Sharper lower bounds on the performance of the empirical risk minimization algorithm

We present an argument based on the multidimensional and the uniform central limit theorems, proving that, under some geometrical assumptions between the target function $T$ and the learning class $F$, the excess risk of the empirical risk minimization algorithm is lower bounded by \[\frac{\mathbb{E}\sup_{q\in Q}G_q}{\sqrt{n}}\delta,\] where $(G_q)_{q\in Q}$ is a canonical Gaussian process associated with $Q$ (a well chosen subset of $F$) and $\delta$ is a parameter governing the oscillations of the empirical excess risk function over a small ball in $F$.