Stochastic Online Learning with Probabilistic Graph Feedback

We consider a problem of stochastic online learning with general probabilistic graph feedback, where each directed edge in the feedback graph has probability pij. Two cases are covered. (a) The one-step case, where after playing arm i the learner observes a sample reward feedback of arm j with independent probability pij. (b) The cascade case where after playing arm i the learner observes feedback of all arms j in a probabilistic cascade starting from i -- for each (i,j) with probability pij, if arm i is played or observed, then a reward sample of arm j would be observed with independent probability pij. Previous works mainly focus on deterministic graphs which corresponds to one-step case with pij ∈ 0,1, an adversarial sequence of graphs with certain topology guarantees, or a specific type of random graphs. We analyze the asymptotic lower bounds and design algorithms in both cases. The regret upper bounds of the algorithms match the lower bounds with high probability.