Deep Learning PyTorch Course, Restricted Boltzmann Machine

The Restricted Boltzmann Machine (RBM) is a type of unsupervised learning algorithm, also known as a generative model. RBMs can effectively learn from large amounts of input data and are utilized in various fields. This document aims to provide a deep understanding of the fundamental principles of RBMs, how to implement them in Python, and examples using the PyTorch framework.

1. Understanding Restricted Boltzmann Machines (RBM)

RBM is a model that originated from statistical physics, based on the concept of ‘Boltzmann Machines’. An RBM consists of two types of nodes: visible nodes and hidden nodes. There are connections between these two nodes, but there are no connections between the hidden nodes, resulting in a restricted structure. This structure allows for more efficient learning in RBMs.

1.1 Structure of RBM

RBM consists of the following components:

Visible Units: Represents the characteristics of the input data.
Hidden Units: Learns the underlying characteristics of the data.
Weights: Represents the strength of the connections between visible and hidden nodes.
Bias: Represents the bias values for each node.

1.2 Energy Function

The learning of RBM occurs through the process of minimizing the Energy Function. The energy function is defined based on the states of the visible and hidden nodes as follows:

E(v, h) = -∑ v_i * b_i - ∑ h_j * c_j - ∑ v_i * h_j * w_ij

Here, \( v \) represents the visible node, \( h \) represents the hidden node, \( b \) is the bias of the visible node, \( c \) is the bias of the hidden node, and \( w \) is the weight.

2. Learning Process of Restricted Boltzmann Machines

The learning process of RBM proceeds as follows:

Initialize the visible nodes from the dataset.
Calculate the probabilities of the hidden nodes.
Sample the hidden nodes.
Calculate the probabilities of the new visible nodes through the reconstruction of visible nodes.
Calculate the probabilities of the new hidden nodes through the reconstruction of hidden nodes.
Update weights and biases.

2.1 Contrastive Divergence Algorithm

The learning of RBM occurs through the Contrastive Divergence (CD) algorithm. CD consists of two main phases:

Positive Phase: Identify the activations of the hidden nodes from the input data and update the weights based on this value.
Pseudo Negative Phase: Reconstruct visible nodes from the sampled hidden nodes and then sample hidden nodes again to update weights in a way that reduces similarity.

3. Implementing RBM with PyTorch

This section explains how to implement RBM using PyTorch. First, let’s install the required libraries and prepare the dataset.

3.1 Install Libraries and Prepare Dataset

!pip install torch torchvision

We will use the MNIST dataset to train the RBM. This dataset consists of handwritten digit images.

import torch
from torchvision import datasets, transforms

# Downloading and transforming MNIST dataset
transform = transforms.Compose([transforms.ToTensor(), transforms.Lambda(lambda x: x.view(-1))])
mnist = datasets.MNIST(root='./data', train=True, download=True, transform=transform)
train_loader = torch.utils.data.DataLoader(dataset=mnist, batch_size=64, shuffle=True)

3.2 Define RBM Class

Now let’s define the RBM class. The class should include methods for weight initialization, weight updates, and training.

class RBM:
    def __init__(self, visible_units, hidden_units, learning_rate=0.1):
        self.visible_units = visible_units
        self.hidden_units = hidden_units
        self.learning_rate = learning_rate
        self.weights = torch.randn(visible_units, hidden_units) * 0.1
        self.visible_bias = torch.zeros(visible_units)
        self.hidden_bias = torch.zeros(hidden_units)

    def sample_hidden(self, visible):
        activation = torch.mm(visible, self.weights) + self.hidden_bias
        probabilities = torch.sigmoid(activation)
        return probabilities, torch.bernoulli(probabilities)

    def sample_visible(self, hidden):
        activation = torch.mm(hidden, self.weights.t()) + self.visible_bias
        probabilities = torch.sigmoid(activation)
        return probabilities, torch.bernoulli(probabilities)

    def train(self, train_loader, num_epochs=10):
        for epoch in range(num_epochs):
            for data, _ in train_loader:
                # Sample visible nodes
                v0 = data
                h0, h0_sample = self.sample_hidden(v0)

                # Negative phase
                v1, v1_sample = self.sample_visible(h0_sample)
                h1, _ = self.sample_hidden(v1_sample)

                # Update weights
                self.weights += self.learning_rate * (torch.mm(v0.t(), h0) - torch.mm(v1.t(), h1)) / v0.size(0)
                self.visible_bias += self.learning_rate * (v0 - v1).mean(0)
                self.hidden_bias += self.learning_rate * (h0 - h1).mean(0)

                print('Epoch: {} - Loss: {:.4f}'.format(epoch, torch.mean((v0 - v1) ** 2).item()))

3.3 Perform RBM Training

Now let’s train the model using the defined RBM class.

visible_units = 784  # For MNIST, 28x28 pixels
hidden_units = 256    # Number of hidden nodes
rbm = RBM(visible_units, hidden_units)
rbm.train(train_loader, num_epochs=10)

4. Results and Interpretation

As training progresses, the loss value is printed for each epoch. The loss value indicates how similar the reconstruction of visible nodes is to the hidden state, so a decrease in the loss value signifies an improvement in model performance. Notably, the Boltzmann Machine forms the basis of many other algorithms and is combined with various deep learning models.

5. Conclusion

In this post, we addressed the concept of restricted Boltzmann machines, the learning process, and a practical implementation example using PyTorch. RBM is a highly effective tool for learning the underlying structure of data. Nevertheless, it is primarily used for pre-training or in combination with other architectures in current deep learning frameworks. Further research on various generative models is expected in the future.