Deep Learning – Backpropagation Algorithm Basics

Backpropagation Algorithm – An important mathematical tool for making better and higher-accuracy predictions in machine learning. This algorithm uses supervised learning methods for training artificial neural networks.

Backpropagation, short for “backward propagation of errors,” is a fundamental algorithm in training neural networks. It computes the gradient of the loss function with respect to the weights of the network’s layers, allowing for the optimization of these weights through gradient descent or its variants.This algorithm, rooted in linear algebraic operations, plays a pivotal role in optimizing the error function by leveraging its intelligence to iteratively adjust weights and minimize errors. Through this iterative process, backpropagation refines the model’s parameters, enhancing its ability to accurately capture underlying patterns and make informed predictions, thereby driving effective learning and optimization in neural networks.

In training multi-layer perceptrons, the core objective lies in computing derivatives of the error function or gradient descent concerning weights, a process facilitated by the backpropagation algorithm. In this post, you and I will focus on backpropagation and basic details around it on a high level in simple English.

According to AILabPage, backpropagation is a method utilized in artificial neural networks to calculate the gradient required for adjusting the network’s weights. It’s a technique that relies on supervised learning to determine the appropriate modifications to the weights within the system.

Artificial Neural Networks

AILabPage defines artificial neural networks (ANNs) as “Biologically inspired computing code with a number of simple, highly interconnected processing elements for simulating (only an attempt) human brain working and processing information models.” It’s way different than a computer program, though. There are several kinds of Neural Networks in deep learning. Neural networks consist of input and output layers and at least one hidden layer.

Neural Network Algorithms are like tech-savvy problem-solvers that use radial basis functions as their secret sauce. Think of them as super-smart tools that can be strategically applied to tackle different challenges.

There are other models of neural networks out there, each with its own bag of tricks. If you’re curious about how these brainy algorithms operate and solve real mathematical problems, keep reading this post. It’s like peeking behind the curtain to understand their inner workings and see how they come to the rescue in the tech world.

What is the Backpropagation Algorithm

Backpropagation serves as a foundational concept in training neural networks, pivotal for refining parameters through supervised learning. It enables networks to iteratively adjust weights based on prediction errors, enhancing accuracy and performance.

Backpropagation is essential in neural networks for optimizing parameters and minimizing errors, ultimately leading to more accurate predictions and improved model performance. Its significance lies in its ability to iteratively refine weights based on prediction errors, driving effective learning and optimization in neural networks.

As per wiki – “Backpropagation is a method used in artificial neural networks to calculate a gradient that is needed in the calculation of the weights to be used in the network.”

Explanation

Backpropagation is a fundamental concept in training neural networks, enabling the iterative adjustment of weights based on prediction errors. It plays a vital role in supervised learning by propagating errors backward through the network, allowing for the optimization of parameters and the refinement of predictions. Through this process, backpropagation enables neural networks to learn from data and iteratively improve their performance over time.

Importance in Neural Networks

Backpropagation is crucial for optimizing the parameters of neural networks, leading to more accurate predictions and better performance. By iteratively adjusting weights based on prediction errors, backpropagation enables networks to learn from data and improve their ability to capture underlying patterns.

Basics of Backpropagation

Backpropagation lies at the heart of training neural networks, particularly in supervised learning scenarios. It operates by minimizing the error between predicted and actual outputs, achieved through iterative adjustments to the model’s parameters.

Central to this process is the optimization algorithm, often gradient descent, which systematically updates these parameters to reduce the error. By grasping the fundamentals of supervised learning, error minimization, and optimization algorithms, one can unravel the essence of backpropagation in the realm of neural network training.

• Supervised Learning: In supervised learning, the algorithm is provided with labeled training data, where each data point is paired with the corresponding correct output. The algorithm learns from these examples to make predictions on unseen data, thereby inferring the relationship between inputs and outputs. This paradigm enables the algorithm to generalize its predictions to new, unseen instances by learning patterns from the labeled data.

• Error Minimization: Backpropagation strives to minimize the discrepancy between the predicted outputs generated by the model and the actual outputs observed in the training data. By iteratively adjusting the model’s parameters, such as weights and biases, based on the computed errors, backpropagation fine-tunes the model to reduce prediction errors. This process of error minimization is crucial for enhancing the model’s accuracy and improving its ability to make reliable predictions on unseen data.
• Optimization Algorithm: Gradient descent serves as the optimization algorithm in backpropagation, driving the iterative process of updating the model’s parameters to minimize the prediction error. It operates by calculating the gradient of the error function with respect to each parameter, indicating the direction of steepest descent in the parameter space. By taking small steps in the opposite direction of the gradient, the algorithm gradually converges towards the optimal values of the parameters, leading to a reduction in prediction errors.

