Overview of Contrastive Divergence (CD) and examples of algorithms and implementations

Overview of Contrastive Divergence (CD)

Contrastive Divergence (CD) is a learning algorithm used primarily for training Restricted Boltzmann Machines (RBMs), which are generative models for modelling the probability distribution of data, and CD is an efficient method for learning its parameters. An overview of CD is given below.

1. restricted Boltzmann machines (RBM):

RBMs are probabilistic graphical models and consist of an observed layer (visible layer) and a hidden layer (hidden layer). The visible layer represents the observed data and the hidden layer is used to learn the features of the data, with total coupling between the two layers but no coupling within the layers.

2. Contrastive Divergence (CD):

– Purpose: The purpose of CD would be to optimise the parameters (weights and biases) of the RBM to learn a model that fits the data distribution. Specifically, the goal is to ensure that the distribution of data produced by the model is close to the distribution of the actual data.

– Basic idea: the CD minimises the differences between the data and the samples generated by the model in order to update the model parameters. To achieve this, stochastic gradient descent (SGD) is used.

3. procedure for CD:

1. initialisation: feed the visible layer with real data and probabilistically update the state of the hidden layer (sampling).

2. sampling: sampling the state of the visible layer again using samples generated from the hidden layer. This process is repeated several times to reconstruct (or generate) the data.

3. gradient calculation: the difference between the actual data and the generated data is calculated and the gradients of the parameters are calculated based on this difference.

4. parameter updating: using the gradients, the parameters of the model are updated. This ensures that the model is closer to the data distribution.

4. features and benefits:

Efficient training: CD is computationally less expensive than calculating the full log-likelihood, and therefore allows for efficient training of RBMs.
Simplicity of sampling: sampling is relatively simple in CD and does not require complex sampling algorithms such as Gibbs sampling.

Algorithms related to Contrastive Divergence (CD)

Several algorithms related to Contrastive Divergence (CD) exist. These are mainly related to training and generative models of RBMs (Restricted Boltzmann Machines), which are used as improvements or variants of CD. Typical algorithms are described below.

1. Gibbs Sampling:

– Abstract: Gibbs sampling is a sampling technique that is a prerequisite for CD. It is used in the training of probabilistic models such as Boltzmann machines and RBMs to update the state of the model by alternately sampling the states of the hidden and visible layers.

– Improvement: CD uses some of the Gibbs sampling to generate samples from the model for training.

2. Persistent Contrastive Divergence (PCD):

– Abstract: PCD is a variant of CD designed to make sampling more efficient; in PCD, samples are stored and used persistently during training, resulting in more stable parameter updates.

– Improvement: in regular CD, each new sampling is initialised, whereas in PCD the sample state is preserved and improves as the training progresses.

3. fast persistent contrastive divergence (FPCD):

– Abstract: FPCD is a more computationally efficient version of PCD. It aims to reduce the computational load, especially for large datasets and high-dimensional data.

– Improvements: faster convergence than PCD and more efficient training on large datasets.

4. Improved Contrastive Divergence (ICD):

– Abstract: ICD is an algorithm designed to improve the sampling accuracy of CD. In particular, it provides better parameter updates by increasing sampling accuracy.

– Improvements: by increasing the number of sampling times (K) of the CD, the performance of the model is improved, but with improvements to cope with the increased computational costs.

5. cyclic contrastive divergence (CCL):

– Abstract: CCL is a method that speeds up the convergence of a model by repeating the training process of CD in cyclic fashion. In particular, it aims to improve the stability and efficiency of training in multi-layer RBMs.

– Improvements: cyclic sampling improves training stability and convergence speed.

6. hybrid contrastive divergence (HCD):

– Abstract: HCD aims to combine CD with other learning algorithms (e.g. variational inference) to improve the performance of the model. By combining different methods, more accurate learning can be achieved.

– Improvements: it is a hybrid approach that incorporates the advantages of other algorithms to compensate for the limitations of CD.

7. marginalised contrastive divergence (MCD):

– Abstract: MCD uses marginalisation to provide more accurate parameter updates when sampling for CD. In particular, marginalisation for hidden layer states is used to improve sampling accuracy.

– Improvement: more stable parameter updates are achieved by improving sampling accuracy.

Contrastive Divergence (CD) application examples

Contrastive Divergence (CD) is an algorithm used primarily for training Restricted Boltzmann Machines (RBMs), but has several other applications. Typical applications of CD are described below.

1. training of restricted Boltzmann machines (RBMs):

– Abstract: CD is widely used in RBM training: an RBM is a generative model that models the probability distribution of data, and CD is used to efficiently learn its parameters (weights and biases).

– Applications: feature extraction, dimensionality reduction and data generation using RBM; CD allows the parameters of the model to be learnt to better fit the data distribution.

2. training Deep Belief Networks (DBN):

– Abstract: a DBN is a deep generative model consisting of multiple RBMs stacked in layers; CD is used to train the RBMs in each layer of the DBN.

