#The Hopfield net as a distribution
#The Helmholtz Free Energy of a System
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At any time, the probability of finding the system in state $s$ at temperature $T$ is $P_T(s)$
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At each state it has a potential energy $E_s$
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The internal energy of the system, representing its capacity to do work, is the average
- $$ U_{T}=\sum_{S} P_{T}(s) E_{S} $$
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The capacity to do work is counteracted by the internal disorder of the system, i.e. its entropy
- $$ H_{T}=-\sum_{S} P_{T}(s) \log P_{T}(s) $$
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The Helmholtz free energy of the system measures the useful work derivable from it and combines the two terms
- $$ F_{T}=U_{T}+k T H_{T} $$
- $$ =\sum_{S} P_{T}(s) E_{S}-k T \sum_{S} P_{T}(s) \log P_{T}(s) $$
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The probability distribution of the states at steady state is known as the Boltzmann distribution
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Minimizing this w.r.t $P_T(s)$, we get
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$$ P_{T}(s)=\frac{1}{Z} \exp \left(\frac{-E_{S}}{k T}\right) $$
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$Z$ is a normalizing constant
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#Hopfield net as a distribution
- $E(S)=-\sum_{i<j} w_{i j} s_{i} s_{j}-b_{i} s_{i}$
- $P(S)=\frac{\exp (-E(S))}{\sum_{S^{\prime}} \exp \left(-E\left(S^{\prime}\right)\right)}$
- The stochastic Hopfield network models a probability distribution over states
- It is a generative model: generates states according to $P(S)$
#The field at a single node
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Let's take one node as example
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Let $S$ and $S^\prime$ be the states with the +1 and -1 states
- $P(S)=P\left(s_{i}=1 \mid s_{j \neq i}\right) P\left(s_{j \neq i}\right)$
- $P\left(S^{\prime}\right)=P\left(s_{i}=-1 \mid s_{j \neq i}\right) P\left(s_{j \neq i}\right)$
- $\log P(S)-\log P\left(S^{\prime}\right)=\log P\left(s_{i}=1 \mid s_{j \neq i}\right)-\log P\left(s_{i}=-1 \mid s_{j \neq i}\right)$
- $\log P(S)-\log P\left(S^{\prime}\right)=\log \frac{P\left(s_{i}=1 \mid s_{j \neq i}\right)}{1-P\left(s_{i}=1 \mid s_{j \neq i}\right)}$
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$\log P(S)=-E(S)+C$
- $E(S)=-\frac{1}{2}\left(E_{\text {not } i}+\sum_{j \neq i} w_{i j} s_{j}+b_{i}\right)$
- $E\left(S^{\prime}\right)=-\frac{1}{2}\left(E_{\text {not } i}-\sum_{j \neq i} w_{i j} s_{j}-b_{i}\right)$
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$\log P(S)-\log P\left(S^{\prime}\right)=E\left(S^{\prime}\right)-E(S)=\sum_{j \neq i} w_{i j} S_{j}+b_{i}$
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$\log \left(\frac{P\left(s_{i}=1 \mid s_{j \neq i}\right)}{1-P\left(s_{i}=1 \mid s_{j \neq i}\right)}\right)=\sum_{j \neq i} w_{i j} s_{j}+b_{i}$
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$P\left(s_{i}=1 \mid s_{j \neq i}\right)=\frac{1}{1+e^{-\left(\sum_{j \neq i} w_{i j} s_{j}+b_{i}\right)}}$
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The probability of any node taking value 1 given other node values is a logistic
#Redefining the network
- Redefine a regular Hopfield net as a stochastic system
- Each neuron is now a stochastic unit with a binary state $s_i$, which can take value 0 or 1 with a probability that depends on the local field
