# A Unified Confidence Sequence for Generalized Linear Models, with Applications to Bandits

arXiv: https://arxiv.org/abs/2407.13977 · HTML: https://mlsanity.com/paper/2407.13977/

## Key facts

- Cited by 29 papers in the map (data through 14 Oct 2026).
- Builds on 46 of its 99 references that are in the map.
- First posted on arXiv 19 Jul 2024.

## Builds on (most cited first, 30 of 46)

- [Training language models to follow instructions with human feedback](https://mlsanity.com/paper/2203.02155/) (2022)
- [Deep reinforcement learning from human preferences](https://mlsanity.com/paper/1706.03741/) (2017)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [A Tutorial on Thompson Sampling](https://mlsanity.com/paper/1707.02038/) (2017)
- [An Introduction to Matrix Concentration Inequalities](https://mlsanity.com/paper/1501.01571/) (2015)
- [On Kernelized Multi-armed Bandits](https://mlsanity.com/paper/1704.00445/) (2017)
- [User-friendly introduction to PAC-Bayes bounds](https://mlsanity.com/paper/2110.11216/) (2021)
- [Learning to Optimize via Information-Directed Sampling](https://mlsanity.com/paper/1403.5556/) (2014)
- [Game-theoretic statistics and safe anytime-valid inference](https://mlsanity.com/paper/2210.01948/) (2022)
- [Improved Optimistic Algorithms for Logistic Bandits](https://mlsanity.com/paper/2002.07530/) (2020)
- [A Short Note on Concentration Inequalities for Random Vectors with SubGaussian Norm](https://mlsanity.com/paper/1902.03736/) (2019)
- [Information Directed Sampling and Bandits with Heteroscedastic Noise](https://mlsanity.com/paper/1801.09667/) (2018)
- [Mixture Martingales Revisited with Applications to Sequential Tests and Confidence Intervals](https://mlsanity.com/paper/1811.11419/) (2018)
- [Randomized Exploration in Generalized Linear Bandits](https://mlsanity.com/paper/1906.08947/) (2019)
- [Feel-Good Thompson Sampling for Contextual Bandits and Reinforcement Learning](https://mlsanity.com/paper/2110.00871/) (2021)
- [Scalable Generalized Linear Bandits: Online Computation and Hashing](https://mlsanity.com/paper/1706.00136/) (2017)
- [Active Preference Optimization for Sample Efficient RLHF](https://mlsanity.com/paper/2402.10500/) (2024)
- [Instance-Wise Minimax-Optimal Algorithms for Logistic Bandits](https://mlsanity.com/paper/2010.12642/) (2020)
- [Fast rates in statistical and online learning](https://mlsanity.com/paper/1507.02592/) (2015)
- [Universal Inference](https://mlsanity.com/paper/1912.11436/) (2019)
- [Improved Regret Analysis for Variance-Adaptive Linear Bandits and Horizon-Free Linear Mixture MDPs](https://mlsanity.com/paper/2111.03289/) (2021)
- [Logistic Regression: The Importance of Being Improper](https://mlsanity.com/paper/1803.09349/) (2018)
- [Jointly Efficient and Optimal Algorithms for Logistic Bandits](https://mlsanity.com/paper/2201.01985/) (2022)
- [Efficient Prompt Optimization Through the Lens of Best Arm Identification](https://mlsanity.com/paper/2402.09723/) (2024)
- [On the Performance of Thompson Sampling on Logistic Bandits](https://mlsanity.com/paper/1905.04654/) (2019)
- [Improved Confidence Bounds for the Linear Logistic Model and Applications to Linear Bandits](https://mlsanity.com/paper/2011.11222/) (2020)
- [Self-Concordant Analysis of Generalized Linear Bandits with Forgetting](https://mlsanity.com/paper/2011.00819/) (2020)
- [Fast Rates for General Unbounded Loss Functions: from ERM to Generalized Bayes](https://mlsanity.com/paper/1605.00252/) (2016)
- [Feel-Good Thompson Sampling for Contextual Dueling Bandits](https://mlsanity.com/paper/2404.06013/) (2024)
- [A unified recipe for deriving (time-uniform) PAC-Bayes bounds](https://mlsanity.com/paper/2302.03421/) (2023)

