call this the mildly overparameterized regime.

### Very wide regime

The second range we consider is the case that not only is the network overparameterized in the sense of having more parameters than data points, but that furthermore the width $m$ of the network is larger than the number of data points $n$. We don’t consider arbitrary width here, but rather the case that $m>n$ but that $m$ is still linear in $n$. In this case, the number of parameters $d$ is quadratic in the number of data points $n$.

We call this the very wide regime. It is wider than the networks used in practice, but not in an extreme way, and there is better theoretical understanding of the geometry of $L$ in this range than in the mildly overparameterized regime.

### Extremely wide regime

The final range we describe is the most heavily overparameterized. Here, we assume that the width $m$ of the neural network is polynomial in the number of data points $n$. In this setting, many authors even take the infinite width limit and let $m$ go to $\infty$.

We call this the extremely wide regime, and it is much more overparameterized than any neural network 