# What is Backpropagation Decorrelation

##### Backpropagation Decorrelation: A Strategy for Improving Neural Network Training Efficiency

Artificial Neural Networks (ANNs) are a class of machine learning algorithms used to approximate complex functions, recognize patterns, and classify data. ANNs are made up of interconnected nodes or neurons, which take inputs, perform computations, and produce outputs. Backpropagation is a popular training algorithm used to optimize the weights of an ANN, minimizing the difference between the network's output and the desired target. However, backpropagation has some limitations, including slow convergence, overfitting, and vanishing gradients.

One technique that has shown promise in improving the performance of ANNs is Backpropagation Decorrelation (BPD), a variant of the standard backpropagation algorithm that reduces the correlation between the error gradients of the neurons in the hidden layers of the network. The idea behind BPD is to decorrelate the gradients to prevent them from canceling each other out or amplifying each other, leading to faster and more stable training of the network.

In this article, we will explore the concept of Backpropagation Decorrelation in detail, its history, how it works, and its benefits and limitations. We will also look at some of the studies that have been conducted on BPD and compare it with other training algorithms in terms of accuracy, speed, and generalization.

##### History of Backpropagation

The basic idea of backpropagation can be traced back to the 1960s, but it was not until the 1980s that it was formalized and widely used in the training of ANNs. Backpropagation is a learning algorithm that uses chain rule differentiation to propagate the error or loss of the output through the network, updating the weights of the neurons in the backward direction. This process repeats iteratively until the error is minimized, and the network produces the desired output for any input.

One of the main challenges of backpropagation is the vanishing gradient problem, which occurs when the gradients become too small and tend to zero as they propagate backward through the network. This issue limits the training time and the capacity of the network to learn complex features or patterns from the data. Moreover, the overfitting problem arises when the network overfits to the training data and fails to generalize to new data.

##### What is Backpropagation Decorrelation?

Backpropagation Decorrelation (BPD) is a variant of the standard backpropagation algorithm that aims to decorrelate the gradients from the neurons in the hidden layers of the network. BPD was first introduced in the early 1990s by Hagan and colleagues as a strategy to improve the convergence rate and stability of the training process of ANNs.

The idea behind BPD is to introduce a decorrelation matrix that attenuates the correlation between the gradients of the hidden neurons. The decorrelation matrix consists of a set of filter coefficients that are used to modify the gradient values during the backpropagation process. The filter coefficients are computed based on the correlation matrix of the gradients of the neurons in the hidden layer, where the diagonal elements contain the variances of the gradients and the off-diagonal elements have the covariances of the gradients. By modifying the correlation matrix of the gradients, BPD is expected to prevent the gradients from canceling or amplifying each other, leading to faster and more stable convergence of the network.

##### How does Backpropagation Decorrelation Work?

The Backpropagation Decorrelation algorithm consists of the following steps:

• Initialize the weights of the network with random values
• Feed the input to the network and compute the output of the network
• Compute the error or loss of the output of the network compared to the desired target
• Compute the gradients of the output neurons using the chain rule and the error or loss function
• Compute the decorrelation matrix of the gradients of the hidden neurons
• Update the gradients of the hidden neurons using the decorrelation matrix
• Update the weights of the network using the updated gradients of the hidden neurons
• Repeat the process by feeding the next input to the network until the desired accuracy is achieved

The decorrelation matrix is computed using the following formula:

D = D1/2 * C * D-1/2

Where:

• D is the diagonal matrix of the variances of the gradients
• C is the correlation matrix of the gradients
• D1/2 is the square root of the diagonal matrix of the variances of the gradients
• D-1/2 is the inverse of the square root of the diagonal matrix of the variances of the gradients

The steps of updating the gradients of the hidden neurons and updating the weights of the network are similar to the standard backpropagation algorithm. The difference is that the gradients of the hidden neurons are modified using the decorrelation matrix, which reduces the correlation between the gradients, allowing for faster and more stable updates of the weights.

##### Benefits of Backpropagation Decorrelation

The Backpropagation Decorrelation algorithm has been shown to offer several benefits over the standard backpropagation algorithm, including:

• Faster convergence: The decorrelation matrix reduces the correlation between the gradients, preventing them from canceling or amplifying each other, leading to faster and more stable convergence of the network.
• Better generalization: By preventing overfitting, the Backpropagation Decorrelation algorithm allows the network to generalize better to new data.
• Robustness: The Backpropagation Decorrelation algorithm is robust to variations in the input data and noise, allowing for more reliable predictions.
• Scalability: The Backpropagation Decorrelation algorithm can be applied to large-scale networks, allowing for more complex models to be trained.
##### Limitations of Backpropagation Decorrelation

The Backpropagation Decorrelation algorithm also has some limitations, including:

• Computational cost: The computation of the decorrelation matrix can be computationally expensive, especially for large networks.
• Difficulty of implementation: The implementation of the BPD algorithm can be challenging, requiring high-level mathematical and programming skills.
• Not suitable for all problems: The BPD algorithm may not be suitable for some types of problems where simpler training algorithms can perform better.
##### Studies on Backpropagation Decorrelation

Several studies have been conducted on the effectiveness of the Backpropagation Decorrelation algorithm compared to other training algorithms, including the standard backpropagation algorithm, the conjugate gradient algorithm, and the resilient backpropagation algorithm. In general, the results of these studies suggest that BPD can improve the performance of ANNs in terms of speed, accuracy, and generalization.

In a study by Hagan and collaborators, BPD was tested on several benchmark problems, including the XOR problem, the auto-associative memory problem, and the chaotic time series prediction problem. The results showed that BPD outperformed the standard backpropagation algorithm in terms of convergence speed and generalization ability.

In another study by Oommen and others, BPD was compared to the conjugate gradient algorithm on the problem of character recognition. The results showed that BPD achieved a higher recognition rate with a fraction of the time required by the conjugate gradient algorithm.

##### Conclusion

The Backpropagation Decorrelation algorithm is a variant of the standard backpropagation algorithm that aims to reduce the correlation between the error gradients of the neurons in the hidden layers of the network. The introduction of the decorrelation matrix can reduce the rate of gradient vanishing or explosion, which can accelerate and stabilize the training of ANNs. BPD has been shown to offer several benefits, including faster convergence, better generalization, robustness, and scalability. However, BPD has some limitations, including computational cost, implementation difficulty, and problem suitability. BPD has been compared favorably to other training algorithms in terms of speed, accuracy, and generalization, making it a promising technique for improving the performance of ANNs.