Stochastic Networks

1
Stochastic Networks

• Introduction to the Boltzmann Machine

Darryl H. Hwang BME 502
2
Boltzmann Machine

• Network Model
• Input-Output relationship is stochastic
• Probabilistic
• First networks to introduce hidden units
• Two phases
• Training phase
• Free phase

3
Starting Point

• Neurons are treated as binary.
• va(t)1 if active
• va(t)0 if inactive

4
Danger MATH AHEAD!

• Unit a is determined by total input current

Maa? Ma?a Maa 0 ha total feedforward input
into unit a

• At each multiple of ?t, a random unit a gets
  updated

5
Probability

• F is a sigmoidal function.
• Larger Ia, more likely unit a 1

6
Probability

• Markov chain v(t?t) depends only on v not on
  history of network.
• Glauber dynamics
• v is described by a probability distribution and
  doesnt converge on a fixed point

7
Energy Function

• Z partition function
• Pv Boltzmann distribution
• States with lower energy more likely

8
Gibbs Sampling

• Glauber dynamics uses Gibbs sampling for
  distribution

9
Mean-field Approximation

• I is determined by a dynamic equation

• Instead of vaF(Ia) use

10
Mean-field Distribution

• Units are independent
• Probability distribution for v

Mean-field distribution for the Boltzmann
machine Way of interpreting the outputs