  In the negative perceptron problem we are given $n$ data points
$({\boldsymbol x}_i,y_i)$, where ${\boldsymbol x}_i$ is a $d$-dimensional
vector and $y_i\in\{+1,-1\}$ is a binary label. The data are not linearly
separable and hence we content ourselves to find a linear classifier with the
largest possible \emph{negative} margin. In other words, we want to find a unit
norm vector ${\boldsymbol \theta}$ that maximizes $\min_{i\le n}y_i\langle
{\boldsymbol \theta},{\boldsymbol x}_i\rangle$. This is a non-convex
optimization problem (it is equivalent to finding a maximum norm vector in a
polytope), and we study its typical properties under two random models for the
data.
  We consider the proportional asymptotics in which $n,d\to \infty$ with
$n/d\to\delta$, and prove upper and lower bounds on the maximum margin
$\kappa_{\text{s}}(\delta)$ or -- equivalently -- on its inverse function
$\delta_{\text{s}}(\kappa)$. In other words, $\delta_{\text{s}}(\kappa)$ is the
overparametrization threshold: for $n/d\le
\delta_{\text{s}}(\kappa)-\varepsilon$ a classifier achieving vanishing
training error exists with high probability, while for $n/d\ge
\delta_{\text{s}}(\kappa)+\varepsilon$ it does not. Our bounds on
$\delta_{\text{s}}(\kappa)$ match to the leading order as $\kappa\to -\infty$.
We then analyze a linear programming algorithm to find a solution, and
characterize the corresponding threshold $\delta_{\text{lin}}(\kappa)$. We
observe a gap between the interpolation threshold $\delta_{\text{s}}(\kappa)$
and the linear programming threshold $\delta_{\text{lin}}(\kappa)$, raising the
question of the behavior of other algorithms.
