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Understanding the Backpropagation Algorithm

A backpropagation algorithm is a tool for improving the neural network during the training process. With the help of this algorithm, the parameters of the individual neurons are modified in such a way that the prediction of the model and the actual value match as quickly as possible. This allows a neural network to deliver good results even after a relatively short training period.

This article joins the other posts on Machine Learning basics. If you don’t have any background on the topic of neural networks and gradient descent, you should take a look at the linked posts before reading on. We will only be able to touch on these concepts briefly. Machine Learning in general is a very mathematical topic. For this explanation of backpropagation, we will try to avoid mathematical derivations as much as possible and just give a basic understanding of the approach.

How do Neural Networks work?

Neural networks consist of many neurons organized in different layers that communicate and are linked to each other. In the input layer, the neurons are given various inputs for computation. The network should be trained so that the output layer, the last layer, makes a prediction based on the input that is as close as possible to the actual result.

Das Bild zeigt den Grundaufbau eines Künstlichen Neuronalen Netzwerkes mit den verschiedenen Schichten.
Structure of a Neural Network | Source: Author

The so-called hidden layers are used for this purpose. These layers also contain many neurons that communicate with the previous and subsequent layers. During training, the weighting parameters of each neuron are changed so that the prediction approximates reality as closely as possible. The backpropagation algorithm helps us decide which parameter to change so that the loss function is minimized.

What is the Gradient Descent Method?

The gradient method is an algorithm from mathematical optimization problems, which helps to approach the minimum of a function as fast as possible. One calculates the derivative of a function, the so-called gradient, and goes in the opposite direction of this vector because there is the steepest descent of the function.

Das Diagramm zeigt das Gradientenverfahren am Beispiel. Es ist der Funktionsgraph von f(x) = x² zu sehen und die Tangenten an verschiedenen Stellen.
Gradient Descent for the function f(x) = x² | Source: Author

If this is too mathematical, you may be familiar with this approach from hiking in the mountains. You’ve finally climbed the mountaintop, taken the obligatory pictures, and enjoyed the view sufficiently, and now you want to get back to the valley and back home as quickly as possible. So you look for the fastest way from the mountain down to the valley, i.e. the minimum of the function. Intuitively, one will simply take the path that has the steepest descent for the eye, because one assumes that this is the fastest way back downhill. Of course, this is only figuratively speaking, since no one will dare to go down the steepest cliff on the mountain.

The gradient method does the same with a function. We are somewhere in the function graph and try to find the minimum of the function. Contrary to the mountain example, we have only two possibilities to move in this situation. Either in the positive or negative x-direction (with more than one variable, i.e. a multi-dimensional space, there are of course correspondingly more directions). The gradient helps us by knowing that the negative direction of it is the steepest function descent.

What is the Gradient Descent in Machine Learning?

The function that interests us in machine learning is the loss function. It measures the difference between the prediction of the neural network and the actual result of the training data point. We also want to minimize this function, since we will then have a difference of 0. This means our model can accurately predict the results of the data set. The adjusting screws to get there are the weights of the neurons, which we can change to get closer to the goal.

In short: During the training, we get the loss function whose minimum we try to find. For this purpose, we calculate the gradient of the function after each repetition and go in the direction of the steepest descent of the function. Unfortunately, we do not know yet which parameter we have to change and by how much to minimize the loss function. The problem here is that the gradient procedure can only be performed for the previous layer and its parameters. However, a deep neural network consists of many different hidden layers whose parameters can theoretically all be responsible for the overall error.

So in large networks, we also need to determine how much influence each parameter had on the eventual error and how much we need to modify them. This is the second pillar of backpropagation, where the name comes from.

How does the Backpropagation Algorithm work?

We’ll try to keep this post as non-mathematical as possible, but unfortunately, we can’t do it completely without it. Each layer in our network is defined by an input, either from the previous layer or from the training dataset, and by an output, which is passed to the next layer. The difference between input and output comes from the weights and activation functions of the neurons.

The problem is that the influence a layer has on the final error also depends on how the following layer passes on this error. Even if a neuron “miscalculates”, this is not so important if the neuron in the next layer simply ignores this input, i.e. sets its weighting to 0. This is mathematically shown by the fact that the gradient of one layer also contains the parameters of the following layer. Therefore, we have to start the backpropagation in the last layer, the output layer, and then optimize the parameters layer by layer forward using the gradient method.

