# Improved Confidence Bounds for the Linear Logistic Model and Applications to Linear Bandits

arXiv: https://arxiv.org/abs/2011.11222 · HTML: https://mlsanity.com/paper/2011.11222/

## Key facts

- Cited by 31 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Nov 2020.

## Built on by (most cited first, 30 of 31)

- [Jointly Efficient and Optimal Algorithms for Logistic Bandits](https://mlsanity.com/paper/2201.01985/) (2022)
- [A Unified Confidence Sequence for Generalized Linear Models, with Applications to Bandits](https://mlsanity.com/paper/2407.13977/) (2024)
- [Improved Regret Bounds of (Multinomial) Logistic Bandits via Regret-to-Confidence-Set Conversion](https://mlsanity.com/paper/2310.18554/) (2023)
- [Parallelizing Thompson Sampling](https://mlsanity.com/paper/2106.01420/) (2021)
- [Instance-optimal PAC Algorithms for Contextual Bandits](https://mlsanity.com/paper/2207.02357/) (2022)
- [Generalized Linear Bandits: Almost Optimal Regret with One-Pass Update](https://mlsanity.com/paper/2507.11847/) (2025)
- [Double Doubly Robust Thompson Sampling for Generalized Linear Contextual Bandits](https://mlsanity.com/paper/2209.06983/) (2022)
- [Learning to be Fair: A Consequentialist Approach to Equitable Decision-Making](https://mlsanity.com/paper/2109.08792/) (2021)
- [Nearly Optimal Algorithms for Contextual Dueling Bandits from Adversarial Feedback](https://mlsanity.com/paper/2404.10776/) (2024)
- [Almost Free: Self-concordance in Natural Exponential Families and an Application to Bandits](https://mlsanity.com/paper/2410.01112/) (2024)
- [An Experimental Design Approach for Regret Minimization in Logistic Bandits](https://mlsanity.com/paper/2202.02407/) (2022)
- [One for All: Simultaneous Metric and Preference Learning over Multiple Users](https://mlsanity.com/paper/2207.03609/) (2022)
- [Active Preference Learning for Ordering Items In- and Out-of-sample](https://mlsanity.com/paper/2405.03059/) (2024)
- [Neural Logistic Bandits](https://mlsanity.com/paper/2505.02069/) (2025)
- [Reinforcement Learning with a Terminator](https://mlsanity.com/paper/2205.15376/) (2022)
- [Queue Length Regret Bounds for Contextual Queueing Bandits](https://mlsanity.com/paper/2601.19300/) (2026)
- [Apple Tasting Revisited: Bayesian Approaches to Partially Monitored Online Binary Classification](https://mlsanity.com/paper/2109.14412/) (2021)
- [Optimal Exploration is no harder than Thompson Sampling](https://mlsanity.com/paper/2310.06069/) (2023)
- [Enhancing Preference-based Linear Bandits via Human Response Time](https://mlsanity.com/paper/2409.05798/) (2024)
- [Enjoying Non-linearity in Multinomial Logistic Bandits: A Minimax-Optimal Algorithm](https://mlsanity.com/paper/2507.05306/) (2025)
- [Near Optimal Pure Exploration in Logistic Bandits](https://mlsanity.com/paper/2410.20640/) (2024)
- [Provable Anytime Ensemble Sampling Algorithms in Nonlinear Contextual Bandits](https://mlsanity.com/paper/2510.10730/) (2025)
- [Algorithm for Contextual Queueing Bandits with Rate-Optimal Queue Length Regret](https://mlsanity.com/paper/2606.09668/) (2026)
- [Choosing Answers in varepsilon-Best-Answer Identification for Linear Bandits](https://mlsanity.com/paper/2206.04456/) (2022)
- [Multi-User Contextual Cascading Bandits for Personalized Recommendation](https://mlsanity.com/paper/2508.13981/) (2025)
- [Logistic Bandits with tildeO(√dT) Regret without Context Diversity Assumptions](https://mlsanity.com/paper/2604.22161/) (2026)
- [Best Arm Identification in Generalized Linear Bandits via Hybrid Feedback](https://mlsanity.com/paper/2605.05745/) (2026)
- [Exploration via Feature Perturbation in Contextual Bandits](https://mlsanity.com/paper/2510.17390/) (2025)
- [Nearly Optimal Active Preference Learning and Its Application to LLM Alignment](https://mlsanity.com/paper/2602.01581/) (2026)
- [Combinatorial Allocation Bandits with Nonlinear Arm Utility](https://mlsanity.com/paper/2603.07005/) (2026)

## Abstract (arXiv)

We propose improved fixed-design confidence bounds for the linear logistic model. Our bounds significantly improve upon the state-of-the-art bound by Li et al. (2017) via recent developments of the self-concordant analysis of the logistic loss (Faury et al., 2020). Specifically, our confidence bound avoids a direct dependence on 1/kappa, where kappa is the minimal variance over all arms' reward distributions. In general, 1/kappa scales exponentially with the norm of the unknown linear parameter θ^*. Instead of relying on this worst-case quantity, our confidence bound for the reward of any given arm depends directly on the variance of that arm's reward distribution. We present two applications of our novel bounds to pure exploration and regret minimization logistic bandits improving upon state-of-the-art performance guarantees. For pure exploration, we also provide a lower bound highlighting a dependence on 1/kappa for a family of instances.