Stochastic Optimization

Chapters

• General Terms and tools
• PCA
  • PCA
  • Hebbian Learning
  • Kernel-PCA
• Source Separation
  • ICA
  • Infomax ICA
  • Second Order Source Separation
  • FastICA
• Stochastic Optimization
• Clustering
  • k-means Clustering
  • Pairwise Clustering
  • Self-Organising Maps
  • Locally Linear Embedding
• Estimation Theory
  • Density Estimation
    • Kernel Density Estimation
    • Parametric Density Estimation
  • Mixture Models - Estimation Models

Simulated Annealing

Simulated annealing is oriented in crystallization procedures in nature where the lowest energy state is achieved only when the temperature is lowered very slowly.

The temperature in nature is equivalent to fast the change in the system decreases. Parameters

• Cost function $E:\underset{―}{s}\mapsto E_{(\underset{―}{s})}\in |R$

Pros

• Easy to implement
• Converges

Drawbacks

• expensive (takes forever)

Algorithm

Note that the original algorithm has an inner loop where ${\beta }_t$ is not changed. I do not see a reason for it.

• while true
  • choose new state randomly
  • calculate difference in energy levels: $\mathrm{\Delta }E=E_{(\underset{―}{s})}-E_{({\underset{―}{s}}_t)}$
  • change state with probability: $\frac{1}{1+e^{({\beta }_t\mathrm{\Delta }E)}}$
  • Update ${\beta }_t=\tau {\beta }_{t-1}$ occasionally

Mean-field Annealing

The idea of mean-field annealing is to estimate $P$ with $Q$. $Q$ is updated with every temperature level $\beta$.

The transition probabilities between to states are symmetric.

Is Markov Process.

Gibbs-Boltzmann-distribution (is symmetric)

Factorizing distribution

Algorithm

• while true
  • calculate mean-fields $e_k\text{, }k=1,...,N$
  • calculate moments $(s_k{)}_Q\text{, }k=1,...,N$
  • until $|e_k^{old}-e_k^{new}|<\epsilon$

Footnotes