# Minimax Optimal Regret for Causal Logistic Bandits with Counterfactual Fairness

arXiv: https://arxiv.org/abs/2610.01377 · HTML: https://mlsanity.com/paper/2610.01377/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 30 of its 42 references that are in the map.
- First posted on arXiv 1 Oct 2026.

## Builds on (most cited first, 30 of 30)

- [Counterfactual Fairness](https://mlsanity.com/paper/1703.06856/) (2017)
- [An Introduction to Matrix Concentration Inequalities](https://mlsanity.com/paper/1501.01571/) (2015)
- [Path-Specific Counterfactual Fairness](https://mlsanity.com/paper/1802.08139/) (2018)
- [Improved Optimistic Algorithms for Logistic Bandits](https://mlsanity.com/paper/2002.07530/) (2020)
- [Causal Bandits: Learning Good Interventions via Causal Inference](https://mlsanity.com/paper/1606.03203/) (2016)
- [Sequential Experimental Design for Transductive Linear Bandits](https://mlsanity.com/paper/1906.08399/) (2019)
- [Linear Stochastic Bandits Under Safety Constraints](https://mlsanity.com/paper/1908.05814/) (2019)
- [Mostly Exploration-Free Algorithms for Contextual Bandits](https://mlsanity.com/paper/1704.09011/) (2017)
- [Stochastic Bandits with Linear Constraints](https://mlsanity.com/paper/2006.10185/) (2020)
- [Adaptive Algorithms for Online Convex Optimization with Long-term Constraints](https://mlsanity.com/paper/1512.07422/) (2015)
- [High-Dimensional Sparse Linear Bandits](https://mlsanity.com/paper/2011.04020/) (2020)
- [Instance-Wise Minimax-Optimal Algorithms for Logistic Bandits](https://mlsanity.com/paper/2010.12642/) (2020)
- [Jointly Efficient and Optimal Algorithms for Logistic Bandits](https://mlsanity.com/paper/2201.01985/) (2022)
- [Information Directed Sampling for Linear Partial Monitoring](https://mlsanity.com/paper/2002.11182/) (2020)
- [Identifying Best Interventions through Online Importance Sampling](https://mlsanity.com/paper/1701.02789/) (2017)
- [Improved Confidence Bounds for the Linear Logistic Model and Applications to Linear Bandits](https://mlsanity.com/paper/2011.11222/) (2020)
- [An Efficient Pessimistic-Optimistic Algorithm for Stochastic Linear Bandits with General Constraints](https://mlsanity.com/paper/2102.05295/) (2021)
- [A Unified Confidence Sequence for Generalized Linear Models, with Applications to Bandits](https://mlsanity.com/paper/2407.13977/) (2024)
- [Stage-wise Conservative Linear Bandits](https://mlsanity.com/paper/2010.00081/) (2020)
- [Safe Linear Stochastic Bandits](https://mlsanity.com/paper/1911.09501/) (2019)
- [On Kernelized Multi-Armed Bandits with Constraints](https://mlsanity.com/paper/2203.15589/) (2022)
- [Stochastic Linear Contextual Bandits with Diverse Contexts](https://mlsanity.com/paper/2003.02681/) (2020)
- [Generalized Linear Bandits: Almost Optimal Regret with One-Pass Update](https://mlsanity.com/paper/2507.11847/) (2025)
- [Achieving Counterfactual Fairness for Causal Bandit](https://mlsanity.com/paper/2109.10458/) (2021)
- [Experimental Design for Regret Minimization in Linear Bandits](https://mlsanity.com/paper/2011.00576/) (2020)
- [Best-of-Both-Worlds Algorithms for Linear Contextual Bandits](https://mlsanity.com/paper/2312.15433/) (2023)
- [Causal Fairness for Outcome Control](https://mlsanity.com/paper/2306.05066/) (2023)
- [An Experimental Design Approach for Regret Minimization in Logistic Bandits](https://mlsanity.com/paper/2202.02407/) (2022)
- [Neural Logistic Bandits](https://mlsanity.com/paper/2505.02069/) (2025)
- [A Causal Lens for Learning Long-term Fair Policies](https://mlsanity.com/paper/2506.11242/) (2025)

## Abstract (arXiv)

We study causal logistic bandits with counterfactual fairness constraints. The causal structure is given through known factual and counterfactual feature maps that share an unknown logistic reward parameter, but the learner observes only factual rewards. Consequently, the directions determining counterfactual feasibility need not be identifiable from the available feedback. The closest prior analyses either omit a coverage condition or impose a comparatively strong one, and do not establish matching lower bounds. We first show that some coverage condition is necessary: without a coverage-type restriction, factually indistinguishable environments with different optimal fair actions force Omega(T) expected joint loss. Under a weaker full-rank condition on the factual covariance pooled across actions, we identify a target-specific information scale V_star that measures the difficulty of estimating rewards and counterfactual effects from factual feedback. We construct worst-case families satisfying this condition on which every policy incurs expected joint loss Omegaleft(left[V_starminlog K,dright]1/3T2/3right). We also give an explore--then--exploit procedure tuned using V_star and an adaptive algorithm that does not require its value. Both algorithms achieve maxRT,VT=widetildeOleft(left[V_starminlog K,dright]1/3T2/3+kappa d/σ02right), where RT is regret relative to the best fair action and VT denotes the cumulative stage-wise positive violations. Thus the upper and lower bounds match in their leading dependence on T, V_star, and minlog K,d, up to logarithmic factors.