This iterative process drives effective learning and enables neural networks to adapt to complex datasets, making it a cornerstone in the field of machine learning.

Components of Backpropagation

Backpropagation involves input, hidden, and output layers in neural networks. Inputs enter the input layer, hidden layers process data through weighted sums and activation functions, and the output layer generates predictions. Activation functions introduce non-linearities, while weights and biases adjust to minimize errors during training.

• Input Layer: The input layer of a neural network serves as the entry point for the raw data or features to be processed. Each neuron in the input layer corresponds to a feature, and the values of these neurons represent the input data. The input layer plays a crucial role in transmitting the data to the subsequent layers for further processing.

• Hidden Layers: Hidden layers are intermediate layers within a neural network that perform complex transformations on the input data. These transformations involve weighted sums of the inputs followed by the application of activation functions, which introduce non-linearities into the network’s computations. Hidden layers enable the network to learn hierarchical representations of the input data, extracting relevant features for making accurate predictions.
• Output Layer: The output layer is the final layer of a neural network responsible for producing the network’s predictions or outputs. The number of neurons in the output layer depends on the nature of the prediction task, with each neuron typically corresponding to a different class or target variable. The output layer synthesizes the information processed by the preceding layers to generate the network’s final predictions.
• Activation Functions: Activation functions are mathematical functions applied to the weighted sum of inputs at each neuron in a neural network. These functions introduce non-linearities into the network’s computations, enabling it to model complex relationships in the data. Popular activation functions include the sigmoid function, which maps inputs to values between 0 and 1, the Rectified Linear Unit (ReLU) function, which outputs the input if positive and zero otherwise, and the softmax function, which normalizes the outputs to represent probabilities.
• Weights and Biases: Weights and biases are adjustable parameters in neural networks that govern the strength of connections between neurons and the neuron’s propensity to activate, respectively. During the training process, backpropagation adjusts these parameters to minimize prediction errors by iteratively updating their values based on the computed gradients of the error function. By fine-tuning the weights and biases, the network learns to make more accurate predictions and generalize better to unseen data.

Backpropagation utilizes input, hidden, and output layers in neural networks. Inputs pass through the input layer, hidden layers process data via weighted sums and activation functions, and the output layer generates predictions. Activation functions introduce non-linearities, while weights and biases adjust to minimize training errors.

Understanding the Backpropagation Process

The backpropagation process involves iteratively adjusting neural network weights and biases to minimize prediction errors. It starts with forward propagation, where inputs are processed through layers to produce predictions.

Errors are then calculated and propagated backward through the network to update parameters using gradient descent optimization.

• Forward Pass: During the forward pass, input data is fed into the neural network, traversing through the layers. Each neuron’s activation function processes the input, producing predictions at the output layer.
• Error Calculation: Following the forward pass, the error or loss is computed by comparing the predicted outputs with the actual labels in the training dataset. This quantifies the disparity between the predicted and expected values.

• Backward Pass: In the backward pass, the calculated error is propagated backward through the network. This involves applying the chain rule of calculus to distribute the error contribution of each neuron back through the layers.
• Weight Update: The weight and bias parameters of the neural network are adjusted during the backward pass to minimize the error. This adjustment is achieved by updating the weights in the direction opposite to the gradient of the error with respect to the weights.
• Gradient Descent: Gradient descent is an optimization algorithm used to update the weights iteratively. It involves calculating the gradient of the error function with respect to the weights and adjusting the weights in the direction that minimizes the error. Techniques like learning rates control the size of the weight updates, ensuring convergence to an optimal solution.

Backpropagation iteratively updates neural network parameters to minimize prediction errors. It begins with forward propagation, processing inputs through layers to generate predictions. Errors are calculated and propagated backward to adjust parameters using gradient descent optimization, enhancing the model’s predictive accuracy over successive iterations.

Backpropagation: Optimizing Neural Networks

This algorithm is used for finding the minimum value error function in the neural network during the training model stage. The core idea of backpropagation is to find out what impact it would have on the overall cost of the neural network if we played around with weights.

Weights are used to minimise the error function, so where it minimises that point is considered as the solution to our learning problem. To understand this better we can take an example below.

Let’s take below table to demonstrate our weights importance

So now if we start playing with weights we will see the real game. With weight as 4, we will have below output. Point to note in below table the difference between the actual and the desired output:

Now let’s compare the two tables above. The weighted output has a huge error margin of 6, 18, and 54 for 3 input values, and only one value is correct. When we do the square, we will notice a further increase. Let’s change our weight value to 3 from 4. The error margin reduces to 0, 3, 9, and 27, but it’s still not optimal. So one thing is clear: our approach is in the correct direction, i.e., reducing weights is the correct decision here. Let’s decrease it further to 2. With a weight value of 2, our desired output value is on the spot with zero error margin.

