Skimmed the beginning of the paper, ISTM that a more accurate, less clickbait description of the work would be: For l-lipshitz-smooth convex functions, there is an existing best known bound on convergence rate with constant-step sizes. Paper proves a better bound with a variant gradient descent algorithm with occasional big steps. Unclear to me if the existing bound is known to be tight (i.e. if there exist known worst-case functions that are right at the bound).

Convex functions do not have suboptimal local optima, so this is seemingly unrelated to this as a known effective technique for escaping those, which other commenters are talking about.