# DCM Bandits: Learning to Rank with Multiple Clicks

arXiv: https://arxiv.org/abs/1602.03146 · HTML: https://mlsanity.com/paper/1602.03146/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Feb 2016.

## Built on by (most cited first, 17 of 17)

- [Hierarchical Bayesian Bandits](https://mlsanity.com/paper/2111.06929/) (2021)
- [Robust Generalization and Safe Query-Specialization in Counterfactual Learning to Rank](https://mlsanity.com/paper/2102.05990/) (2021)
- [PairRank: Online Pairwise Learning to Rank by Divide-and-Conquer](https://mlsanity.com/paper/2103.00368/) (2021)
- [Adversarial Attacks on Online Learning to Rank with Click Feedback](https://mlsanity.com/paper/2305.17071/) (2023)
- [Adversarial Attacks on Online Learning to Rank with Stochastic Click Models](https://mlsanity.com/paper/2305.19218/) (2023)
- [Probabilistic Permutation Graph Search: Black-Box Optimization for Fairness in Ranking](https://mlsanity.com/paper/2204.13765/) (2022)
- [Cascading Reinforcement Learning](https://mlsanity.com/paper/2401.08961/) (2024)
- [Product Ranking for Revenue Maximization with Multiple Purchases](https://mlsanity.com/paper/2210.08268/) (2022)
- [Pessimistic Off-Policy Optimization for Learning to Rank](https://mlsanity.com/paper/2206.02593/) (2022)
- [Online Learning of Independent Cascade Models with Node-level Feedback](https://mlsanity.com/paper/2109.02519/) (2021)
- [Calibrating Explore-Exploit Trade-off for Fair Online Learning to Rank](https://mlsanity.com/paper/2111.00735/) (2021)
- [Misalignment, Learning, and Ranking: Harnessing Users Limited Attention](https://mlsanity.com/paper/2402.14013/) (2024)
- [UniRank: Unimodal Bandit Algorithm for Online Ranking](https://mlsanity.com/paper/2208.01515/) (2022)
- [Adaptively Learning to Select-Rank in Online Platforms](https://mlsanity.com/paper/2406.05017/) (2024)
- [Online Learning to Rank under Corruption: A Robust Cascading Bandits Approach](https://mlsanity.com/paper/2511.03074/) (2025)
- [A Linear Matching Bandit Approach to Online Multi-Human Multi-Robot Teaming](https://mlsanity.com/paper/2606.29221/) (2026)
- [DCM Bandits: Multiplayer Information Asymmetric Cascading Bandits for Multiple Clicks](https://mlsanity.com/paper/2608.11873/) (2026)

## Abstract (arXiv)

A search engine recommends to the user a list of web pages. The user examines this list, from the first page to the last, and clicks on all attractive pages until the user is satisfied. This behavior of the user can be described by the dependent click model (DCM). We propose DCM bandits, an online learning variant of the DCM where the goal is to maximize the probability of recommending satisfactory items, such as web pages. The main challenge of our learning problem is that we do not observe which attractive item is satisfactory. We propose a computationally-efficient learning algorithm for solving our problem, dcmKL-UCB; derive gap-dependent upper bounds on its regret under reasonable assumptions; and also prove a matching lower bound up to logarithmic factors. We evaluate our algorithm on synthetic and real-world problems, and show that it performs well even when our model is misspecified. This work presents the first practical and regret-optimal online algorithm for learning to rank with multiple clicks in a cascade-like click model.