Nearly Linear-Time Follow-the-Perturbed-Leader for Weakly Observable Graph Bandits
We study online learning with directed weakly observable feedback graphs and propose a Follow-the-Perturbed-Leader (FTPL) policy that achieves best-of-both-worlds guarantee with nearly linear expected time complexity. To preserve the simple structure of FTPL, we develop a novel graph-based loss estimator that combines forced exploration, conditional geometric resampling and adaptive learning rates with surrogate probabilities. Our policy achieves {{O}}(δ_*^{1/3} T^{2/3}) regret for adversarial regimes as well as {{O}}(δ_* T / Δ_{}^2) regret for stochastic regimes, where δ_* and Δ, respectively, denote the fractional weak domination number and the suboptimality gap. These results show that refined guarantees for problems with minimax regret of Θ(T^{2/3}) can be obtained through appropriately designed adaptive learning rates and loss estimators without sacrificing the simplicity of FTPL.