Restricted Boltzmann machines RBM

RBM §

Architecture §

Is a neural network architecture on a complete bipartite graph. The two layer are called visible and hiddent, in total we have

$$N=N_v+N_h$$

total neurons. Indichiamo lo stato dei neuroni visibili con $v\in \{-1,1\}$ e $h\in \{-1,1\}$ for the hidden neurons.

The hamiltonian is the usual of neural networks models

$${\mathcal{H}}^{RBM}=-\sum _{ij}{J}_{ij}{s}_i{s}_j-\sum _i{\theta }_i{s}_i$$

Supervised learning §

The goal is to learn the joint distribution $q(x,y)$ where $x$ is the input and $y$ the label.

The idea is to use the visible neurons $v$ to encode $x$, and the hidden to encode the label (class).

We need to find parameters $J,\theta$ such that the equilibrium distribution of the network is as close as possible to the data

$$P_{J,\theta }(v,h)\sim q(v,h)$$

Then we can use the network to predict the correct label of unseen data, the best guess

$$h^{\ast }=\mathbb{E}[h|v=v^{\ast }]$$

Unsupervides learning §

Input: some data $D=\{x_i\}$. Task: learn the data distribution $q(x)$.

Again we encode our data $x$ in the visible layer neurons $v$. So we need to find the best parameters $J,\theta$ such that

$$P_{J,\theta }(v)\sim q(v)$$

Once the network is trained, it can be used to generate new data, denoise, and to reconstruct.

Learning §

The task is to learn a distribution. The obvious choice is to define a metric (loss) beetween the network distribution and the goal and try to minimize that using gradient descent.

We will use the KL-divergence $D(q\Vert P_{J,\theta })$. Using gradient descent we obtain update rules for the parameters:

$$\Delta J_{ij}=-\eta {\partial }_{J_{ij}}D(q\Vert P_{J,\theta })$$ $$\Delta {\theta }_i=-\eta {\partial }_{{\theta }_i}D(q\Vert P_{J,\theta })$$

thus we need to compute this derivatives. Let’s consider the RBM in the unsupervised learning setting.

$$P_{J,\theta }(v)=\frac{{\sum }_h\exp [-\beta \mathcal{H}(v,h)]}{Z}=\frac{Z(v)}{Z}$$

since we only care about visibile neurons, we had to marginalize.

$${\partial }_{\lambda }D(q\Vert P_{\lambda })=-\sum _{v}q(v){\partial }_{\lambda }[\ln q(v)-\ln P_{\lambda }(q)]$$ $$=-\sum _{v}q(v){\partial }_{\lambda }[-\ln Z(q)+\ln Z]$$ $$=\sum _{v}q(v)\left[\frac{{\partial }_{\lambda }Z(v)}{Z(v)}-\frac{{\partial }_{\lambda }Z}{Z}\right]$$

calcoliamo le derivate che compaiono al numeratore

$${\partial }_{\lambda }Z(v)=-\beta \sum _h{\partial }_{\lambda }H(v,h)e^{-\beta H}$$ $${\partial }_{\lambda }Z=-\beta \sum _{v,h}{\partial }_{\lambda }H(v,h)e^{-\beta H}$$

rimettendo insieme…

$$\Delta J_{ij}=-\frac{\eta \beta }{\ln 2}\left[⟨s_i{s}_j{⟩}_{free}-⟨s_i{s}_j{⟩}_{clamped}\right]$$ $$\Delta {\theta }_i=-\frac{\eta \beta }{\ln 2}\left[⟨s_i{⟩}_{free}-⟨s_i{⟩}_{clamped}\right]$$

per la procedura di training devo far termalizzare la rete… Computazionalmente non trattabile per reti non miniuscole.

Teniche per approssimare:

• MCMC Monte Carlo Markov Chain, più passi faccio meglio è (empiricamente ne basta anche uno solo!)
• Approssimazione mean field usando le equazioni di autoconsistenza.