Tractability from overparametrization: The example of the negative perceptron

Abstract

In the negative perceptron problem we are given $n$ data points $({\mathbf{x}}_i,y_i)$, where ${\mathbf{x}}_i$ 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 $\mathbf{\theta }$ that maximizes ${\min }_{i\le n}{y}_i⟨\mathbf{\theta },{\mathbf{x}}_i⟩$. 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 $n/d\le {\delta }_{\text{s}}(\kappa )-\epsilon$ a classifier achieving vanishing training error exists with high probability, while for $n/d\ge {\delta }_{\text{s}}(\kappa )+\epsilon$ 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.

107 pages; 7 pdf figures