# Causal Bandits for Linear Structural Equation Models

arXiv: https://arxiv.org/abs/2208.12764 · HTML: https://mlsanity.com/paper/2208.12764/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 32 references that are in the map.
- First posted on arXiv 26 Aug 2022.

## Builds on (most cited first, 11 of 11)

- [Causal Bandits: Learning Good Interventions via Causal Inference](https://mlsanity.com/paper/1606.03203/) (2016)
- [Identifying Best Interventions through Online Importance Sampling](https://mlsanity.com/paper/1701.02789/) (2017)
- [Budgeted and Non-budgeted Causal Bandits](https://mlsanity.com/paper/2012.07058/) (2020)
- [Causal Bandits with Unknown Graph Structure](https://mlsanity.com/paper/2106.02988/) (2021)
- [Causal Bandits with Propagating Inference](https://mlsanity.com/paper/1806.02252/) (2018)
- [Adaptively Exploiting d-Separators with Causal Bandits](https://mlsanity.com/paper/2202.05100/) (2022)
- [Contextual Multi-Armed Bandits for Causal Marketing](https://mlsanity.com/paper/1810.01859/) (2018)
- [Causal Bandits without prior knowledge using separating sets](https://mlsanity.com/paper/2009.07916/) (2020)
- [Combinatorial Causal Bandits](https://mlsanity.com/paper/2206.01995/) (2022)
- [A Causal Bandit Approach to Learning Good Atomic Interventions in Presence of Unobserved Confounders](https://mlsanity.com/paper/2107.02772/) (2021)
- [Scalable Intervention Target Estimation in Linear Models](https://mlsanity.com/paper/2111.07512/) (2021)

## Built on by (most cited first, 19 of 19)

- [Active Learning for Optimal Intervention Design in Causal Models](https://mlsanity.com/paper/2209.04744/) (2022)
- [Model-based Causal Bayesian Optimization](https://mlsanity.com/paper/2211.10257/) (2022)
- [Combinatorial Causal Bandits without Graph Skeleton](https://mlsanity.com/paper/2301.13392/) (2023)
- [Causal Bandits with General Causal Models and Interventions](https://mlsanity.com/paper/2403.00233/) (2024)
- [Learning Good Interventions in Causal Graphs via Covering](https://mlsanity.com/paper/2305.04638/) (2023)
- [Linear Causal Bandits: Unknown Graph and Soft Interventions](https://mlsanity.com/paper/2411.02383/) (2024)
- [Reinforcement Learning in the Real World: A Survey of Statistical Challenges and Future Directions](https://mlsanity.com/paper/2601.15353/) (2026)
- [Robust Causal Bandits for Linear Models](https://mlsanity.com/paper/2310.19794/) (2023)
- [Improved Bound for Robust Causal Bandits with Linear Models](https://mlsanity.com/paper/2405.07795/) (2024)
- [Asymmetric Graph Error Control with Low Complexity in Causal Bandits](https://mlsanity.com/paper/2408.11240/) (2024)
- [Graph Learning Is Suboptimal in Causal Bandits](https://mlsanity.com/paper/2510.16811/) (2025)
- [Causal Contextual Bandits with Adaptive Context](https://mlsanity.com/paper/2405.18626/) (2024)
- [Causal Bandit Over Unknown Graphs: Upper Confidence Bounds With Backdoor Adjustment](https://mlsanity.com/paper/2502.02020/) (2025)
- [On Transportability for Structural Causal Bandits](https://mlsanity.com/paper/2511.17953/) (2025)
- [Optimal Observation-Intervention Trade-Off in Optimisation Problems with Causal Structure](https://mlsanity.com/paper/2309.02287/) (2023)
- [Incentive-Aware Synthetic Control: Accurate Counterfactual Estimation via Incentivized Exploration](https://mlsanity.com/paper/2312.16307/) (2023)
- [Evaluating and Learning Robust Bandit Policies Under Uncertain Causal Mechanisms](https://mlsanity.com/paper/2508.02812/) (2025)
- [Graph Structured Combinatorial Semi-Bandit with Nonlinear Reward Associations through Separable Signals](https://mlsanity.com/paper/2606.14650/) (2026)
- [On the Identifiability of Mixed Ordinal and Exponential Family Causal DAGs under Linear Parametric Models](https://mlsanity.com/paper/2609.17942/) (2026)

## Abstract (arXiv)

This paper studies the problem of designing an optimal sequence of interventions in a causal graphical model to minimize cumulative regret with respect to the best intervention in hindsight. This is, naturally, posed as a causal bandit problem. The focus is on causal bandits for linear structural equation models (SEMs) and soft interventions. It is assumed that the graph's structure is known and has N nodes. Two linear mechanisms, one soft intervention and one observational, are assumed for each node, giving rise to 2N possible interventions. Majority of the existing causal bandit algorithms assume that at least the interventional distributions of the reward node's parents are fully specified. However, there are 2N such distributions (one corresponding to each intervention), acquiring which becomes prohibitive even in moderate-sized graphs. This paper dispenses with the assumption of knowing these distributions or their marginals. Two algorithms are proposed for the frequentist (UCB-based) and Bayesian (Thompson Sampling-based) settings. The key idea of these algorithms is to avoid directly estimating the 2N reward distributions and instead estimate the parameters that fully specify the SEMs (linear in N) and use them to compute the rewards. In both algorithms, under boundedness assumptions on noise and the parameter space, the cumulative regrets scale as tildecal O (d^L+frac12 √NT), where d is the graph's maximum degree, and L is the length of its longest causal path. Additionally, a minimax lower of Omega(d^fracL2-2√T) is presented, which suggests that the achievable and lower bounds conform in their scaling behavior with respect to the horizon T and graph parameters d and L.