Nearly Linear-Time Follow-the-Perturbed-Leader for Weakly Observable Graph Bandits

We study first-order regret guarantees for adversarial bandits with a fixed, known directed feedback graph. For strongly observable graphs, [ACBDK15] establishes a minimax regret of $\widetildeΘ(\sqrt{αT})$, where $α$ denotes the independence number of the graph and $T$ denotes the horizon. On the other hand, the best existing first-order bound scales as $\widetilde{O}(\sqrt{(κ+1) L^\star})$ [LLZ20], where $κ$ denotes the clique partition number, satisfying $κ+1\geqα$, and $L^\star$ denotes the cumulative loss of the best arm, leaving a gap between the graph dependence in worst-case and first-order guarantees. In this paper, we bridge this gap by obtaining an expected regret bound of $\widetilde{O}(\sqrt{αL^\star}+α)$. Our first result is an algorithm for the special case of self-aware graphs, in which every vertex has a self-loop. The algorithm combines the update of EXP3 [ACBFS02] with a sampling rule inspired by exploration by optimization [LS20]. More specifically, we construct a map whose fixed points yield sampling distributions that control the one-round regret remainder in terms of the learner's loss and the graph's independence number, which is crucial in deriving optimal first-order guarantees. We further show that a sufficiently accurate fixed point can be computed in polynomial time by a geometrically convergent iteration. Finally, we extend the algorithm to general strongly observable graphs and achieve the same $\widetilde{O}(\sqrt{αL^\star}+α)$ guarantee by adapting the two-level aggregation approach of [LLZ20].

Follow-the-regularized-leader framework has shown effectiveness and flexibility in online learning problems, where the choice of learning rates are known to be crucial. Recently, adaptive learning rates defined in terms of the arm-selection probabilities, obtained by solving convex optimization, have achieved improved best-of-both-worlds (BOBW) guarantees in various bandit problems. In contrast, BOBW guarantees for its computationally efficient alternative, follow-the-perturbed-leader (FTPL), remain relatively limited since its optimization-free nature ironically makes the design of adaptive, probability-dependent learning rates non-trivial. To address this challenge, we propose an adaptive learning rate for FTPL by introducing surrogate probability functions that can be computed only from the available quantities, without requiring the exact probabilities. Based on these learning rates with surrogate functions, we provide the BOBW guarantee for FTPL with Pareto perturbations for any shape parameter $α>1$, generalizing prior results restricted to specific choices of $α=2$. We further show the BOBW guarantees for FTPL with adaptive learning rates in the bandit problem with expert advices. Our approach preserves the computational simplicity of FTPL while enabling probability-dependent adaptivity, and the surrogate-based methodology may be of independent interest in other algorithmic frameworks beyond FTPL and learning rate designs.

We give the first adversarial linear bandit algorithm based on Follow-the-Perturbed-Leader (FTPL) that achieves $\sqrt{T}$ regret over general action sets. Our algorithm attains $\widetilde{\mathcal{O}}(d^{5/4}\sqrt{T})$ regret using only linear optimization oracles, and the near-optimal $\widetilde{\mathcal{O}}(d\sqrt{T})$ 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 action sets, and generalizes the $\sqrt{T}$ regret of Lévy 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 past reward estimators. 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.