Step size adaptation in first-order method for stochastic strongly convex programming

where f(.) : Γf → R is an unknown, not necessarily smooth, multivariate and λ-strongly convex function, with Γf its convex definition domain. The algorithm is not allowed to accurately sample f(.) by any means since f(.) itself is unknown. Instead the algorithm can call stochastic oracles ω̃(.) at chosen points x̃1, . . . , x̃n, which are unbiased and independent probabilistic estimators of the first-order local information of f(.) in the vicinity of each xi: