We prove a Chernoff-type bound for sums of matrix-valued random variables sampled viaaregular (aperiodic and irreducible) finite Markovchain.

Inourlatestversion,wehaveallowedtheMarkov22 chain to start from an arbitrary initial distributionφ rather than the stationary distributionπ. To verify that is a meaning-34 ful range for tuningL, we enumerate trajectory lengthL from {104,,1010}, estimate the co-occurrence ma-35 trix with the single trajectory sampled from BlogCatalog, convert the co-occurrence matrix to the one required36 by NetMF, and factorize it with SVD.

Return ˆ α We show the following generalization of Proposition 2.1. Moreover, Alg. 4 has sample complexity The sample complexity is clear so we focus on the first statement. Theorem 4.5 in [MU17]) on these events as i varies and noting that Hence recalling (A.2) above, we conclude that The other direction is similar. Using (A.2) in the same way as above, we find First we analyze the expected sample complexity. Finally Alg. 4 has sample complexity We do this using Bayes' rule.

We study pure exploration with infinitely many bandit arms generated i.i.d.