What was done here 

• With initial random value for “Weight” (W), we actually used forward propagate method. This is actually the first step in any neural networks. Forward propagate helps to get the output to be compared with the desired output real value to get the error.
• We got our error values as 0, 6, 18 and 54 which were really not appealing values for obvious reasons. To reduce the error, the backwards propagation method was used i.e. reduced the value of ‘W’.
• After reducing there was still an error (0, 3, 9, and 27) though it was decreased, it was not our desired result. Learning was made as “Reducing the value of “W” is in the correct direction and any increment will never yield desired output“.
• Again we propagated backwards and reduced the value of  ‘W’ to 2 from 3.

The whole idea of forward/backward propagation and playing with weights is to reduce, minimize, or optimize the error value. After a couple of iterations, the network learns which side of the number scale it needs to move until the error gets minimized. There is a sort of breakpoint where any further update to the weight results in an increase in error and an indication to stop; take it as a final weight value.

Challenges in Backpropagation

Backpropagation faces hurdles like the vanishing and exploding gradient problems, hindering deep neural network training. Vanishing gradients diminish updates to early layers, while exploding gradients cause instability. Overfitting occurs when models memorize data, affecting generalization.

• Vanishing Gradient Problem: The vanishing gradient problem occurs when gradients become extremely small during backpropagation, hindering the update of early layers’ weights. This phenomenon leads to slow learning and prevents deep neural networks from effectively capturing complex relationships in the data.

• Exploding Gradient Problem: Conversely, the exploding gradient problem arises when gradients grow exponentially during backpropagation, causing weight updates to become too large. This instability disrupts the training process, leading to divergent behavior and making it challenging to converge to an optimal solution.
• Overfitting: Overfitting happens when a model learns to fit the training data too closely, capturing noise or irrelevant patterns that do not generalize well to unseen data. This results in poor performance on new data and indicates that the model has memorized the training set rather than learning underlying patterns.

It confronts obstacles such as gradient vanishing/exploding and overfitting, impeding effective training and generalization in deep neural networks. Addressing these challenges is vital for improving model performance and advancing artificial intelligence applications.

Backpropagation algorithm Step by Step

In neural networks, learning about how to make neurons intelligent depends on the activation process, i.e., when to get activated and when to remain mum. The human brain is not designed to accommodate or allow any of the backpropagation principles. 

The basic steps in the artificial neural network for  backpropagation used for calculating derivatives in a much faster manner:

• Set inputs and desired outputs – Choose inputs and set the desired outputs
• Set random weights – This is needed for manipulating the output values.
• Calculating the error – Calculating error helps to check how far is the required output from the actual. How good/bad is the model output from the actual output.
• Minimising the error – Now at this step, we need to check the error rate for its minimization
• Updating the parameters – In case the error has a huge gap then, change/update the parameters i.e. weights and biases to reduce it. Repeating this check and update process until error gets minimised is the motive here.
• Model readiness for a prediction – After the last step, we get our error optimised and once it’s done, we can now test our output with some testing inputs.

The human brain is a deep and complex recurrent neural network. Deep learning allows computational models that are composed of multiple processing layers to learn representations of data with multiple levels of abstraction. In very simple words and not to confuse anything/anyone here, we can define both models as below.

• Feedforward propagation – Type of Neural Network architecture where the connections are “fed forward”only i.e. input to hidden to output The values are “fed forward”.
• Backpropagation (supervised learning algorithm) is a training algorithm with 2 steps:
  • Feedforward the values
  • Calculate the error and propagate it back to the layer before.

Propagating forward help to see the behaviour of neural network i.e how well the performance is. Observe the error and then backpropagation comes in to reduce the error (also update the bias and weight)  in the gradient descent manner. In short, forward-propagation is part of the backpropagation algorithm but comes before back-propagating.

The Need of Backpropagation

Backpropagation or backward propagation comes in as a very handy, important and useful mathematical tool when it’s about improving the accuracy of our prediction in machine learning. As mentioned above as well it is used in neural networks as the learning algorithm for computing the gradient descent by playing around with weights.

Backpropagation is a very efficient learning algorithm for multi-layer neural networks as compared with reinforcement learning. In perturbation, we try to randomly perturb one weight at a time to measure the change in performance, and the saving of any improvement is seen as quite inefficient.

In backpropagation, the computation of efficient error derivatives is possible, and that too for all hidden units at the same time. So in this regard, backpropagation is far better, as you don’t need to randomly change one word and do the whole forward propagation. This is a kind of supervised machine learning algorithm that requires a known, desired output for each input value. This way, it calculates the loss function gradient descent. This algorithm is emerging as an important machine-learning tool for predictive analytics.

