Boltzmann Machines 1

#Training hopfield nets

#Geometric approach

  • Behavior of $\mathbf{E}(\mathbf{y})=\mathbf{y}^{T} \mathbf{W y}$ with $\mathbf{W}=\mathbf{Y} \mathbf{Y}^{T}-N_{p} \mathbf{I}$ is identical to behavior with $W=YY^T$

    • Energy landscape only differs by an additive constant
    • Gradients and location of minima remain same (Have the same eigen vectors)
  • Sine : $\mathbf{y}^{T}\left(\mathbf{Y} \mathbf{Y}^{T}-N_{p} \mathbf{I}\right) \mathbf{y}=\mathbf{y}^{T} \mathbf{Y} \mathbf{Y}^{T} \mathbf{y}-N N_{p}$

  • We use $\mathbf{y}^{T} \mathbf{Y} \mathbf{Y}^{T} \mathbf{y}$ for analyze

  • A pattern $y_p$ is stored if:

    • $\operatorname{sign}\left(\mathbf{W} \mathbf{y}_{p}\right)=\mathbf{y}_{p}$ for all target patterns
  • Training: Design $W$ such that this holds

  • Simple solution: $y_p$ is an Eigenvector of $W$

#Storing k orthogonal patterns

  • Let $\mathbf{Y}=\left[\mathbf{y}_{1} \mathbf{y}_{2} \ldots \mathbf{y}_{K}\right]$
    • $\mathbf{W}=\mathbf{Y} \Lambda \mathbf{Y}^{T}$
    • $\lambda_1,...,\lambda_k$ are positive
    • for $\lambda_1= \lambda_2=\lambda_k= 1$ this is exactly the Hebbian rule
  • Any pattern $y$ can be written as
    • $\mathbf{y}=a_{1} \mathbf{y}_{1}+a_{2} \mathbf{y}_{2}+\cdots+a_{N} \mathbf{y}_{N}$
    • $\mathbf{W y}=a_{1} \mathbf{W y}_{1}+a_{2} \mathbf{W y}_{2}+\cdots+a_{N} \mathbf{W y}_{N} = y$
  • All patterns are stable
    • Remembers everything
    • Completely useless network
  • Even if we store fewer than $N$ patterns
    • Let $Y=\left[\mathbf{y}_{1} \mathbf{y}_{2} \ldots \mathbf{y}_{K} \mathbf{r}_{K+1} \mathbf{r}_{K+2} \ldots \mathbf{r}_{N}\right]$
    • $W=Y \Lambda Y^{T}$
    • $\mathbf{r}_{K+1} \mathbf{r}_{K+2} \ldots \mathbf{r}_{N}$ are orthogonal to $\mathbf{y}_1 \mathbf{y}_2 \ldots \mathbf{y}_K$
    • $\lambda_1= \lambda_2=\lambda_k= 1$
    • Problem arise because eigen values are all 1.0
      • Ensures stationarity of vectors in the subspace
      • All stored patterns are equally important

#General (nonorthogonal) vectors

  • $w_{j i}=\sum_{p \in{p}} y_{i}^{p} y_{j}^{p}$
  • The maximum number of stationary patterns is actually exponential in $N$ (McElice and Posner, 84’)
  • For a specific set of $K$ patterns, we can always build a network for which all $K$ patterns are stable provided $k \le N$
    • But this may come with many β€œparasitic” memories

#Optimization

  • Energy function
    • $E=-\frac{1}{2} \mathbf{y}^{T} \mathbf{W} \mathbf{y}-\mathbf{b}^{T} \mathbf{y}$
    • This must be maximally low for target patterns
    • Must be maximally high for all other patterns
      • So that they are unstable and evolve into one of the target patterns
  • Estimate $W$ such that
    • $E$ is minimized for $y_1,...,y_P$
    • $E$ is maximized for all other $y$
  • Minimize total energy of target patterns
    • $E(\mathbf{y})=-\frac{1}{2} \mathbf{y}^{T} \mathbf{W y} \quad \widehat{\mathbf{W}}=\underset{\mathbf{W}}{\operatorname{argmin}} \sum_{\mathbf{y} \in \mathbf{Y}_{P}} E(\mathbf{y})$
    • However, might also pull all the neighborhood states down
  • Maximize the total energy of all non-target patterns
    • $E(\mathbf{y})=-\frac{1}{2} \mathbf{y}^{T} \mathbf{W} \mathbf{y}$
    • $\widehat{\mathbf{W}}=\underset{\mathbf{W}}{\operatorname{argmin}} \sum_{\mathbf{y} \in \mathbf{Y}_{P}} E(\mathbf{y})-\sum_{\mathbf{y} \notin \mathbf{Y}_{P}} E(\mathbf{y})$
  • Simple gradient descent
    • $\mathbf{W}=\mathbf{w}+\eta\left(\sum_{\mathbf{y} \in \mathbf{Y}_{P}} \mathbf{y} \mathbf{y}^{T}-\sum_{\mathbf{y} \notin \mathbf{Y}_{P}} \mathbf{y} \mathbf{y}^{T}\right)$
    • minimize the energy at target patterns
    • raise all non-target patterns
      • Do we need to raise everything?

