Follow the Perturbed Leader for Linear Bandits: sqrt{T} Regret over General Convex Sets

arXiv:2610.17818v1 Announce Type: new Abstract: We give the first adversarial linear bandit algorithm based on Follow-the-Perturbed-Leader (FTPL) that achieves sqrt{T} regret over general convex action sets. Our algorithm attains widetilde{O}(d^{5/4}sqrt{T}) regret using only linear optimization oracles, and the near-optimal widetilde{O}(dsqrt{T}) regret when an optimal-design exploration distribution is further available. This improves on the T^{2/3} regret of the FTPL algorithm of Dani and Hayes (2006) for general convex sets, and generalizes the sqrt{T} regret of Levy et al. (2026) over two special action sets (hypercube and Euclidean ball). Compared to previous oracle-efficient linear bandit algorithms based on continuous exponential weights, the FTPL approach generally has better computational efficiency, and is much simpler to implement. The key algorithmic element is a perturbation distribution that adapts to the estimated rewards. Its simplest form, an adaptive Gaussian perturbation, is straightforward to implement and already attains the regret bound above. We further extend it to two general constructions, random-weighting and elliptical perturbations, each covering distributions from light tails (Gaussian) to heavy tails (Student-t_2). On the technical side, we develop a systematic way to analyze the stability term in the FTPL regret through the Fisher information of the perturbation distribution. This builds a general framework for FTPL beyond existing ones such as the hazard rate and the FTRL-FTPL duality.

Source: arXiv cs.LG | 2026-10-14