The Transition to Perfect Generalization in Perceptrons

Several recent papers (Gardner and Derrida 1989; Györgyi 1990; Sompolinsky et al. 1990) have found, using methods of statistical physics, that a transition to perfect generalization occurs in training a simple perceptron whose weights can only take values ±1. We give a rigorous proof of such a phenomena. That is, we show, for α = 2.0821, that if at least αn examples are drawn from the uniform distribution on {+1, −1}ⁿ and classified according to a target perceptron w_t ∈ {+1, −1}ⁿ as positive or negative according to whether w_t·x is nonnegative or negative, then the probability is 2^−(√n) that there is any other such perceptron consistent with the examples. Numerical results indicate further that perfect generalization holds for α as low as 1.5.