# FOCUS: Fixed-Confidence Online Causal Learning Using Sequential Adaptive Interventions

arXiv: https://arxiv.org/abs/2609.32165 · HTML: https://mlsanity.com/paper/2609.32165/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 34 references that are in the map.
- First posted on arXiv 26 Sep 2026.

## Builds on (most cited first, 20 of 20)

- [DAGs with NO TEARS: Continuous Optimization for Structure Learning](https://mlsanity.com/paper/1803.01422/) (2018)
- [On the Complexity of Best Arm Identification in Multi-Armed Bandit Models](https://mlsanity.com/paper/1407.4443/) (2014)
- [Optimal Best Arm Identification with Fixed Confidence](https://mlsanity.com/paper/1602.04589/) (2016)
- [Differentiable Causal Discovery from Interventional Data](https://mlsanity.com/paper/2007.01754/) (2020)
- [Follow the Leader If You Can, Hedge If You Must](https://mlsanity.com/paper/1301.0534/) (2013)
- [Learning Causal Graphs with Small Interventions](https://mlsanity.com/paper/1511.00041/) (2015)
- [Non-Asymptotic Pure Exploration by Solving Games](https://mlsanity.com/paper/1906.10431/) (2019)
- [ABCD-Strategy: Budgeted Experimental Design for Targeted Causal Structure Discovery](https://mlsanity.com/paper/1902.10347/) (2019)
- [Interventions, Where and How? Experimental Design for Causal Models at Scale](https://mlsanity.com/paper/2203.02016/) (2022)
- [Cost-Optimal Learning of Causal Graphs](https://mlsanity.com/paper/1703.02645/) (2017)
- [Learning Neural Causal Models with Active Interventions](https://mlsanity.com/paper/2109.02429/) (2021)
- [Budgeted Experiment Design for Causal Structure Learning](https://mlsanity.com/paper/1709.03625/) (2017)
- [Active Structure Learning of Causal DAGs via Directed Clique Tree](https://mlsanity.com/paper/2011.00641/) (2020)
- [Active Bayesian Causal Inference](https://mlsanity.com/paper/2206.02063/) (2022)
- [Experimental Design for Cost-Aware Learning of Causal Graphs](https://mlsanity.com/paper/1810.11867/) (2018)
- [Verification and search algorithms for causal DAGs](https://mlsanity.com/paper/2206.15374/) (2022)
- [Near-Optimal Multi-Perturbation Experimental Design for Causal Structure Learning](https://mlsanity.com/paper/2105.14024/) (2021)
- [Adaptivity Complexity for Causal Graph Discovery](https://mlsanity.com/paper/2306.05781/) (2023)
- [Trust Your nabla: Gradient-based Intervention Targeting for Causal Discovery](https://mlsanity.com/paper/2211.13715/) (2022)
- [Adaptive Online Experimental Design for Causal Discovery](https://mlsanity.com/paper/2405.11548/) (2024)

## Abstract (arXiv)

We study fully online fixed-confidence causal discovery without any historical observational data. Starting from zero samples, the learner sequentially selects interventions to recover both the causal DAG and its edge weights under a linear-Gaussian structural equation model. We establish an instance-dependent lower bound for any (ε,δ)-correct algorithm and propose textscFOCUS, which adaptively allocates interventions through an online max--min game. A key contribution is a computable concentration inequality for the accumulated KL divergence involving causal parameters shared across interventions. We prove that textscFOCUS is (ε,δ)-correct and that its expected stopping time matches the lower bound in its Theta(log(1/δ)) dependence up to an instance-dependent constant. Experiments demonstrate improved structure and edge-weight recovery and confirm the predicted stopping-time trend. Our codes are available on https://anonymous.4open.science/r/FOCUScode-76E5