Almost Free: Self-Concordance in Natural Exponential Families and an Application to Bandits
Abstract
We prove that single-parameter natural exponential families with subexponential tails are self-concordant with polynomial-sized parameters. For subgaussian natural exponential families we establish an exact characterization of the growth rate of the self-concordance parameter. Applying these findings to bandits allows us to fill gaps in the literature: We show that optimistic algorithms for generalized linear bandits enjoy regret bounds that are both second-order (scale with the variance of the optimal arm's reward distribution) and free of an exponential dependence on the bound of the problem parameter in the leading term. To the best of our knowledge, ours is the first regret bound for generalized linear bandits with subexponential tails, broadening the class of problems to include Poisson, exponential and gamma bandits.
Cite
Text
Liu et al. "Almost Free: Self-Concordance in Natural Exponential Families and an Application to Bandits." Neural Information Processing Systems, 2024. doi:10.52202/079017-2263Markdown
[Liu et al. "Almost Free: Self-Concordance in Natural Exponential Families and an Application to Bandits." Neural Information Processing Systems, 2024.](https://mlanthology.org/neurips/2024/liu2024neurips-almost/) doi:10.52202/079017-2263BibTeX
@inproceedings{liu2024neurips-almost,
title = {{Almost Free: Self-Concordance in Natural Exponential Families and an Application to Bandits}},
author = {Liu, Shuai and Ayoub, Alex and Sentenac, Flore and Tan, Xiaoqi and Szepesvári, Csaba},
booktitle = {Neural Information Processing Systems},
year = {2024},
doi = {10.52202/079017-2263},
url = {https://mlanthology.org/neurips/2024/liu2024neurips-almost/}
}