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In terms of the consensus function, the stationary distribution of a Boltzmann machine can be written as:

where Csmax is the maximum consensus over all states. Annealing refers to T → 0 in the stationary distribution πs. This reduces πs to a uniform distribution over all the maximum consensus states which is the reason for the Boltzmann machine being capable of performing global optimization.

If the solution of binary output is described probabilistically, the output value oi is set to one with the probability regardless of the current state. Further:

where ΔEi is the change in energy identifiable as NETi and T is the pseudo-temperature. Given a current slate i with energy Ei, then a subsequent state j is generated by applying a small disturbance to transform the current state into a next state with energy Ej. If the difference (Ej - Ei) = ΔEi is less than or equal to zero, the state j is accepted as the current state. If ΔEj > 0, the state j is accepted with a probability as given above. This rate of probability acceptance is also known as the Metropolis criterion. Explicitly, this acceptance criterion determines whether j is accepted from i with a probability:

where f(i), f(j) are cost-functions (equivalent of energy of a state) in respect to solutions i and j; and Cp denotes a control parameter (equivalent to the role played by temperature). The Metropolis algorithm indicated above differs from the Boltzmann machine in that the transition matrix is defined in terms of some (positive) energy function Es over S. Assuming Es ≥ Es′:

It should be noted that pss′ ≠ ps′s. The difference is determined by the intrinsic ordering on {s, s′} induced by the energy function.

Following the approach due to Akiyama et al. [54], a machine can in general be specified by three system parameters, namely, a reference activation level, ao; pseudo-temperature, T; and discrete time-step, Δt. Thus, the system parameter space for a Boltzmann machine is S(ao = 0, T, Δt = 1), with the output being a unit step-function. The distribution of the output oi is specified by the following moments:

where Φ(x) is the standard cumulative gaussian distribution defined by:

It may be noted that oi being binary, <oi> refers to the probability of oi equal to 1; and the cumulative gaussian distribution is a sigmoid which matches the probability function of the Boltzmann machine, defined by Equation (4.3).

As indicated before, the Boltzmann machine is a basic model that enables a solution to a combinational problem of finding the optimal (the best under constrained circumstances) solution among a “countably infinite” number of alternative solutions. (Example: The traveling salesman problem*.)


*The traveling salesman problem: A salesman, starting from his headquarters, is to visit each town in a prescribed list of towns exactly once and return to the headquarters in such a way that the length of his tour is minimal.

In the Metropolis algorithm (as the Boltzmann’s acceptance rule), if the lowering of the temperature is done adequately slowly, the network reaches thermal equilibrium at each pseudo-temperature as a result of a large number of transitions being generated at a given temperature value. This “thermal” equilibrium condition is decided by Boltzmann distribution, which as indicated earlier refers to the probability of state i with energy Ei at a pseudo-temperature T. It is given by the following conjecture:

where Z(T) is the partition function defined as Z(T) = Σexp[Ej/(kBT)] with the summation over all possible states. It serves as the normalization constant in Equation (4.13).

4.3.2 McCulloch-Pitts Machine

Since the McCulloch-Pitts model has a binary output with a deterministic decision and instantaneous activation in time, its machine parameter space can be defined by: Sm (ao = 0, T = 0, Δt = 1). The corresponding output is a unit step function, assuming a totally deterministic decision.

4.3.3 Hopfield Machine

Contrary to the McCulloch-Pitts model, the Hopfield machine has a graded output with a deterministic decision but with continuous (monotonic) activation in time. Therefore, its machine parameter is Sm (ao, 0, Δt). If the system gain approaches infinity, then the machine parameter becomes Sm(0, 0, 0). The neuron model employed in the generalized delta rule* is described by Sm(ao, 0, 1) , since it is a discrete time version of the Hopfield machine.


*Delta rule (Widrow-Hoff rule): The delta rule is a training algorithm which modifies weights appropriately for target and actual outputs (of either polarity) and for both continuous and binary inputs and outputs. Symbolically denoting the correction associated with the ith input xi by Δi, the difference between the target (or desired output) and the actual output by δo and a learning rate coefficient by , the delta rule specifies Δi as equal to ( )(δo)(xi). Further, if the value of ith weight after adjustment is Wi(n + 1), it can be related to the value of ith weight before adjustment, namely, Wi(n) by the equation Wi(n + 1) = Wi(n) + Δi.


Figure 4.2  Parametric space of the machines
BM: Boltzmann machine; MPM: McCulloch-Pitts machine HM: Hopfield machine (Adapted from [54])

4.3.4 Gaussian Machine

Akiyama et al. [54] proposed a machine representation of a neuron and termed it as a gaussian machine which has a graded response like the Hopfield machine, and behaves stochastically as the Boltzmann machine. Its output is influenced by a random noise added to each input and as a result forms a probabilistic distribution. The relevant machine parameters allow the system to escape from local minima.

The properties of the gaussian machine are derived from the normal distribution of random noise added to the neural input. The machine parameters are specified by Sm(ao, T, Δt.). The other three machines discussed earlier are special cases of the Gaussian machine as depicted by the system parameter space shown in Figure 4.2.


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