Why a sign of gradient (plus or minus) is not enough for finding a steepest ascend?
Using only sign of gradient is a way to go, but might result in slow convergence. Nevertheless it is a valid variation of the method.

The Geometry of Sign Gradient Descent


Sign-based optimization methods have become popular in machine
learning due to their favorable communication cost in distributed
optimization and their surprisingly good performance in neural network
training. Furthermore, they are closely connected to so-called
adaptive gradient methods like Adam. Recent works on signSGD have used
a non-standard &quot;separable smoothness&quot; assumption, whereas some older
works study sign gradient descent as steepest descent with respect to
the ℓ∞-norm. In this work, we unify these existing results by showing
a close connection between separable smoothness and ℓ∞-smoothness and
argue that the latter is the weaker and more natural assumption. We
then proceed to study the smoothness constant with respect to the
ℓ∞-norm and thereby isolate geometric properties of the objective
function which affect the performance of sign-based methods. In short,
we find sign-based methods to be preferable over gradient descent if
(i) the Hessian is to some degree concentrated on its diagonal, and
(ii) its maximal eigenvalue is much larger than the average
eigenvalue. Both properties are common in deep networks.


Properties of the sign gradient descent algorithms


The aim of this article is to study the properties of the sign
gradient descent algorithms involving the sign of the gradient instead
of the gradient itself and first introduced in the RPROP algorithm.
This article provides two results of convergence for local
optimization, a first one for nominal systems without uncertainty and
a second one for systems with uncertainties. New sign gradient descent
algorithms including the dichotomy algorithm DICHO are applied on
several examples to show their effectiveness in terms of speed of
convergence. As a novelty, the sign gradient descent algorithms can
allow to converge in practice towards other minima than the closest
minimum of the initial condition making these algorithms suitable for
global optimization as a new metaheuristic method.