  We prove general nonlinear large deviation estimates similar to
Chatterjee-Dembo's original bounds except that we do not require any second
order smoothness. Our approach relies on convex analysis arguments and is valid
for a broad class of distributions. Our results are then applied in three
different setups. Our first application consists in the mean-field
approximation of the partition function of the Ising model under an optimal
assumption on the spectra of the adjacency matrices of the sequence of graphs.
Next, we apply our general large deviation bound to investigate the large
deviation of the traces of powers of Wigner matrices with sub-Gaussian entries,
and the upper tail of cycles counts in sparse Erd\"os-Renyi graphs down to the
connectivity threshold $n^{-1/2}$.
