Stochastic Online Learning with Probabilistic Graph Feedback

We consider a problem of stochastic online learning with general probabilistic graph feedback. Two cases are covered. (a) The one-step case where for each edge $(i,j)$ with probability $p_{ij}$ in the probabilistic feedback graph. After playing arm $i$ the learner observes a sample reward feedback of arm $j$ with independent probability $p_{ij}$. (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 $p_{ij}$, if arm $i$ is played or observed, then a reward sample of arm $j$ would be observed with independent probability $p_{ij}$. Previous works mainly focus on deterministic graphs which corresponds to one-step case with $p_{ij} \in \{0,1\}$, 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.