Learning on the Edge: Online Learning with Stochastic Feedback Graphs

The framework of feedback graphs is a generalization of sequential decisionmaking
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

minε
p
(αε/ε)T, minε(δε/ε)1/3T2/3
(ignoring logarithmic factors),
where αε and δε 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.