– Applications: DBNs are used to achieve high performance in tasks such as image and speech recognition, where the RBMs in each layer are efficiently trained by CD to build a strong model as a whole.

3. training of the generative model:

– Abstract: CD applies not only to RBMs but also to other generative models. For example, it may be used for variants of Boltzmann machines and algorithms that are predecessors of generative adversarial networks (GANs).

– Applications: in generative tasks such as image generation, speech generation and text generation, CD is used to streamline the training of models and improve the quality of the generated results.

4. model-based approaches in reinforcement learning:

– Abstract: In reinforcement learning, CD is used to train models of the environment. Here, CD is used to train the parameters of a model that generates state transitions in the environment.

– Applications: CD is used for more efficient learning and simulation using models of the environment, e.g. in robot control and training of self-driving vehicles.

5. speech processing:

– Abstract: CD has also been applied to speech processing tasks. In particular, it is used for feature extraction from speech data and for training generative models.

– Applications: in speech synthesis and speech recognition, CD is used to efficiently learn features of speech data, resulting in more accurate speech processing.

6. document classification and recommendation systems:

– Abstract: CD is used to learn feature extraction and generative models for document classification and recommendation systems.

– Applications: CD is used to train topic models of documents and generative models to model user preferences, thereby enabling more accurate classification and recommendation.

7. anomaly detection:

– Abstract: CD is used to train generative models to detect anomalous data patterns.

– Applications: CD is used in anomaly detection systems in cyber security and manufacturing to model normal data patterns and detect anomalous data.

Examples of Contrastive Divergence (CD) implementations

Examples of Contrastive Divergence (CD) implementations are described. In the following, examples of CD implementations in basic RBM training using Python, TensorFlow and PyTorch are given.

Examples of basic implementations of Contrastive Divergence in Python:

1. definition of a Restricted Boltzmann Machine (RBM): first, a basic definition of an RBM is given. Here, a simple RBM class is defined.

import numpy as np

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 = np.random.randn(visible_units, hidden_units) * 0.1
        self.visible_bias = np.zeros(visible_units)
        self.hidden_bias = np.zeros(hidden_units)

    def sigmoid(self, x):
        return 1 / (1 + np.exp(-x))

    def sample_hidden(self, visible_data):
        activation = np.dot(visible_data, self.weights) + self.hidden_bias
        return self.sigmoid(activation)

    def sample_visible(self, hidden_data):
        activation = np.dot(hidden_data, self.weights.T) + self.visible_bias
        return self.sigmoid(activation)

    def contrastive_divergence(self, visible_data, k=1):
        # Positive phase
        hidden_probs = self.sample_hidden(visible_data)
        hidden_states = (hidden_probs > np.random.rand(*hidden_probs.shape)).astype(np.float32)

        # Negative phase
        for _ in range(k):
            visible_probs = self.sample_visible(hidden_states)
            hidden_probs = self.sample_hidden(visible_probs)
            hidden_states = (hidden_probs > np.random.rand(*hidden_probs.shape)).astype(np.float32)

        # Update weights and biases
        positive_gradient = np.dot(visible_data.T, hidden_probs)
        negative_gradient = np.dot(visible_probs.T, hidden_probs)
        
        self.weights += self.learning_rate * (positive_gradient - negative_gradient) / visible_data.shape[0]
        self.visible_bias += self.learning_rate * np.mean(visible_data - visible_probs, axis=0)
        self.hidden_bias += self.learning_rate * np.mean(hidden_probs - hidden_probs, axis=0)

2. training and testing: the RBM is then trained and tested.

import numpy as np

# Data generation (e.g. using random data)
data = np.random.rand(100, 6)  # 100 samples, 6 dimensional visible layers.

# Instantiation of RBMs.
rbm = RBM(visible_units=6, hidden_units=3, learning_rate=0.1)

# training
epochs = 10
for epoch in range(epochs):
    rbm.contrastive_divergence(data, k=1)
    print(f'Epoch {epoch + 1}/{epochs} completed.')

Example implementation of Contrastive Divergence with PyTorch: This is an example implementation of RBM using PyTorch.

1. definition of RBM:.

import torch
import torch.nn as nn
import torch.optim as optim

class RBM(nn.Module):
    def __init__(self, visible_units, hidden_units):
        super(RBM, self).__init__()
        self.visible_units = visible_units
        self.hidden_units = hidden_units
        
        self.W = nn.Parameter(torch.randn(visible_units, hidden_units) * 0.1)
        self.visible_bias = nn.Parameter(torch.zeros(visible_units))
        self.hidden_bias = nn.Parameter(torch.zeros(hidden_units))

    def sample_hidden(self, visible_data):
        activation = torch.mm(visible_data, self.W) + self.hidden_bias
        return torch.sigmoid(activation)

    def sample_visible(self, hidden_data):
        activation = torch.mm(hidden_data, self.W.t()) + self.visible_bias
        return torch.sigmoid(activation)

    def contrastive_divergence(self, visible_data, k=1):
        # Positive phase
        hidden_probs = self.sample_hidden(visible_data)
        hidden_states = (hidden_probs > torch.rand_like(hidden_probs)).float()

