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We present the complete, equilibrium solution of a neural-network model in which each neuron is connected to a small fraction of the others by symmetric, Hebb-rule synapses. At first replica symmetry is assumed, but the results are then corrected for full symmetry breaking, which leads to a substantial increase in storage capacity.A breakthrough in the field of neural networks was made by Hopfield [l], who pointed out a mapping between them and spin-glasses. Under this mapping, the average steadystate neural firing patterns correspond to equilibrium states of the corresponding spin system. Spin-glasses themselves are solved using an iterative procedure E21 which is *believed to be exact>> for the SK spin glass [3] (for a review of the technique in this and other disordered problems see [4]). When the method was applied to highly connected neural networks, however, it transpired that only the first, simple steps need be taken to obtain a good result, although a complete solution may be very difficult [5,6].

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We introduce optimal learning with a neural network, which we define as minimising the expectation generalisation error. We find that the optimally-trained spherical perceptron may learn a linearly-separable rule as well as any possible network. We sketch an algorithm to generate optimal learning, and simulation results support our conclusions. Optimal learning of a well-known, significant unlearnable problem, the <