- $z_{i}=\sum_{j} w_{i j} s_{j}+b_{i}$
- $P\left(s_{i}=1 \mid s_{j \neq i}\right)=\frac{1}{1+e^{-z_{i}}}$
- Note
- The Hopfield net is a probability distribution over binary sequences (Boltzmann distribution)
- The conditional distribution of individual bits in the sequence is a logistic
- The evolution of the Hopfield net can be made stochastic
- Instead of deterministically responding to the sign of the local field, each neuron responds probabilistically
- Recall patterns
β Annealing
#The Boltzmann Machine
- The entire model can be viewed as a generative model
- Has a probability of producing any binary vector $y$
- $E(\mathbf{y})=-\frac{1}{2} \mathbf{y}^{T} \mathbf{W} \mathbf{y}$
- $P(\mathbf{y})=\operatorname{Cexp}\left(-\frac{E(\mathbf{y})}{T}\right)$
- Training a Hopfield net: Must learn weights to βrememberβ target states and βdislikeβ other states
- Must learn weights to assign a desired probability distribution to states
- Just maximize likelihood
#Maximum Likelihood Training
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$\log (P(S))=\left(\sum_{i<j} w_{i j} s_{i} s_{j}\right)-\log \left(\sum_{S^{\prime}} \exp \left(\sum_{i<j} w_{i j} s_{i}^{\prime} s_{j}^{\prime}\right)\right)$
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$\mathcal{L}=\frac{1}{N} \sum_{S \in \mathbf{S}} \log (P(S)) =\frac{1}{N} \sum_{S}\left(\sum_{i<j} w_{i j} s_{i} s_{j}\right)-\log \left(\sum_{S^{\prime}} \exp \left(\sum_{i<j} w_{i j} s_{i}^{\prime} s_{j}^{\prime}\right)\right)$
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Second term derivation
- $\frac{d \log \left(\sum_{S^{\prime}} \exp \left(\sum_{i<j} w_{i j} s_{i}^{\prime} s_{j}^{\prime}\right)\right)}{d w_{i j}}=\sum_{S^{\prime}} \frac{\exp \left(\sum_{i<j} w_{i j} s_{i}^{\prime} s_{j}^{\prime}\right)}{\sum_{S^{\prime}} \exp \left(\sum_{i<j} w_{i j} s_{i}^{\prime \prime} s_{j}^{\prime}\right)} s_{i}^{\prime} s_{j}^{\prime}$
- $\frac{d \log \left(\sum_{S^{\prime}} \exp \left(\sum_{i<j} w_{i j} s_{i}^{\prime} s_{j}^{\prime}\right)\right)}{d w_{i j}}=\sum_{S_{\prime}} P\left(S^{\prime}\right) s_{i}^{\prime} s_{j}^{\prime}$
- The second term is simply the expected value of $s_iS_j$, over all possible values of the state
- We cannot compute it exhaustively, but we can compute it by sampling!
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Overall gradient ascent rule
- $w_{i j}=w_{i j}+\eta \frac{d\langle\log (P(\mathbf{S}))\rangle}{d w_{i j}}$
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Overall Training
- Initialize weights
- Let the network run to obtain simulated state samples
- Compute gradient and update weights
- Iterate
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Note the similarity to the update rule for the Hopfield network
- The only difference is how we got the samples
#Adding Capacity
β Expanding the network
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Visible neurons
- The neurons that store the actual patterns of interest
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Hidden neurons
- The neurons that only serve to increase the capacity but whose actual values are not important
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We could have multiple hidden patterns coupled with any visible pattern
- These would be multiple stored patterns that all give the same visible output
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We are interested in the marginal probabilities over visible bits
- $S=(V,H)$
- $P(S)=\frac{\exp (-E(S))}{\sum_{S^{\prime}} \exp \left(-E\left(S^{\prime}\right)\right)}$