        # Negative phase
        for _ in range(k):
            visible_probs = self.sample_visible(hidden_states)
            hidden_probs = self.sample_hidden(visible_probs)
            hidden_states = (hidden_probs > torch.rand_like(hidden_probs)).float()

        # Update weights and biases
        positive_gradient = torch.mm(visible_data.t(), hidden_probs)
        negative_gradient = torch.mm(visible_probs.t(), hidden_probs)
        
        self.W.data += 0.1 * (positive_gradient - negative_gradient) / visible_data.size(0)
        self.visible_bias.data += 0.1 * torch.mean(visible_data - visible_probs, dim=0)
        self.hidden_bias.data += 0.1 * torch.mean(hidden_probs - hidden_probs, dim=0)

2. training and testing:

import torch
from torch.utils.data import DataLoader, TensorDataset

# Data generation (random data)
data = torch.rand(100, 6)  # 100 samples, 6 dimensional visible layers.

# data loader
dataset = TensorDataset(data)
dataloader = DataLoader(dataset, batch_size=10, shuffle=True)

# Instantiation of RBMs.
rbm = RBM(visible_units=6, hidden_units=3)

# training
epochs = 10
for epoch in range(epochs):
    for batch in dataloader:
        visible_data, = batch
        rbm.contrastive_divergence(visible_data, k=1)
    print(f'Epoch {epoch + 1}/{epochs} completed.')

Contrastive Divergence (CD) challenges and measures to address them

Contrastive Divergence (CD) is a widely used technique in Restricted Boltzmann Machine (RBM) training, but several challenges exist. The main challenges of CD and measures to address them are described below.

1. sampling accuracy issues:

Challenges:
In CD, when generating samples from training data, poor sampling accuracy leads to inaccurate parameter updates. Particularly in the early stages of training, sample convergence is slow, leading to inaccurate CD estimation.

Solution:
– Increase the number of samplings (K): increasing the number of times the CD is sampled can improve the accuracy of the samples. However, it also increases the computational cost.
– Persistent Contrastive Divergence (PCD): PCD improves sampling accuracy by sampling continuously and reusing each sample.
– Improved Contrastive Divergence (ICD): in ICD, the approach is to increase sampling accuracy by increasing K, but this increases the computational load.

2. training convergence issues:

Challenges:
CD takes a long time for training to converge, especially if the sample quality is low. Slow convergence may result in inefficient training of the model.

Solution:
– Adjust the learning rate: training convergence can be improved by adjusting the learning rate appropriately. It is important to strike a balance, as too high a learning rate will result in oscillations and too low a rate will result in slower convergence.
– Introducing regularisation: training stability can be improved by introducing L1 or L2 regularisation to prevent over-learning.

3. increased computational costs:

Challenge:
Increasing the number of CD sampling times significantly increases the computational cost. Computational resources can be enormous, especially for large datasets and high dimensional data.

Solution:
– Mini-batch training: dividing the dataset into mini-batches and training each mini-batch separately can reduce computational costs.
– Use of parallel processing and GPUs: the use of parallel processing and GPUs can significantly increase computational speed.

4. setting up the energy function:

Challenge:
Inadequate setting of the energy function of the RBM leads to poor CD performance. In particular, the definition and parameters of the energy function have a significant impact on the performance of the model.

Solution:
– Model selection and adjustment: setting the energy function appropriately and adjusting the parameters of the model will improve CD performance. In particular, it is important to adjust the energy function to fit the data.

5. scaling of training data:

Challenge:
Inadequate scaling of training data leads to inefficient training of models. In particular, different data scaling can make training difficult.

Solution:
– Data pre-processing: proper scaling and normalisation of the data can improve training efficiency.

Reference information and reference books

References and books on Contrastive Divergence (CD) include literature and resources mainly related to the field of machine learning, in particular Restricted Boltzmann Machines (RBM) and generative modelling. They are described below.

Reference books:

1. “Deep Learning” by Ian Goodfellow, Yoshua Bengio, and Aaron Courville

2. “Pattern Recognition and Machine Learning” by Christopher M. Bishop

3. “Machine Learning: A Probabilistic Perspective” by Kevin P. Murphy

4. “The Deep Learning Revolution” by Terrence J. Sejnowski

References and articles:

1. “Training Products of Experts by Minimizing Contrastive Divergence“** by Geoffrey E. Hinton

2. “A Fast Learning Algorithm for Deep Belief Nets” by Geoffrey Hinton, Simon Osindero, and Yee-Whye Teh

3. “Efficient Training of Restricted Boltzmann Machines” by Geoffrey E. Hinton

4. “On the Properties of Contrastive Divergence Learning” by Ian Goodfellow

5. “Learning Deep Architectures for AI“** by Yoshua Bengio