  A class of algorithms for unconstrained nonconvex optimization is considered
where the value of the objective function is never computed. The class contains
a deterministic version of the first-order Adagrad method typically used for
minimization of noisy function, but also allows the use of second-order
information when available. The rate of convergence of methods in the class is
analyzed and is shown to be identical to that known for first-order
optimization methods using both function and gradients values. The result is
essentially sharp and improves on previously known complexity bounds (in the
stochastic context) by Defossez et al. (2020) and Gratton et al. (2022). A new
class of methods is designed, for which a slightly worse and essentially sharp
complexity result holds. Limited numerical experiments show that the new
methods' performance may be comparable to that of standard steepest descent,
despite using significantly less information, and that this performance is
relatively insensitive to noise.
