Learning on the Edge: Online Learning with Stochastic Feedback Graphs
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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{min_ε√((α_ε/ε) T), min_ε (δ_ε/ε)^1/3 T^2/3} (ignoring logarithmic factors), where α_ε and δ_ε are graph-theoretic quantities measured on the support of the stochastic feedback graph 𝒢 with edge probabilities thresholded at ε. Our result, which holds without any preliminary knowledge about 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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