SupplementaryMaterial

Proof of Proposition 2. If s = 0, the result is trivial. Hence, using the alternative formulationµs(X) = lnkeXks, we get that s 7 µs(X) is nondecreasing, andlims + µs(X) = ln(esssup(eX)) = esssupX. By definition of IX, φs0(X) and φs1(X) are integrable. The result follows from standard analysis of non-convex gradient descent. Hence, f(x) is inferior to the sum of both terms.