  A fully stochastic second-order adaptive-regularization method for
unconstrained nonconvex optimization is presented which never computes the
objective-function value, but yet achieves the optimal
$\mathcal{O}(\epsilon^{-3/2})$ complexity bound for finding first-order
critical points. The method is noise-tolerant and the inexactness conditions
required for convergence depend on the history of past steps. Applications to
cases where derivative evaluation is inexact and to minimization of finite sums
by sampling are discussed. Numerical experiments on large binary classification
problems illustrate the potential of the new method.
