A perceptron is a linear classifier. The standard method for training a perceptron involves maximizing the minimum margin. That is, we are trying to find:
$x^* = \text{argmax}_{x\in \text{unit vectors}} \min_{i \in 1..n} A_i x $
where $x$ is a normal vector to the separating hyperplane, $A$ is the $n\times d$ matrix of input values, $n$ is the number of input data points, $d$ is their diminsion, and $A_i x$ is the dot product between $x$ and the $i$th data point.
My question is: has there been any work on perceptrons whose hyperplane maximizes the average margin? That is,
$x'^* = \text{argmax}_{x\in \text{unit vectors}} \sum_{i \in 1..n} A_i x $
This is a much easier quantity to calculate because it does not require gradient descent. The problem is that it is not guaranteed to find separating hyperplanes even when they do exist---the points with low margin aren't given enough weight to do this.