- $P(S) = P(V,H)$
- $P(V)=\sum_{H} P(S)$
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Train to maximize probability of desired patterns of visible bits
- $E(S)=-\sum_{i<j} w_{i j} s_{i} s_{j}$
- $P(S)=\frac{\exp \left(\sum_{i<j} w_{i j} s_{i} s_{j}\right)}{\sum_{S^{\prime}} \exp \left(\sum_{i<j} w_{i j} s_{i}^{\prime} s_{j}^{\prime}\right)}$
- $P(V)=\sum_{H} \frac{\exp \left(\sum_{i<j} w_{i j} s_{i} s_{j}\right)}{\sum_{S^{\prime}} \exp \left(\sum_{i<j} w_{i j} s_{i}^{\prime} s_{j}^{\prime}\right)}$
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Maximum Likelihood Training
$$\log (P(V))=\log \left(\sum_{H} \exp \left(\sum_{i<j} w_{i j} s_{i} s_{j}\right)\right)-\log \left(\sum_{S_{\prime}} \exp \left(\sum_{i<j} w_{i j} s_{i}^{\prime} s_{j}^{\prime}\right)\right)$$
$$\mathcal{L}=\frac{1}{N} \sum_{V \in \mathbf{V}} \log (P(V))$$ $$ \frac{d \mathcal{L}}{d w_{i j}}=\frac{1}{N} \sum_{V \in \mathbf{V}} \sum_{H} P(S \mid V) s_{i} s_{j}-\sum_{S !} P\left(S^{\prime}\right) s_{i}^{\prime} s_{j}^{\prime} $$
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$\sum_{H} P(S \mid V) s_{i} s_{j} \approx \frac{1}{K} \sum_{H \in \mathbf{H}_{s i m u l}} s_{i} S_{j}$
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Computed as the average sampled hidden state with the visible bits fixed
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$\sum_{S^{\prime}} P\left(S^{\prime}\right) s_{i}^{\prime} s_{j}^{\prime} \approx \frac{1}{M} \sum_{S_{i} \in \mathbf{S}_{s i m u l}} s_{i}^{\prime} S_{j}^{\prime}$
- Computed as the average of sampled states when the network is running βfreelyβ
#Training
Step1
- For each training pattern $V_i$
- Fix the visible units to $V_i$
- Let the hidden neurons evolve from a random initial point to generate $H_i$
- Generate $S_i = [V_i,H_i]$
- Repeat K times to generate synthetic training
$$ \mathbf{S}={S_{1,1}, S_{1,2}, \ldots, S_{1 K}, S_{2,1}, \ldots, S_{N, K}} $$
Step2
- Now unclamp the visible units and let the entire network evolve several times to generate
$$ \mathbf{S}_{simul}=S_{simul, 1}, S_{simul, 2}, \ldots, S_{simul, M} $$
Gradients $$ \frac{d\langle\log (P(\mathbf{S}))\rangle}{d w_{i j}}=\frac{1}{N K} \sum_{\boldsymbol{S}} s_{i} s_{j}-\frac{1}{M} \sum_{S_{i} \in \mathbf{S}_{\text {simul }}} s_{i}^{\prime} s_{j}^{\prime} $$
$$ w_{i j}=w_{i j}-\eta \frac{d\langle\log (P(\mathbf{S}))\rangle}{d w_{i j}} $$
- Gradients are computed as before, except that the first term is now computed over the expanded training data
Issues
- Training takes for ever
- Doesnβt really work for large problems
- A small number of training instances over a small number of bits
#Restricted Boltzmann Machines
β Restricted Boltzmann Machines
- Partition visible and hidden units
- Visible units ONLY talk to hidden units
- Hidden units ONLY talk to visible units
#Training
Step1
- For each sample
- Anchor visible units
- Sample from hidden units
- No looping!!
Step2
- Now unclamp the visible units and let the entire network evolve several times to generate
$$ \mathbf{S}_{simul}=S_{simul, 1}, S_{simul, 2}, \ldots, S_{simul, M} $$
β Sampling
- For each sample
- Initialize $V_0$ (visible) to training instance value
- Iteratively generate hidden and visible units
- Gradient
β Training
$$ \frac{\partial \log p(v)}{\partial w_{i j}}=<v_{i} h_{j}>^{0}-<v_{i} h_{j}>^{\infty} $$
#A Shortcut: Contrastive Divergence
- Recall: Raise the neighborhood of each target memory
- Sufficient to run one iteration to give a good estimate of the gradient
$$ \frac{\partial \log p(v)}{\partial w_{i j}}=< v_{i} h_{j}>^{0}-<v_{i} h_{j}>^{1} $$