Reading

	CS 420/594: Flake, chs. 19 (ÒPostscript: Complex SystemsÓ) & 20 (ÒGenetic and EvolutionÓ)
	CS 594: Bar-Yam, ch. 6 (ÒLife I: Evolution Ñ Origin of Complex OrganismsÓ)

Pseudo-Temperature

	Temperature = measure of thermal energy (heat)
	Thermal energy = vibrational energy of molecules
	A source of random motion
	Pseudo-temperature = a measure of nondirected (random) change
	Logistic sigmoid gives same equilibrium probabilities as Boltzmann-Gibbs distribution

Transition Probability

Stability

	Are stochastic Hopfield nets stable?
	Thermal noise prevents absolute stability
	But with symmetric weights:

Does ÒThermal NoiseÓ Improve memory Performance?

	Experiments by Bar-Yam (pp. 316-20):
		n = 100
		p = 8
	Random initial state
	To allow convergence, after 20 cycles set T = 0
	How often does it converge to an imprinted pattern?

Probability of Random State Converging on Imprinted State (n=100, p=8)

Probability of Random State Converging on Imprinted State (n=100, p=8)

Analysis of Stochastic Hopfield Network

	Complete analysis by Daniel J. Amit & colleagues in mid-80s
	See D. J. Amit, Modeling Brain Function: The World of Attractor Neural Networks, Cambridge Univ. Press, 1989.
	The analysis is beyond the scope of this course

Phase Diagram

Conceptual Diagrams of Energy Landscape

Phase Diagram Detail

Simulated Annealing

	(Kirkpatrick, Gelatt & Vecchi, 1983)

Dilemma

	In the early stages of search, we want a high temperature, so that we will explore the space and find the basins of the global minimum
	In the later stages we want a low temperature, so that we will relax into the global minimum and not wander away from it
	Solution: decrease the temperature gradually during search

Quenching vs. Annealing

	Quenching:
		rapid cooling of a hot material
		may result in defects & brittleness
		local order but global disorder
		locally low-energy, globally frustrated
	Annealing:
		slow cooling (or alternate heating & cooling)
		reaches equilibrium at each temperature
		allows global order to emerge
		achieves global low-energy state

Multiple Domains

Moving Domain Boundaries

Effect of Moderate Temperature

Effect of High Temperature

Effect of Low Temperature

Annealing Schedule

	Controlled decrease of temperature
	Should be sufficiently slow to allow equilibrium to be reached at each temperature
	With sufficiently slow annealing, the global minimum will be found with probability 1
	Design of schedules is a topic of research

Typical Practical Annealing Schedule

	Initial temperature T0 sufficiently high so all transitions allowed
	Exponential cooling: Tk+1 = aTk
		typical 0.8 < a < 0.99
		at least 10 accepted transitions at each temp.
	Final temperature: three successive temperatures without required number of accepted transitions

Demonstration of Boltzmann Machine & Necker Cube Example

	Run ~mclennan/pub/cube/cubedemo

Necker Cube

Biased Necker Cube

Summary

	Non-directed change (random motion) permits escape from local optima and spurious states
	Pseudo-temperature can be controlled to adjust relative degree of exploration and exploitation

Additional Bibliography

	Kandel, E.R., & Schwartz, J.H.  Principles of Neural Science, Elsevier, 1981.
	Peters, A., Palay, S. L., & Webster, H. d.  The Fine Structure of the Nervous System, 3rd ed., Oxford, 1991.
	Anderson, J.A.  An Introduction to Neural Networks, MIT, 1995.
	Arbib, M. (ed.) Handbook of Brain Theory & Neural Networks, MIT, 1995.
	Hertz, J., Krogh, A., & Palmer, R. G.  Introduction to the Theory of Neural Computation, Addison-Wesley, 1991.