Markov Chain Monte Carlo Methods for Neural Networks

1
Markov Chain Monte Carlo Methods for Neural
Networks

• Raul Cruz-Cano
• University of Texas at El Paso

2
Outline

• Introduction
• Bayesian Methods for Neural Networks
• Approximation
• Markov Chain Monte Carlo Methods
• The Metropolis Algorithm
• The Hybrid Monte Carlo Algorithm
• Results and Advantages
• Conclusions
• References

3
Introduction

• Best Single value guess for , being
• (X,Y) the training set and the desired
  output for the input .

4
Bayesian Methods for Neural Networks

• Distribution of interest
• Likelihood function
• Prior Distribution

5
Approximation
In the NN if we can produce a sample from the
distribution , then we can
find an approximation of
6
Markov Chain Monte Carlo Methods

• Metropolis Algorithm Hybrid MC

7
Results Advantages

• Learning to predict the position of a robotic arm
• 2 Outputs -3.3 to 3.3
• Training Error .0050
• Test Error .0057
• Predicting the quality of cement
• Experts 27 inputs
• After ARD 9 inputs
• Possibility of leave local minimums

8
Conclusions

• Connection with genetic algorithms
• Biological connection
• Stopping criteria
• Hardware implementation
• Physics papers
• MCMC provide advantages that make them desirable
  for certain situations

9
References

• Duane, S., Kennedy, A.D., Pendleton, B.J., and
  Roweth, D. Hybrid Monte Carlo, Physics Letters,
  Vol. 195, pag. 216-222 (1987).
• Horowitz, A. M. A generalized guided Monte Carlo
  Algorithm, Physics Letters B, Vol. 268, pag.
  247-252 (1991).
• Lampinen, J. and Vehtari, A. Bayesian approach
  for neural networks-review and case studies,
  Neural Networks, Vol. 14, pag. 257-274 (2001).
• MacKay, D. J. C. A practical Bayesian Framework
  for backpropagation Networks, Neural
  Computation, Vol. 4, pag. 448-472 (1992).
• Neal, R.M. Bayesian Training of Backpropagation
  Networks by the Hybrid Monte Carlo Method,
  Technical Report CRG-TR-92-1 (1992).
• Neal, R.M. Probabilistic Inference Using Markov
  Chain Monte Carlo Methods, Technical Report
  CRG-TR-93-1 (1993).
• Neal, R.M. An improved acceptance procedure for
  the hybrid Monte Carlo algorithm, Journal of
  Computational Physics (1993).
• Rice, J. A. Mathematical Statistics and Data
  Analysis, Duxbury Press (1995).