This paper presents a novel way to approximate a distribution governing a system of coupled particles with a product of independent distributions. The approach is an extension of mean field theory that allows the independent distributions to live in a different space from the system, and thereby capture statistical dependencies in that system. It also allows different Hamiltonians for each independent distribution, to facilitate Monte Carlo estimation of those distributions. The approach leads to a novel energy-minimization algorithm in which each coordinate Monte Carlo estimates an associated spectrum, and then independently sets its state by sampling a Boltzmann distribution across that spectrum. It can also be used for high-dimensional numerical integration, (constrained) combinatorial optimization, and adaptive distributed control. This approach also provides a simple, physics-based derivation of the powerful approximate energy-minimization algorithms semi-formally derived in \cite{wowh00, wotu02c, wolp03a}. In addition it suggests many improvements to those algorithms, and motivates a new (bounded rationality) game theory equilibrium concept.
[1]
Edward W. Piotrowski,et al.
An Invitation to Quantum Game Theory
,
2002,
ArXiv.
[2]
Damien Challet,et al.
Optimal combinations of imperfect objects.
,
2002,
Physical review letters.
[3]
H. Nishimori.
Statistical Physics of Spin Glasses and Information Processing
,
2001
.
[4]
David H. Wolpert,et al.
Beyond Mechanism Design
,
2004
.
[5]
Kagan Tumer,et al.
Collective Intelligence for Control of Distributed Dynamical Systems
,
1999,
ArXiv.
[6]
S. Hart,et al.
Handbook of Game Theory with Economic Applications
,
1992
.
[7]
Carsten Peterson,et al.
A Mean Field Theory Learning Algorithm for Neural Networks
,
1987,
Complex Syst..
[8]
E. Jaynes.
Information Theory and Statistical Mechanics
,
1957
.