A Learning Algorithm for Boltzmann Machines**The research reported here was supported by grants from the System Development Foundation. We thank Peter Brown, Francis Crick, Mark Derthick, Scott Fahlman, Jerry Feldman, Stuart Geman, Gail Gong, John Hopfield, Jay McClelland, Barak Pearlmutter, Harry Printz, Dave Rumelhart, Tim Shallice, Paul Smolensky, Rick Szeliski, and Venkataraman Venkatasubramanian for helpful discussions.Reprint requests should be addressed to David Ackley, Computer Science Department, Carnegie-Mellon University, Pittsburgh, PA 15213.

Computational properties of use of biological organisms or to the construction of computers can emerge as collective properties of systems having a large number of simple equivalent components (or neurons). The physical meaning of content-addressable memory is described by an appropriate phase space flow of the state of a system. A model of such a system is given, based on aspects of neurobiology but readily adapted to integrated circuits. The collective properties of this model produce a content-addressable memory which correctly yields an entire memory from any subpart of sufficient size. The algorithm for the time evolution of the state of the system is based on asynchronous parallel processing. Additional emergent collective properties include some capacity for generalization, familiarity recognition, categorization, error correction, and time sequence retention. The collective properties are only weakly sensitive to details of the modeling or the failure of individual devices.

There is a deep and useful connection between statistical mechanics (the behavior of systems with many degrees of freedom in thermal equilibrium at a finite temperature) and multivariate or combinatorial optimization (finding the minimum of a given function depending on many parameters). A detailed analogy with annealing in solids provides a framework for optimization of the properties of very large and complex systems. This connection to statistical mechanics exposes new information and provides an unfamiliar perspective on traditional optimization problems and methods.