What Size Net Gives Valid Generalization?

TLDR

The study investigates how many random training examples are required for a network to reliably generalize to future samples drawn from the same distribution. The analysis assumes an error tolerance ε ≤ 1/8 and derives sample size bounds for feedforward networks of linear threshold functions with N nodes and W weights. The authors show that m ≥ O((W/ε) log(N/ε)) examples suffice for such networks to achieve 1 − ε accuracy on future data, whereas fewer than Ω(W/ε) examples can lead to failure to reach that accuracy on some distributions.

Abstract

We address the question of when a network can be expected to generalize from m random training examples chosen from some arbitrary probability distribution, assuming that future test examples are drawn from the same distribution. Among our results are the following bounds on appropriate sample vs. network size. Assume 0 < ∊ ≤ 1/8. We show that if m ≥ O(W/∊ log N/∊) random examples can be loaded on a feedforward network of linear threshold functions with N nodes and W weights, so that at least a fraction 1 − ∊/2 of the examples are correctly classified, then one has confidence approaching certainty that the network will correctly classify a fraction 1 − ∊ of future test examples drawn from the same distribution. Conversely, for fully-connected feedforward nets with one hidden layer, any learning algorithm using fewer than Ω(W/∊) random training examples will, for some distributions of examples consistent with an appropriate weight choice, fail at least some fixed fraction of the time to find a weight choice that will correctly classify more than a 1 − ∊ fraction of the future test examples.