Introduced by Chatterjee and Dembo, the nonlinear large deviation theory aims at unifying under the same paradigm certain large deviations problems such as the problem of the upper tail of sub-graph counts in Erdös-Rényi graphs, or of the traces of powers of Wigner matrices. This paradigm consists in the fact that for these large deviations problems, the optimal large deviations strategy corresponds to changes of measure which have an affine log-density with respect to the background measure.The goal of the nonlinear large deviations theory is to find a sufficient criterion for this particular strategy to be optimal in a given large deviation problem, and to propose quantitative estimates. We will discuss some improvements on this question which will lead us to develop transportation tools to prove in the case of the Gaussian measure and the uniform measure on discrete hypercube dimension-free estimates.