  Measurements of the cosmological parameter $S_8$ provided by cosmic microwave
background and large scale structure data reveal some tension between them,
suggesting that the clustering features of matter in these early and late
cosmological tracers could be different. In this work, we use a supervised
learning method designed to solve Bayesian approach to regression, known as
Gaussian Processes regression, to quantify the cosmic evolution of $S_8$ up to
$z \sim 1.5$. For this, we propose a novel approach to find firstly the
evolution of the function $\sigma_8(z)$, then we find the function $S_8(z)$. As
a sub-product we obtain a minimal cosmological model-dependent $\sigma_8(z=0)$
and $S_8(z=0)$ estimates. We select independent data measurements of the growth
rate $f(z)$ and of $[f\sigma_8](z)$ according to criteria of non-correlated
data, then we perform the Gaussian reconstruction of these data sets to obtain
the cosmic evolution of $\sigma_8(z)$, $S_8(z)$, and the growth index
$\gamma(z)$. Our statistical analyses show that $S_8(z)$ is compatible with
Planck $\Lambda$CDM cosmology; when evaluated at the present time we find
$\sigma_8(z=0) = 0.766 \pm 0.116$ and $S_8(z=0) = 0.732 \pm 0.115$. Applying
our methodology to the growth index, we find $\gamma(z=0) = 0.465 \pm 0.140$.
Moreover, we compare our results with others recently obtained in the
literature. In none of these functions, i.e. $\sigma_8(z)$, $S_8(z)$, and
$\gamma(z)$, do we find significant deviations from the standard cosmology
predictions.
