Here is a useful website about Batch Norm: https://www.cerebras.net/neurips-2019-online-normalization-for-training-neural-networks/

NeurIPS 2019: Online Normalization for Training Neural Networks

- It makes your hyperparameter search problem much easier.
- It makes the neural network more robust to the choice of hyperparameters in a much bigger range of hyperparameters.
- It makes training even very deep networks easier.
- It has a slight regularization effect.

Benefits of Batch Norm in Deep Learning

In each layer, before we put our input $z^{(i)}$ to the activation function, we first normalize it by Batch Norm.  That is,  we first calculate 
$\mu = \frac{1}{n}\sum_{i} z^{(i)}$
$\sigma^2=\frac{1}{n}\sum_{i}(z^{(i)}-\mu)^2$
$z_{norm}^{(i)} = \frac{z^{(i)}-\mu}{\sqrt{\sigma^2+\epsilon}}$
$\tilde{z}^{(i)}=\gamma^{(i)} z_{norm}^{(i)} + \beta^{(i)}$
We introduce $\gamma^{(i)}$ and $\beta^{(i)}$ because we don't want all the inputs to neurons in hidden layers to always have mean 0 and variance 1.
$\Rightarrow a^{(i)} = g(\tilde{z}^{(i)})$,
where g is some activation function.
Recall in each layer,
$z^{(i)}= W^{(i)} a^{(i-1)}+b^{(i)}$
When we calculate $\tilde{z}^{(i)}$, we first normalize $z^{(i)}$ by subtracting the mean. So the value of $b^{(i)}$ has no influence on the result. $\Rightarrow$ In each layer, we have three parameters  $W^{(i)}, \gamma^{(i)},  \beta^{(i)}$.
Frequently, we combine this with mini-batches; i.e., we train $W^{(i)}$, $\gamma^{(i)}$,  $\beta^{(i)}$ with respect to $X^{\{1\}}, ..., X^{\{n\}}$, where $X^{\{i\}}$ is $i^{th}$ batch of the training data.

Batch Norm in Deep Learning Implementation

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.

University of Michigan - Ann Arbor

In the chain structure of a feedforward network, Let's assume the input example is $x \in \mathbb{R}^d$, then the first layer is given by
$h^{(1)} = g^{(1)} (W^{(1)T}x + b^{(1)})$
the second layer is given by 
$h^{(2)} = g^{(2)} (W^{(2)T} h^{(1)}+ b^{(2)})$
and so on.
For the output layer,
$\hat{y} = g^{(n)}(W^{(n)T} h^{(n-1)}+ b^{(n)})$

where the matrix $W^{(k)}\in \mathbb{R}^{d\times h_k}$ is the weight parameter for layer $k$ and $g^{k}$ is the activation function for layer $k$. Whether transposing $W^{(k)}$ depends on the shape of the vector.

List of Common Hyperparameters in Deep Learning

Normalization of training data may help reducing the bias and variance, and speeds up Gradient Descent, and consequently, training of the deep learning model, as shown in the figure.

Normalization Helps Deep Learning

https://www.coursera.org/learn/deep-neural-network?specialization=deep-learning

Improving Deep Neural Networks: Hyperparameter tuning, Regularization and Optimization

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

Batch Normalization in Deep Learning

Matrix degeneration is a phenomenon where the rank of a matrix is reduced after it undergoes some form of processing.

Matrix Degeneration

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.


Depth and Width for Neural Networks

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

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

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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