  A ray-tracing method inspired by ergodic billiards is used to estimate the
theoretically best decision rule for a set of linear separable examples. While
the Bayes-optimum requires a majority decision over all Perceptrons separating
the example set, the problem considered here corresponds to finding the single
Perceptron with best average generalization probability. For randomly
distributed examples the billiard estimate agrees with known analytic results.
In real-life classification problems the generalization error is consistently
reduced compared to the maximal stability Perceptron.
