Supplementary Material for Risk and Regret of Hierarchical Bayesian Learners

Proposition. The Bayesian cumulative loss is bounded as LBayes(ZT ) ≤ LQ(ZT ) + KL(Q||P0). (A.1) Proof of Theorem 2.4. Fix a choice of θ∗ and φ and write Q = Qθ∗,φ. Take a second-order Taylor expansion of fy about z ∗, yielding fy(z) = fy(z ∗) + f ′ y(z ∗)>(z − z∗) + 1 2 (z − z∗)>f ′′ y (ζ(z))(z − z∗), for some function ζ. Let z = (ξx,ψ) with θ ∼ Q and let z∗ = E[z] = (ξ∗x,ψ∗). Hence, Ez[fy(z)] = fy(z) + f ′ y(z)0 + 1 2 Ez [ (z − z∗)>f ′′ y (ζ(z))(z − z∗) ]