## Built on by (most cited first, 29 of 29)

- [Generalized Linear Bandits: Almost Optimal Regret with One-Pass Update](https://mlsanity.com/paper/2507.11847/) (2025)
- [Improved Online Confidence Bounds for Multinomial Logistic Bandits](https://mlsanity.com/paper/2502.10020/) (2025)
- [Confidence Sequences for Generalized Linear Models via Regret Analysis](https://mlsanity.com/paper/2504.16555/) (2025)
- [Neural Logistic Bandits](https://mlsanity.com/paper/2505.02069/) (2025)
- [Provably Efficient Regularized Online RLHF with Generalized Bilinear Preferences](https://mlsanity.com/paper/2602.23116/) (2026)
- [Generalized Kernelized Bandits: A Novel Self-Normalized Bernstein-Like Dimension-Free Inequality and Regret Bounds](https://mlsanity.com/paper/2508.01681/) (2025)
- [Queue Length Regret Bounds for Contextual Queueing Bandits](https://mlsanity.com/paper/2601.19300/) (2026)
- [Efficient Preference-Based Reinforcement Learning: Randomized Exploration Meets Experimental Design](https://mlsanity.com/paper/2506.09508/) (2025)
- [Eluder dimension: localise it!](https://mlsanity.com/paper/2601.09825/) (2026)
- [Provable Anytime Ensemble Sampling Algorithms in Nonlinear Contextual Bandits](https://mlsanity.com/paper/2510.10730/) (2025)
- [Preference-based Reinforcement Learning beyond Pairwise Comparisons: Benefits of Multiple Options](https://mlsanity.com/paper/2510.18713/) (2025)
- [Algorithm for Contextual Queueing Bandits with Rate-Optimal Queue Length Regret](https://mlsanity.com/paper/2606.09668/) (2026)
- [Principled Confidence Estimation for Deep Computed Tomography](https://mlsanity.com/paper/2602.05812/) (2026)
- [A Jointly Efficient and Optimal Algorithm for Heteroskedastic Generalized Linear Bandits with Adversarial Corruptions](https://mlsanity.com/paper/2602.10971/) (2026)
- [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)
- [Generalized Linear Bandits with Memory](https://mlsanity.com/paper/2608.15848/) (2026)
- [DAL: A Practical Prior-Free Black-Box Framework for Piecewise Stationary Bandits](https://mlsanity.com/paper/2501.19401/) (2025)
- [On the Sublinear Regret of Continuous K-Max Bandits](https://mlsanity.com/paper/2502.13467/) (2025)
- [The Good, the Bad, and the Sampled: a No-Regret Approach to Safe Online Classification](https://mlsanity.com/paper/2510.01020/) (2025)
- [Exploration via Feature Perturbation in Contextual Bandits](https://mlsanity.com/paper/2510.17390/) (2025)
- [Combinatorial Allocation Bandits with Nonlinear Arm Utility](https://mlsanity.com/paper/2603.07005/) (2026)
- [Nonstationary Generalized Linear Bandits with Discounted Online Mirror Descent](https://mlsanity.com/paper/2605.25590/) (2026)
- [Optimal Design for Multinomial Logit Model with Applications to Best Assortment Identification](https://mlsanity.com/paper/2605.25592/) (2026)
- [Variance-sensitive Thompson sampling for generalised linear bandits, revisited](https://mlsanity.com/paper/2606.00431/) (2026)
- [Saturation-Insensitive Dueling Bandits with General Function Approximation](https://mlsanity.com/paper/2609.33011/) (2026)
- [On the Complexity of Preference-Based Bandits](https://mlsanity.com/paper/2609.39351/) (2026)
- [Minimax Optimal Regret for Causal Logistic Bandits with Counterfactual Fairness](https://mlsanity.com/paper/2610.01377/) (2026)
- [Learning While Scheduling Jobs under Context-Dependent Service Rates: An Anytime Rate-Optimal Algorithm](https://mlsanity.com/paper/2610.06006/) (2026)

## Abstract (arXiv)

We present a unified likelihood ratio-based confidence sequence (CS) for any (self-concordant) generalized linear model (GLM) that is guaranteed to be convex and numerically tight. We show that this is on par or improves upon known CSs for various GLMs, including Gaussian, Bernoulli, and Poisson. In particular, for the first time, our CS for Bernoulli has a poly(S)-free radius where S is the norm of the unknown parameter. Our first technical novelty is its derivation, which utilizes a time-uniform PAC-Bayesian bound with a uniform prior/posterior, despite the latter being a rather unpopular choice for deriving CSs. As a direct application of our new CS, we propose a simple and natural optimistic algorithm called OFUGLB, applicable to any generalized linear bandits (GLB; Filippi et al. (2010)). Our analysis shows that the celebrated optimistic approach simultaneously attains state-of-the-art regrets for various self-concordant (not necessarily bounded) GLBs, and even poly(S)-free for bounded GLBs, including logistic bandits. The regret analysis, our second technical novelty, follows from combining our new CS with a new proof technique that completely avoids the previously widely used self-concordant control lemma (Faury et al., 2020, Lemma 9). Numerically, OFUGLB outperforms or is at par with prior algorithms for logistic bandits.