Self-Concordant Perturbations for Linear Bandits

We study the adversarial linear bandits problem and present a unified algorithmic framework that bridges Follow-the-Regularized-Leader (FTRL) and Follow-the-Perturbed-Leader (FTPL) methods, extending the known connection between them from the full-information setting. Within this framework, we introduce self-concordant perturbations, a family of probability distributions that mirror the role of self-concordant barriers previously employed in the FTRL-based SCRiBLe algorithm. Using this idea, we design a novel FTPL-based algorithm that combines self-concordant regularization with efficient stochastic exploration. Our approach achieves a regret of $O(d\sqrt{n \ln n})$ on both the $d$-dimensional hypercube and the Euclidean ball. On the Euclidean ball, this matches the rate attained by existing self-concordant FTRL methods. For the hypercube, this represents a $\sqrt{d}$ improvement over these methods and matches the optimal bound up to logarithmic factors.