A Guide to Convolutional Neural Networks for Computer Vision. Salman Khan

is the activation which is the input to the neuron (prior to the activation function), xj’s are the outputs from the previous layer, pi = f(ai) is the output from the neuron (prediction for the case of output layer) and f(·) denotes a nonlinear activation function while f′(·) represents its derivative. The activation function decides whether the neuron will fire or not, in response to a given input activation. Note that the nonlinear activation functions are differentiable so that the parameters of the network can be tuned using error back-propagation.

      One popular activation function is the sigmoid function, given as follows:

      We will discuss other activation functions in detail in Section 4.2.4. The derivative of the sigmoid activation function is ideally suitable because it can be written in terms of the sigmoid function itself (i.e., pi) and is given by:

      Therefore, we can write the gradient equation for the output layer neurons as follows:

      Similarly, we can calculate the error signal for the intermediate hidden layers in a multi-layered neural network architecture by back propagation of errors as follows:

      where l ∈ {1 … L – 1} and L denotes the total number of layers in the network. The above equation applies the chain rule to progressively calculate the gradients of the internal parameters using the gradients of all subsequent layers. The overall update equation for the MLP parameters θij can be written as:

      where is the output from the previous layer and t denotes the number of previous training iteration. The complete learning process usually involves a number of iterations and the parameters are continually updated until the network is optimized (i.e., after a set number of iterations or when does not change).

      Gradient Instability Problem: The generalized delta rule successfully works for the case of shallow networks (ones with one or two hidden layers). However, when the networks are deep (i.e., L is large), the learning process can suffer from the vanishing or exploding gradient problems depending on the choice of the activation function (e.g., sigmoid in above example). This instability relates particularly to the initial layers in a deep network. As a result, the weights of the initial layers cannot be properly tuned. We explain this with an example below.

      Consider a deep network with many layers. The outputs of each weight layer are squashed within a small range using an activation function (e.g., [0,1] for the case of the sigmoid). The gradient of the sigmoid function leads to even smaller values (see Fig. 3.2). To update the initial layer parameters, the derivatives are successively multiplied according to the chain rule (as in Eq. (3.10)). These multiplications exponentially decay the back-propagated signal. If we consider a network depth of 5, and the maximum possible gradient value for the sigmoid (i.e., 0.25), the decaying factor would be (0.25)⁵ = 0.0009. This is called the “vanishing gradient” problem. It is easy to follow that in cases where the gradient of the activation function is large, successive multiplications can lead to the “exploding gradient” problem.

      We will introduce the ReLU activation function in Chapter 4, whose gradient is equal to 1 (when a unit is “on”). Since 1^L = 1, this avoids both the vanishing and the exploding gradient problems.

      Figure 3.2: The sigmoid activation function and its derivative. Note that the range of values for the derivative is relatively small which leads to the vanishing gradient problem.

3.3 RECURRENT NEURAL NETWORKS

      The feed-back networks contain loops in their network architecture, which allows them to process sequential data. In many applications, such as caption generation for an image, we want to make a prediction such that it is consistent with the previously generated outputs (e.g., already generated words in the caption). To accomplish this, the network processes each element in an input sequence in a similar fashion (while considering the previous computational state). For this reason it is also called an RNN.

      Since, RNNs process information in a manner that is dependent on the previous computational states, they provide a mechanism to “remember” previous states. The memory mechanism is usually effective to remember only the short term information that is previously processed by the network. Below, we outline the architectural details of an RNN.

3.3.1 ARCHITECTURE BASICS

      A simple RNN architecture is shown in Fig. 3.3. As described above, it contains a feed-back loop whose working can be visualized by unfolding the recurrent network over time (shown on the right). The unfolded version of the RNN is very similar to a feed-forward neural network described in Section 3.2. We can, therefore, understand RNN as a simple multi-layered neural network, where the information flow happens over time and different layers represent the computational output at different time instances. The RNN operates on sequences and therefore the input and consequently the output at each time instance also varies.

      Figure 3.3: The RNN Architecture. Left: A simple recurrent network with a feed-back loop. Right: An unfolded recurrent architecture at different time-steps.

      We highlight the key features of an RNN architecture below.