Boltzmann Machine Example

This is a simple example of a Boltzmann Machine using Python and the PyTorch library.

Boltzmann Machine Overview

A Boltzmann Machine is a type of generative stochastic artificial neural network that can learn a probability distribution over its set of inputs. It consists of visible and hidden units with symmetric connections between them. Training a Boltzmann Machine involves finding the parameters that maximize the likelihood of the training data. Boltzmann Machines are known for their ability to capture complex dependencies in data and have been used in various applications, including dimensionality reduction and feature learning.

Key concepts of Boltzmann Machines:

• Visible and Hidden Units: Units that represent the input data and are used for capturing complex relationships.
• Energy Function: A function that assigns an energy value to each configuration of visible and hidden units.
• Gibbs Sampling: A Markov Chain Monte Carlo method used for sampling configurations from the probability distribution defined by the Boltzmann Machine.
• Learning Rule: A rule for adjusting the weights of connections to improve the model's likelihood of the training data.

Boltzmann Machines have been extended and modified over the years, with variations such as Restricted Boltzmann Machines (RBMs) being widely used in practice.

Python Source Code:

# Import necessary libraries
import torch
import torch.nn as nn
import torch.optim as optim
import numpy as np

# Define a simple Boltzmann Machine model
class BoltzmannMachine(nn.Module):
    def __init__(self, num_visible, num_hidden):
        super(BoltzmannMachine, self).__init__()
        self.weights = nn.Parameter(torch.randn(num_visible, num_hidden))

    def forward(self, visible_states):
        hidden_activations = torch.sigmoid(torch.matmul(visible_states, self.weights))
        return hidden_activations

# Generate synthetic data
np.random.seed(42)
num_samples = 100
num_visible_units = 5
num_hidden_units = 3
visible_data = torch.tensor(np.random.randint(0, 2, size=(num_samples, num_visible_units)), dtype=torch.float32)

# Instantiate the Boltzmann Machine model
boltzmann_machine = BoltzmannMachine(num_visible_units, num_hidden_units)

# Define the optimizer and loss function
optimizer = optim.SGD(boltzmann_machine.parameters(), lr=0.01)
criterion = nn.BCELoss()

# Training loop
num_epochs = 1000
for epoch in range(num_epochs):
    # Forward pass
    hidden_activations = boltzmann_machine(visible_data)

    # Compute the loss (negative log likelihood)
    loss = criterion(hidden_activations, visible_data)

    # Backward pass and optimization
    optimizer.zero_grad()
    loss.backward()
    optimizer.step()

    # Print the loss every 100 epochs
    if (epoch + 1) % 100 == 0:
        print(f'Epoch [{epoch + 1}/{num_epochs}], Loss: {loss.item():.4f}')

# Print the learned weights
learned_weights = boltzmann_machine.weights.data.numpy()
print(f'Learned Weights:\n{learned_weights}')

Explanation:

• Import Libraries: Import necessary Python libraries, including PyTorch for building and training the Boltzmann Machine.
• Define Boltzmann Machine: Define a simple Boltzmann Machine model as a subclass of the PyTorch nn.Module.
• Generate Synthetic Data: Generate synthetic binary data for training the Boltzmann Machine.
• Instantiate Model: Instantiate the Boltzmann Machine model.
• Define Optimizer and Loss: Define the stochastic gradient descent (SGD) optimizer and binary cross-entropy loss.
• Training Loop: Train the Boltzmann Machine using a simple optimization loop.
• Print Learned Weights: Print the learned weights of the Boltzmann Machine after training.