This is where the name backpropagation, or error backpropagation, comes from, as we propagate the error through the network from behind and optimize the parameters.

How does the forward and backward pass work in detail?

The forward pass is the process of taking an input and propagating it forward through the neural network to generate an output. During the forward pass, each layer of the neural network applies a linear transformation to the inputs and passes the result through a nonlinear activation function. The output of the final layer is then compared to the true output (the target) to calculate the loss or error.

The backward pass is the process of updating the weights of the neural network based on the calculated loss during the forward pass. The goal of the backward pass is to adjust the weights in such a way that the error is minimized, thereby improving the accuracy of the neural network.

To update the weights, the gradient of the loss function with respect to each weight is calculated using the chain rule of calculus. This involves computing the partial derivative of the loss with respect to the output of each neuron in the network and then using these partial derivatives to compute the partial derivative of the loss with respect to each weight.

Once the gradient has been calculated, it is used to update the weights of the neural network using an optimization algorithm such as stochastic gradient descent. The optimization algorithm adjusts the weights in a direction that minimizes the loss function, allowing the network to learn from the training data and improve its performance.

The key to the success of backpropagation is the chain rule, which allows gradients to be efficiently computed by propagating errors backward through the network. By computing gradients efficiently, backpropagation enables neural networks to learn from large amounts of data and achieve high levels of accuracy in a wide range of applications.

However, backpropagation can also be computationally expensive, particularly for large and complex neural networks. To address this issue, various modifications and optimization techniques have been proposed, such as mini-batch training, momentum, and adaptive learning rates.

What are the Advantages of the Backpropagation Algorithm?

This algorithm offers several advantages when training neural networks. These include among others:

  • Ease of use: error feedback can be easily and quickly implemented and incorporated into network training.
  • No additional hyperparameters: Unlike neural networks, backpropagation does not introduce additional hyperparameters that affect the performance of the algorithm.
  • Standard method: The use of error backpropagation has become standard and is therefore already offered by default in many modules without additional programming.

How does backpropagation compare to other models?

Backpropagation is a type of supervised learning algorithm used for training artificial neural networks, while there are other learning algorithms that can also be used for supervised learning, unsupervised learning, and reinforcement learning. Here are some comparisons of backpropagation with other popular learning algorithms:

  1. Decision trees: Decision trees are a type of supervised learning algorithm used for classification and regression tasks. They build a model by recursively splitting the data into subsets based on the value of a particular feature. Decision trees are easy to understand and interpret and can handle both categorical and continuous data. However, they can be prone to overfitting and may not perform well with complex data.
  2. Support vector machines (SVMs): SVMs are a type of supervised learning algorithm used for classification and regression tasks. They find the hyperplane that maximizes the margin between two classes of data and can handle both linear and nonlinear data. SVMs can be computationally expensive and may require careful tuning of hyperparameters.
  3. k-nearest neighbors (k-NN): k-NN is a type of lazy learning algorithm used for classification and regression tasks. It works by finding the k closest data points to a new input and then using those points to predict the output. k-NN can be used with any distance metric and can handle both categorical and continuous data. However, it can be computationally expensive and may not perform well with high-dimensional data.

Compared to these algorithms, backpropagation has some unique features and advantages:

  1. Backpropagation is well-suited for tasks that involve large amounts of data, high-dimensional inputs, and complex relationships between inputs and outputs.
  2. Backpropagation can handle both continuous and categorical data, making it a versatile algorithm for a wide range of applications.
  3. Backpropagation can be used with a variety of activation functions, allowing it to model nonlinear relationships between inputs and outputs.
  4. Backpropagation is capable of learning hierarchical representations of data, making it a powerful tool for tasks such as image recognition and natural language processing.

However, backpropagation can also have some limitations and challenges, such as overfitting, vanishing gradients, and computational complexity. To address these issues, various modifications and optimization techniques have been proposed, such as regularization, dropout, and batch normalization. Overall, the choice of learning algorithm depends on the specific task, the data, and the available resources, and each algorithm has its own strengths and weaknesses.

This is what you should take with you

  • Backpropagation is an algorithm for training neural networks.
  • Among other things, it is the application of the gradient method to the loss function of the network.
  • The name comes from the fact that the error is propagated backward from the end of the model layer by layer.

Other Articles on the Topic of Backpropagation

  • Tensorflow offers a detailed explanation of gradients and backpropagation and also shows directly how the whole thing can be implemented in Python.
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