The backpropagation algorithm aims to minimize the error between the current and the desired output. Since the network is feedforward, the activation flow always proceeds forward from the input units to the output units.

The gradient of the cost function is backpropagated and the network weights get updated; the overall method can be applied to any number of hidden layers recursively. In such a method, the incorporation between two phases is important. In short, the basic steps of the training procedure are as follows:

1. Initialize the network with some random (or more advanced XAVIER) weights
2. For all training cases, follow the steps of forward and backward passes as outlined next

In the forward pass, a number of operations are performed to obtain some predictions or scores. In such an operation, a graph is created, connecting all dependent operations in a top-to-bottom fashion. Then the network's error is computed, which is the difference between the predicted output and the actual output.

On the other hand, the backward pass is involved mainly with mathematical operations, such as creating derivatives for all differential operations (that is auto-differentiation methods), top to bottom (for example, measuring the loss function to update the network weights), for all the operations in the graph, and then using them in chain rule.

In this pass, for all layers starting with the output layer back to the input layer, it shows the network layer's output with the correct input (error function). Then it adapts the weights in the current layer to minimize the error function. This is backpropagation's optimization step. By the way, there are two types of auto-differentiation methods:

1. Reverse mode: Derivation of a single output with respect to all inputs
2. Forward mode: Derivation of all outputs with respect to one input

The backpropagation algorithm processes the information in such a way that the network decreases the global error during the learning iterations; however, this does not guarantee that the global minimum is reached. The presence of hidden units and the nonlinearity of the output function mean that the behavior of the error is very complex and has many local minimas.

This backpropagation step is typically performed thousands or millions of times, using many training batches, until the model parameters converge to values that minimize the cost function. The training process ends when the error on the validation set begins to increase, because this could mark the beginning of a phase overfitting.

Besides the state of a neuron, synaptic weight is considered, which influences the connection within the network. Each weight has a numerical value indicated by W_ij, which is the synaptic weight connecting neuron i to neuron j.

For each neuron (also known as, unit) i, an input vector can be defined by x_i= (x₁, x₂,...x_n) and a weight vector can be defined by w_i= (w_i1, w_i2,...w_in). Now, depending on the position of a neuron, the weights and the output function determine the behavior of an individual neuron. Then during forward propagation, each unit in the hidden layer gets the following signal:

Nevertheless, among the weights, there is also a special type of weight called bias unit b. Technically, bias units aren't connected to any previous layer, so they don't have true activity. But still, the bias b value allows the neural network to shift the activation function to the left or right. Now, taking the bias unit into consideration, the modified network output can be formulated as follows:

The preceding equation signifies that each hidden unit gets the sum of inputs multiplied by the corresponding weight—summing junction. Then the resultant in the summing junction is passed through the activation function, which squashes the output as depicted in the following figure:

Now, a tricky question: how do we initialize the weights? Well, if we initialize all weights to the same value (for example, 0 or 1), each hidden neuron will get exactly the same signal. Let's try to break it down:

• If all weights are initialized to 1, then each unit gets a signal equal to the sum of the inputs
• If all weights are 0, which is even worse, every neuron in a hidden layer will get zero signal

For network weight initialization, Xavier initialization is nowadays used widely. It is similar to random initialization but often turns out to work much better since it can automatically determine the scale of initialization based on the number of input and output neurons.

You may be wondering whether you can get rid of random initialization while training a regular DNN (for example, MLP or DBN). Well, recently, some researchers have been talking about random orthogonal matrix initializations that perform better than just any random initialization for training DNNs.

When it comes to initializing the biases, we can initialize them to be zero. But setting the biases to a small constant value such as 0.01 for all biases ensures that all Rectified Linear Unit (ReLU) units can propagate some gradient. However, it neither performs well nor shows consistent improvement. Therefore, sticking with zero is recommended.

Before the training starts, the network parameters are set randomly. Then to optimize the network weights, an iterative algorithm called Gradient Descent (GD) is used. Using GD optimization, our network computes the cost gradient based on the training set. Then, through an iterative process, the gradient G of the error function E is computed.

In following graph, gradient G of error function E provides the direction in which the error function with current values has the steeper slope. Since the ultimate target is to reduce the network error, GD makes small steps in the opposite direction -G. This iterative process is executed a number of times, so the error E would move down towards the global minima. This way, the ultimate target is to reach a point where G = 0, where no further optimization is possible:

The downside is that it takes too long to converge, which makes it impossible to meet the demand of handling large-scale training data. Therefore, a faster GD called Stochastic Gradient Descent (SDG) is proposed, which is also a widely used optimizer in DNN training. In SGD, we use only one training sample per iteration from the training set to update the network parameters.

By the way, the term stochastic comes from the fact that the gradient based on a single training sample per iteration is a stochastic approximation of the true cost gradient.

To allow a neural network to learn complex decision boundaries, we apply a non-linear activation function to some of its layers. Commonly used functions include Tanh, ReLU, softmax, and variants of these. More technically, each neuron receives as input signal the weighted sum of the synaptic weights and the activation values of the neurons connected. One of the most widely used functions for this purpose is the so-called sigmoid function. It is a special case of the logistic function, which is defined by the following formula:

The domain of this function includes all real numbers, and the co-domain is (0, 1). This means that any value obtained as an output from a neuron (as per the calculation of its activation state), will always be between zero and one. The sigmoid function, as represented in the following diagram, provides an interpretation of the saturation rate of a neuron, from not being active (= 0) to complete saturation, which occurs at a predetermined maximum value (= 1).

On the other hand, a hyperbolic tangent, or tanh, is another form of the activation function. Tanh squashes a real-valued number to the range [-1, 1]. In particular, mathematically, tanh activation function can be expressed as follows:

The preceding equation can be represented in the following figure:

In general, in the last level of an feedforward neural network (FFNN), the softmax function is applied as the decision boundary. This is a common case, especially when solving a classification problem. In probability theory, the output of the softmax function is squashed as the probability distribution over K different possible outcomes. Nevertheless, the softmax function is used in various multiclass classification methods, such that the network's output is distributed across classes (that is, probability distribution over the classes) having a dynamic range between -1 and 1 or 0 and 1.

To conclude, choosing proper activation functions and network weights initialization are two problems that make a network perform at its best and help to obtain good training. We'll discuss more in upcoming chapters; we will see where to use which activation function.