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This paper provides a framework to show the concentration of solutions $Y^*$to convex minimizing problem where the objective function $\phi(X)(Y)$ dependson some random vector $X$ satisfying concentration of measure hypotheses. Moreprecisely, the convex problem translates into a contractive fixed pointequation that ensure the transmission of the concentration from $X$ to $Y^*$.This result is of central interest to characterize many machine learningalgorithms which are defined through implicit equations (e.g., logisticregression, lasso, boosting, etc.). Based on our framework, we provide preciseestimations 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 andperformance (e.g., generalization error) of a wide variety of machine learningclassifiers.

A Concentration of Measure Framework to study convex problems and other implicit formulation problems in machine learning