Nearly Linear-Time Follow-the-Perturbed-Leader for Weakly Observable Graph Bandits
Abstract
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 $\tilde{\mathcal{O}}(\delta_*^{1/3} T^{2/3})$ regret for adversarial regimes as well as $\tilde{\mathcal{O}}(\delta_* \log T / \Delta_{\min}^2)$ regret for stochastic regimes, where $\delta_*$ and $\Delta$, respectively, denote the fractional weak domination number and the suboptimality gap. These results show that refined guarantees for problems with minimax regret of $\Theta(T^{2/3})$ can be obtained through appropriately designed adaptive learning rates and loss estimators without sacrificing the simplicity of FTPL.
