Deep learning systems are able to learn extremely complex patterns, and they accomplish this by adjusting their weights. How are the weights of a deep neural network adjusted exactly? They are adjusted through a process called backpropagation. Without backpropagation, deep neural networks wouldn’t be able to carry out tasks like recognizing images and interpreting natural language. Understanding how backpropagation works are critical to understanding deep neural networks in general, so let’s delve into backpropagation and see how the process is used to adjust a network’s weights.

Backpropagation can be difficult to understand, and the calculations used to carry out backpropagation can be quite complex. This article will endeavour to give you an intuitive understanding of backpropagation, using little in the way of complex math. However, some discussion of the math behind backpropagation is necessary.

## The Goal of Backprop

Let’s start by defining the goal of backpropagation. The weights of a deep neural network are the strength of connections between units of a neural network. When the neural network is established assumptions are made about how the units in one layer are connected to the layers joined with it. As the data moves through the neural network, the weights are calculated and assumptions are made. When the data reaches the final layer of the network, a prediction is made about how the features are related to the classes in the dataset. The difference between the predicted values and the actual values is the loss/error, and the goal of backpropagation is to reduce the loss. This is accomplished by adjusting the weights of the network, making the assumptions more like the true relationships between the input features.

## Training A Deep Neural Network

Before backpropagation can be done on a neural network, the regular/forward training pass of a neural network must be carried out. When a neural network is created, a set of weights is initialized. The value of the weights will be altered as the network is trained. The forward training pass of a neural network can be conceived of as three discrete steps: neuron activation, neuron transfer, and forward propagation.

When training a deep neural network, we need to make use of multiple mathematical functions. Neurons in a deep neural network are comprised of the incoming data and an activation function, which determines the value necessary to activate the node. The activation value of a neuron is calculated with several components, is a weighted sum of the inputs. The weights and input values depending on the index of the nodes being used to calculate the activation. Another number must be taken into account when calculating the activation value, a bias value. Bias values don’t fluctuate, so they aren’t multiplied together with the weight and inputs, they are just added. All of this means that the following equation could be used to calculate the activation value:

Activation = sum(weight * input) + bias

After the neuron is activated, an activation function is used to determine what the output of the actual output of the neuron will be. Different activation functions are optimal for different learning tasks, but commonly used activation functions include the sigmoid function, the Tanh function, and the ReLU function.

Once the outputs of the neuron are calculated by running the activation value through the desired activation function, forward propagation is done. Forward propagation is just taking the outputs of one layer and making them the inputs of the next layer. The new inputs are then used to calculate the new activation functions, and the output of this operation passed on to the following layer. This process continues all the way through to the end of the neural network.

## Backpropagation

The process of backpropagation takes in the final decisions of a model’s training pass, and then it determines the errors in these decisions. The errors are calculated by contrasting the outputs/decisions of the network and the expected/desired outputs of the network.

Once the errors in the network’s decisions have been calculated, this information is backpropagated through the network and the parameters of the network are altered along the way. The method that is used to update the weights of the network is based in calculus, specifically, it’s based in the chain-rule. However, an understanding of calculus isn’t necessary to understand the idea of behind backpropagation. Just know that when output value is provided from a neuron, the slope of the output value is calculated with a transfer function, producing a derived output. When doing backpropagation, the error for a specific neuron is calculated according to the following formula:

error = (expected_output – actual_output) * slope of neuron’s output value

When operating on the neurons in the output layer, the class value is used as the expected value. After the error has been calculated, the error is used as the input for the neurons in the hidden layer, meaning that the error for this hidden layer is the weighted errors of the neurons found within the output layer. The error calculations travel backwards through the network along with the weights network.

After the errors for the network have been calculated, the weights in the network must be updated. As mentioned, calculating the error involves determining the slope of the output value. After the slope has been calculated, a process known as gradient descent can be used to adjust the weights in the network. A gradient is a slope, whose angle/steepness can be measured. The slope is calculated by plotting “y over” or the “rise” over the “run”. In the case of the neural network and the error rate, the “y” is the calculated error, while the “x” is the network’s parameters. The network’s parameters have a relationship to the calculated error values, and as the network’s weights are adjusted the error increases or decreases.

“Gradient descent” is the process of updating the weights so that the error rate decreases. Backpropagation is used to predict the relationship between the neural network’s parameters and the error rate, which sets up the network for gradient descent. Training a network with gradient descent involved calculating the weights through forward propagation, backpropagating the error, and then updating the weights of the network.