Learning on the Edge: Online Learning with Stochastic Feedback Graphs

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 ErdHos-R'enyi 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 minbiglminvarepsilon √(α_varepsilon/varepsilon) T,, minvarepsilon (δ_varepsilon/varepsilon)1/3 T2/3bigr (ignoring logarithmic factors), where αvarepsilon and δvarepsilon are graph-theoretic quantities measured on the support of the stochastic feedback graph G with edge probabilities thresholded at varepsilon. 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.