Performance of empirical risk minimization in linear aggregation

We study conditions under which, given a dictionary $F=\{f_1,\dots ,f_M\}$ and an i.i.d. sample $(X_i,Y_i{)}_{i=1}^N$, the empirical minimizer in $\mathrm{span}(F)$ relative to the squared loss, satisfies that with high probability $$R\left({\tilde{f}}^{\mathrm{E}\mathrm{R}\mathrm{M}}\right)\le \underset{f\in \mathrm{span}(F)}{\inf }R(f)+r_N(M),$$ where $R(\cdot )$ is the squared risk and $r_N(M)$ is of the order of $M\mathrm{/}N$. Among other results, we prove that a uniform small-ball estimate for functions in $\mathrm{span}(F)$ is enough to achieve that goal when the noise is independent of the design.