Fast rates with high probability in exp-concave statistical learning A Proofs for Stochastic Exp-Concave Optimization

This condition is equivalent to stochastic mixability as well as the pseudoprobability convexity (PPC) condition, both defined by Van Erven et al. (2015). To be precise, for stochastic mixability, in Definition 4.1 of Van Erven et al. (2015), take their Fd and F both equal to our F , their P equal to {P}, and ψ(f) = f∗; then strong stochastic mixability holds. Likewise, for the PPC condition, in Definition 3.2 of Van Erven et al. (2015) take the same settings but instead φ(f) = f∗; then the strong PPC condition holds. Now, Theorem 3.10 of Van Erven et al. (2015) states that the PPC condition implies the (strong) central condition. Proof (of Theorem 1) First, from Lemma 1, the convexity of F together with η-exp-concavity implies that (P, `,F) satisfies the η-central condition.