#Raise negative class

  • Focus on raising the valleys
    • If you raise every valley, eventually they’ll all move up above the target patterns, and many will even vanish
  • How do you identify the valleys for the current $W$?
    • Initialize the network randomly and let it evolve
    • It will settle in a valley
  • Should we randomly sample valleys?
    • Are all valleys equally important?
    • Major requirement: memories must be stable
      • They must be broad valleys
  • Solution: initialize the network at valid memories and let it evolve
    • It will settle in a valley
    • If this is not the target pattern, raise it
  • What if there’s another target pattern downvalley
    • no need to raise the entire surface, or even every valley
      • Raise the neighborhood of each target memory

#Storing more than N patterns

  • Visible neurons
    • The neurons that store the actual patterns of interest
  • Hidden neurons
    • The neurons that only serve to increase the capacity but whose actual values are not important
  • The maximum number of patterns the net can store is bounded by the width $N$ of the patterns..
  • So lets pad the patterns with $K$ β€œdon’t care” bits
    • The new width of the patterns is $N+K$
    • Now we can store $N+K$ patterns!
  • Taking advantage of don’t care bits
    • Simple random setting of don’t care bits, and using the usual training and recall strategies for Hopfield nets should work
    • However, to exploit it properly, it helps to view the Hopfield net differently: as a probabilistic machine

#A probabilistic interpretation

  • For binary y the energy of a pattern is the analog of the negative log likelihood of a Boltzmann distribution
    • Minimizing energy maximizes log likelihood
    • $E(\mathbf{y})=-\frac{1}{2} \mathbf{y}^{T} \mathbf{W y} \quad P(\mathbf{y})=\operatorname{Cexp}(-E(\mathbf{y}))$

#Boltzmann Distribution

  • $E(\mathbf{y})=-\frac{1}{2} \mathbf{y}^{T} \mathbf{W} \mathbf{y}-\mathbf{b}^{T} \mathbf{y}$
  • $P(\mathbf{y})=\operatorname{Cexp}\left(\frac{-E(\mathbf{y})}{k T}\right)$
  • $C=\frac{1}{\sum_{\mathrm{y}} \exp \left(\frac{-E(\mathbf{y})}{k T}\right)}$
  • $k$ is the Boltzmann constant, $T$ is the temperature of the system
  • Optimizing $W$
    • $E(\mathbf{y})=-\frac{1}{2} \mathbf{y}^{T} \mathbf{W} \mathbf{y} \quad \widehat{\mathbf{W}}=\underset{\mathbf{W}}{\operatorname{argmin}} \sum_{\mathbf{y} \in \mathbf{Y}_{P}} E(\mathbf{y})-\sum_{\mathbf{y} \notin \mathbf{Y}_{P}} E(\mathbf{y})$
    • Simple gradient descent
    • $\mathbf{W}=\mathbf{W}+\eta\left(\sum_{\mathbf{y} \in \mathbf{Y}_{P}} \alpha_{\mathbf{y}} \mathbf{y} \mathbf{y}^{T}-\sum_{\mathbf{y} \notin \mathbf{Y}_{P}} \beta(E(\mathbf{y})) \mathbf{y} \mathbf{y}^{T}\right)$
    • $\alpha_y$ more importance to more frequently presented memories
    • $\beta (E(y))$ more importance to more attractive spurious memories
    • Looks like an expectation
    • $\mathbf{W}=\mathbf{W}+\eta\left(E_{\mathbf{y} \sim \mathbf{Y}_{P}} \mathbf{y} \mathbf{y}^{T}-E_{\mathbf{y} \sim Y} \mathbf{y} \mathbf{y}^{T}\right)$
  • The behavior of the Hopfield net is analogous to annealed dynamics of a spin glass characterized by a Boltzmann distribution
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