Applications of Backpropagation

Backpropagation is fundamental in training diverse neural network architectures for tasks like image recognition, speech processing, and natural language understanding. It drives the training of convolutional neural networks (CNNs) for image-related tasks and recurrent neural networks (RNNs) for language processing, revolutionizing various domains with advanced machine learning capabilities.

• Training Neural Networks:
  • Backpropagation is the primary mechanism for training neural networks, including feedforward neural networks, convolutional neural networks (CNNs), recurrent neural networks (RNNs), and their variants.
  • During training, backpropagation computes gradients of the loss function with respect to the weights of the network, which are then used to update the weights iteratively through optimization algorithms like stochastic gradient descent (SGD), Adam, RMSprop, etc.
• Learning Representations:
  • Backpropagation enables neural networks to learn hierarchical representations of input data through multiple layers of transformations.
  • Each layer learns to extract and represent features at different levels of abstraction, with backpropagation adjusting weights to minimize differences between predicted and actual outputs.
• Pattern Recognition and Classification:
  • Backpropagation is widely used in tasks such as image classification, object detection, speech recognition, and natural language processing.
  • By adjusting network weights based on computed gradients, backpropagation helps networks recognize patterns and make accurate predictions on unseen data.
• Fine-tuning Pre-trained Models:
  • Backpropagation is employed in transfer learning scenarios, where pre-trained neural networks are fine-tuned on new tasks or datasets.
  • By backpropagating gradients through the network while keeping pre-trained weights fixed (or partially fixed), models can quickly adapt to new tasks or domains without extensive training from scratch.
• Reinforcement Learning:
  • In reinforcement learning, backpropagation is used with policy gradients or Q-learning to train neural network policies or value functions.
  • It optimizes parameters of the policy or value function to maximize expected cumulative reward received by an agent interacting with an environment.
• Generative Models:
  • Backpropagation is essential for training generative models like Generative Adversarial Networks (GANs) and Variational Autoencoders (VAEs).
  • In GANs, it updates generator and discriminator networks simultaneously, while in VAEs, it optimizes encoder and decoder networks to reconstruct input data and generate new samples from a learned distribution.

It enables the training of neural networks across diverse applications, including image recognition, speech processing, and natural language understanding. It underpins the development of advanced models like CNNs for image-related tasks and RNNs for language processing, driving innovation in machine learning across various domains.

How the backpropagation algorithm works

The backpropagation algorithm works like a recipe for changing the weights Wij in any feed-forward network. The idea is to learn the training set of input-output pairs (a1b, a2b). The section below will try to describe the working process of the multi-layer neural network, which employs the backpropagation algorithm.

We will take a three-layer neural network for our example with two inputs and one output, as shown in the slideshow below:

Here in the example above we taking each neuron with two units.

• Unit One – This will adds products of weights coefficients and input signals.
• Second unit – This unit realises nonlinear function, called neuron transfer (activation) function.

In the picture above, signal ‘r’ is an adder output, and b = f(r) is an output nonlinear element signal. As a practice, a neural network needs a training data set to learn and get trained. Our training data set has input signals (a1 and a2) assigned with our desired output as ‘b’. Since training neural networks is an iterative process, in each iteration, the weights and coefficients of nodes are replaced using new data from the training data set.

Conclusion – In the post above, we saw and got a firm understanding of that. The whole idea of backpropagation (a generalization of the Widrow-Hoff learning rule to multiple-layer networks) is to optimize the weights on the connecting neurons and the bias of each hidden layer. Backpropagation is used in neural networks as the learning algorithm for computing the gradient descent by playing with weights. To get correct and accurate results, a backpropagation algorithm is needed, though it’s been said the problems can be solved. One goes from the general to the specific conclusion and vice versa, but, for the sake of best performance for neural networks, backpropagation can’t be divorced from it. So backpropagation is used to train a neural network until it can give the best approximate function.

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Books Referred & Other material referred

• Open Internet reading and research work
• AILabPage (group of self-taught engineers) members hands-on lab work.

Points to Note:

When to use artificial neural networks as oppose to traditional machine learning algorithms is a complex one to answer. It entirely depends upon on the problem in hand to solve. One needs to be patient and experienced enough to have the correct answer. All credits if any remains on the original contributor only. In the next upcoming post will talk about using neural nets to recognize handwritten digits.

Feedback & Further Question

Do you have any questions about  AI,  Machine Learning, Data Science or Big Data Analytics? Leave a question in a comment section or ask via an email. Will try best to answer it.

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