  This paper provides a framework to show the concentration of solutions $Y^*$
to convex minimizing problem where the objective function $\phi(X)(Y)$ depends
on some random vector $X$ satisfying concentration of measure hypotheses. More
precisely, the convex problem translates into a contractive fixed point
equation that ensure the transmission of the concentration from $X$ to $Y^*$.
This result is of central interest to characterize many machine learning
algorithms which are defined through implicit equations (e.g., logistic
regression, lasso, boosting, etc.). Based on our framework, we provide precise
estimations for the first moments of the solution $Y^*$, when $X= (x_1,\ldots,
x_n)$ is a data matrix of independent columns and $\phi(X)(y)$ writes as a sum
$\frac{1}{n}\sum_{i=1}^n h_i(x_i^TY)$. That allows to describe the behavior and
performance (e.g., generalization error) of a wide variety of machine learning
classifiers.
