of data points even once and still achieves indistinguishable results compared to lazy greedy.