Van den Bout North Carolina State University Box 7914 Raleigh, NC 27695-7914 We apply the theory of Tishby, Levin, and Sol1a (TLS) to two problems. First we analyze an elementary problem for which we find the predictions consistent with conventional statistical results. Second we numerically examine the more realistic problem of training a competitive net to learn a probability density from samples. We find TLS useful for predicting average training behavior. . 1 TLS APPLIED TO LEARNING DENSITIES Recently a theory of learning has been constructed which describes the learning of a relation from examples (Tishby, Levin, and Sol1a, 1989), (Schwarb, Samalan, Sol1a, and Denker, 1990). The original derivation relies on a statistical mechanics treatment of the probability of independent events in a system with a specified average value of an additive error function. The resulting theory is not restricted to learning relations and it is not essentially statistical mechanical. The TLS theory can be derived from the principle of mazimum entropy,a general inference tool which produces probabilities characterized by certain values of the averages of specified functions(Jaynes, 1979). A TLS theory can be constructed whenever the specified function is additive and associated with independent examples. In this paper we treat the problem of learning a probability density from samples.

Raleigh, NC 27695-7914 Abstract We apply the theory of Tishby, Levin, and Sol1a (TLS) to two problems. First we analyze an elementary problem for which we find the predictions consistent with conventional statistical results. Second we numerically examine the more realistic problem of training a competitive net to learn a probability density from samples. We find TLS useful for predicting average training behavior.. 1 TLS APPLIED TO LEARNING DENSITIES Recently a theory of learning has been constructed which describes the learning of a relation from examples (Tishby, Levin, and Sol1a, 1989), (Schwarb, Samalan, Sol1a, and Denker, 1990). The original derivation relies on a statistical mechanics treatment of the probability of independent events in a system with a specified average value of an additive error function. The resulting theory is not restricted to learning relations and it is not essentially statistical mechanical.

Nearly optimal solutions to many combinatorial problems can be found using stochastic simulated annealing. This paper extends the concept of simulated annealing from its original formulation as a Markov process to a new formulation based on mean field theory. Mean field annealing essentially replaces the discrete degrees offreedom in simulated annealing with their average values as computed by the mean field approximation. The net result is that equilibrium at a given temperature is achieved 1-2 orders of magnitude faster than with simulated annealing. A general framework forthe mean field annealing algorithm is derived, and its relationship toHopfield networks is shown. The behavior of MFA is examined both analytically and experimentally for a generic combinatorial optimizationproblem: graph bipartitioning. This analysis indicates the presence of critical temperatures which could be important inimproving the performance of neural networks.

Nearly optimal solutions to many combinatorial problems can be found using stochastic simulated annealing. This paper extends the concept of simulated annealing from its original formulation as a Markov process to a new formulation based on mean field theory. Mean field annealing essentially replaces the discrete degrees of freedom in simulated annealing with their average values as computed by the mean field approximation. The net result is that equilibrium at a given temperature is achieved 1-2 orders of magnitude faster than with simulated annealing. A general framework for the mean field annealing algorithm is derived, and its relationship to Hopfield networks is shown. The behavior of MFA is examined both analytically and experimentally for a generic combinatorial optimization problem: graph bipartitioning. This analysis indicates the presence of critical temperatures which could be important in improving the performance of neural networks.