Microscopic Equations and Stability Conditions in Optimal Neural Networks

Using the cavity method, I derive the microscopic equations and their stability condition for information learning in neural networks, optimized with arbitrary performance functions in terms of the aligning fields of the examples. In the thermodynamic limit the aligning fields are well-defined functions of the cavity fields. Iterating the microscopic equations provides a general algorithm for network learning, supported by simulations in the maximally stable perceptron and the committee tree. Macroscopic results agree with the replica theory and the Almeida-Thouless stability condition.