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arXiv · 2210.04229

Learning on the Edge: Online Learning with Stochastic Feedback Graphs

Emmanuel Esposito·Federico Fusco·Dirk van der Hoeven·Nicolò Cesa-Bianchi
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Abstract

The framework of feedback graphs is a generalization of sequential decision-making with bandit or full information feedback. In this work, we study an extension where the directed feedback graph is stochastic, following a distribution similar to the classical Erdős-Rényi model. Specifically, in each round every edge in the graph is either realized or not with a distinct probability for each edge. We prove nearly optimal regret bounds of order $\min\bigl\{\min_{\varepsilon} \sqrt{(α_\varepsilon/\varepsilon) T},\, \min_{\varepsilon} (δ_\varepsilon/\varepsilon)^{1/3} T^{2/3}\bigr\}$ (ignoring logarithmic factors), where $α_{\varepsilon}$ and $δ_{\varepsilon}$ are graph-theoretic quantities measured on the support of the stochastic feedback graph $\mathcal{G}$ with edge probabilities thresholded at $\varepsilon$. Our result, which holds without any preliminary knowledge about $\mathcal{G}$, requires the learner to observe only the realized out-neighborhood of the chosen action. When the learner is allowed to observe the realization of the entire graph (but only the losses in the out-neighborhood of the chosen action), we derive a more efficient algorithm featuring a dependence on weighted versions of the independence and weak domination numbers that exhibits improved bounds for some special cases.

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BibTeXRIS

Emmanuel Esposito, Federico Fusco, Dirk van der Hoeven, Nicolò Cesa-Bianchi. 2022-10-09. Learning on the Edge: Online Learning with Stochastic Feedback Graphs. https://arxiv.org/abs/2210.04229

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