A feedforward network with a single layer is suﬃcient to representany function, but the layer may be infeasibly large and may fail to learn and generalize correctly. In many circumstances, using deeper models can reduce the number of units required to represent the desired function and can reduce the amount of generalization error.

As is shown in the figure, empirical results show that deeper networks generalize better when used to transcribe multidigit numbers from photographs of addresses. Data from Goodfellow et al. (2014d). The test set accuracy consistently increases with increasing depth.


Effect of Depth for Neural Networks

In a neural network with many time steps or layers, a gradient at the early layer is the product of all the terms from the later layers, which leads to an inherently unstable situation. Especially when the value of gradient has become so small, it no longer updates properly or is vanished eventually. Exploding gradient can be considered as the opposite of vanishing process. The updated weights using gradient descent become so large that they cause the whole network to become unstable, which leads to numerical overflow. 

Vanishing/exploding gradient

The depth of a model can be measured in two main approaches:
- Analyzing the length of sequential pathway of an input to the output. The depth can vary by the definition of a unit operation. The figure below illustrates how a logistic regression can be viewed as a model with a depth of 3 or 1.
- Analyzing how the concepts are related to each other.

Measuring the depth of the model

For a feedforward neural network, the depth of the network is the number of hidden layers plus one (as the output layer is also parameterized). The width of the network is the dimensionality of its hidden layers.

The main architectural considerations for designing a neural network are choosing the depth of the network and the width of each layer.


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Feedforward neural networks are called networks because they are typically represented by composing together many different functions. The model is associated with a directed acyclic graph describing how the functions are composed together.

For example, we might have three functions $f^{(1)} , f ^{(2)}$, and $f^{(3)}$ connected in a chain, to form $f (x) = f ^{(3)}(f^{(2)}(f^{(1)}(x)))$. These chain structures are the most commonly used structures of neural networks. 


Overall Structure of Deep Feedforward Networks

Because deeper networks can learn more complicated functions. Exponentially more nodes would be needed in each layer to achieve the same level of complexity if the network was shallower.

Why deep networks instead of shallower networks?

Hyperparameters related to neural network structure:
- Number of hidden layers (Depth)
- Number of hidden units (Width)
- Dropout method
- Activation function for each layer
- Weights Initialization

Hyperparameters related to training algorithm:
- Learning rate $\alpha$
- Momentum parameter $\beta \sim 0.9$
- $\beta_1 \sim 0.9, \beta_2 \sim 0.999, \epsilon \sim 10^{-8}$
- Number of Gradient descent iterations
- Mini-batch size 
- Optimizer algorithm
- Learning rate decay
- Regularization rate $\lambda$ 

List of Common Hyperparameters in Deep Learning

Goodfellow, I., Bengio, Y., & Courville, A. (2016). $\mathit{Deep \ Learning.}$ MIT Press. Retrieved from [www.deeplearningbook.org](https://www.deeplearningbook.org) 

Deep Learning

Wikipedia

Depth and Width for Neural Networks

In the chain structure example $f (x) = f ^{(3)}(f^{(2)}(f^{(1)}(x)))$, $f^{(1)}$ is called
the first layer of the network, f^{(2)} is called the second layer, and so on. The final layer of a feedforward network is called the output layer. 

During neural network training, we drive $f(x)$ to match $f^∗(x)$. The training data provides us with noisy, approximate examples of $f^∗(x)$ evaluated at different training points. Each example x is accompanied by a label $y ≈ f^*(x)$.

The training examples specify directly that the output layer must produce a value that is close to $y$ at each point $x$. However, the behavior of the other layers is not directly specified by the training data. The training data do not say what each individual layer should do. Instead, the learning algorithm must decide how to use these layers to best implement an approximation of $f^*$. Because the training data does not show the desired output for each of these layers, they are called hidden layers.

Layers of a Feed Forward Neural Network

$L$ = # of layers
$m$ = # of training datapoints
$$n^{[l]} = $$ # of units (neurons) in layer $l$
$$A^{[l]} = g^{[l]}(Z^{[l]}) = $$ activations (outputs) in layer $l$
$Z^{[l]}, A^{[l]}: (n^{[l]},m)$
$$X = A^{[0]}$$      $$\hat{Y} = A^{[L]}$$
$$W^{[l]} = $$ Weights for $$Z^{[l]}$$      $$W^{[l]}: (n^{[l]}, n^{[l - 1]})$$
$$b^{[l]} = $$ Biases for $$Z^{[l]}$$      $$b^{[l]}: (n^{[l]}, 1)$$

Formulation of a Feedforward Neural Networks

The universal approximation theorem (Horniket al., 1989; Cybenko, 1989) states that a feedforward network with a linear output layer and at least one hidden layer with any “squashing” activation function (such as the logistic sigmoid activation function) can approximate any Borel measurable function from one ﬁnite-dimensional space to another with any desired nonzero amount of error, provided that the network is given enough hidden units. The derivatives of the feedforward network can also approximate the derivatives of the function arbitrarily well.

