  An Adagrad-inspired class of algorithms for smooth unconstrained optimization
is presented in which the objective function is never evaluated and yet the
gradient norms decrease at least as fast as $\calO(1/\sqrt{k+1})$ while
second-order optimality measures converge to zero at least as fast as
$\calO(1/(k+1)^{1/3})$. This latter rate of convergence is shown to be
essentially sharp and is identical to that known for more standard algorithms
(like trust-region or adaptive-regularization methods) using both function and
derivatives' evaluations. A related "divergent stepsize" method is also
described, whose essentially sharp rate of convergence is slighly inferior. It
is finally discussed how to obtain weaker second-order optimality guarantees at
a (much) reduced computional cost.
