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 $\tilde{\mathcal{O}}(δ_*^{1/3} T^{2/3})$ regret for adversarial regimes as well as $\tilde{\mathcal{O}}(δ_* \log T / Δ_{\min}^2)$ 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.

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