It means that regardless of what function we are trying to learn, we know that a large feedforward network will be able to represent this function. 

Universal Approximation Theorem

https://www.coursera.org/learn/neural-networks-deep-learning/lecture/rz9xJ/why-deep-representations

Why deep representations?

Here You can see an example of some of the filters  that were learned by a CNN and how they have the representation of some shapes inside of them. This keeps on happening with more detailed forms as they move further into the layers.

An example of latent representations in deep networks

Example of how deeper layers of a neural network can learn more complicated functions

Dropout means what percentage of neurons should be randomly “killed” during each epoch to prevent overfitting.

Dropout

Neural network learning rate means how fast the backpropagation algorithm performs gradient descent. A larger learning rate makes the network train faster but might result in missing the minimum of the loss function.

Neural Network Learning Rate

When you try to train a huge amount of data with limited computer memory, you can separate the whole training set to several batches that can fit into your computer memory. Then you feed these batches to your model one by one. After feeding all batches once, you complete one epoch. To successfully train your model, you need multiple epochs.

Epochs in Machine Learning

In a neural network, numeric data points, called inputs, are fed into the neurons in the input layer. Each neuron has a weight, and multiplying the input number with the weight gives the output of the neuron, which is transferred to the next layer.
The activation function is a mathematical “gate” in between the input feeding the current neuron and its output going to the next layer. It can be as simple as a step function that turns the neuron output on and off, depending on a rule or threshold. Or it can be a transformation that maps the input signals into output signals that are needed for the neural network to function.
Activation functions are mathematical equations that determine the output of a neural network. The function is attached to each neuron in the network, and determines whether it should be activated (“fired”) or not, based on whether each neuron’s input is relevant for the model’s prediction. Activation functions also help normalize the output of each neuron to a range between 1 and 0 or between -1 and 1. When used in a multilayer neural network, activation functions can be different for different layers. 

Activation Functions in Neural Networks

When a neural network trains, it uses an algorithm to determine the optimal weights for the model, called an optimizer. 
There are several optimizer algorithms, such as:
- Gradient descent
- Mini-batch gradient descent
- Gradient descent with momentum
- RMSprop
- Adam optimization algorithm
- Nesterov momentum
- AdaGrad

Deep Learning Optimizer Algorithms

Batch normalization is a method that helps to speed neural network training. The basic idea is that we shift and scale the inputs to every layer of neural network. 
For example, for a 3-layer neural network shown below, we want to normalize $z^{[3]}$ to speed the training of $w^{[3]}$ and $b^{[3]}$ in the third layer.

Batch Normalization in Deep Learning

Initial weights are applied to all the neurons.
It is necessary to set initial weights for the first forward pass. Two basic options are to set weights to zero or to randomize them. However, this can result in a vanishing or exploding gradient, which will make it difficult to train the model. To mitigate this problem, you can use a heuristic (a formula tied to the number of neuron layers) to determine the weights. A common heuristic used for the Tanh activation is called Xavier initialization.

On top of that if the architecture being used has already been trained by someone else (such as ImageNet) the node values can be initialized to those values. This is known as transfer learning and it can be very useful to substantially increase training speed and model accuracy.

Deep Learning Weight Initialization

- Grid search
- Randomly choosing
- Coarse to fine

Hyperparameters Tuning Methods in Deep Learning

A model parameter is a configuration variable that is internal to the model and whose value can be estimated from data. It is required by the model when making predictions and estimated or learned from data. It is often not set manually by the practitioner. E.g. The coefficients in a linear regression or logistic regression; the weights in an artificial neural network, etc.

A model hyperparameter is a configuration that is external to the model and whose value cannot be estimated from data. It is often used in processes to help estimate model parameters. It is often specified by the practitioner. We may use rules of thumb, copy values used on other problems, or search for the best value by trial and error. E.g. The learning rate for training a neural network; the K in K-nearest neighbors, etc.

Difference between Model Parameter and Model Hyperparameter

The trade-off between the standard prediction loss and the additive weight decay penalty is characterized by the regularization constant, $$\lambda$$. This nonnegative hyperparameter is fit using validation data and modifies the objective to $$L(\mathbf{w}, b) + \frac{\lambda}{2} \|\mathbf{w}\|^2$$. When $$\lambda = 0$$, the original loss function is recovered. For $$\lambda > 0$$, the size of the weights is restricted, with larger values of $$\lambda$$ constraining the weights more considerably. The penalty term is divided by $$2$$ by convention so that the constant cancels out gracefully when the derivative of the quadratic function is taken.

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