When we use a feedforward neural network to accept an input $x$ from training set and produce an output $\hat{y}$, information ﬂows forward through the network. The input $x$ provides the initial information that then propagates up to the hidden units at each layer and ﬁnally produces $\hat{y}$. This is called forward propagation. 


Forward Propagation

Weights are changed to the optimal values according to the results of the backpropagation algorithm.

Since the weights are updated a small delta step at a time, several iterations are required in order for the network to learn. After each iteration, the gradient descent force updates the weights towards less and less global loss function. The amount of iterations needed to converge depends on the learning rate, the network meta-parameters, and the optimization method used.

Update Weight Iteratively Until Convergence

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

What is the "cache" used for in our implementation of forward propagation and backward propagation?

Which of the following statements are True? (Check all that apply).

Consider the following 1 hidden layer neural network:

Which of the following are true regarding activation outputs and vectors? (Check all that apply.)

Backpropagation is a systematic computational procedure for applying the chain rule to calculate gradients automatically. It operates by traversing a computational graph in a backwards direction—from the output loss back to the input parameters—multiplying matrices of partial derivatives at each step to determine how parameters affect the final output.

Backpropagation

To build a formal mathematical system for learning, machine learning algorithms rely on objective functions, which are formal mathematical measures that quantify how well or poorly a model is performing its task. These functions provide the fundamental evaluation metric that optimization algorithms seek to improve.

Objective Function

A Feed forward neural network model training occurs in six stages:

- Iterate until convergence
   - Initialization
   - Forward propagation
   - Error function (Objective function)
   - Backpropagation
   - Weight update

At the end of this process, the model is ready to make predictions for unknown input data. New data can be fed to the model, a forward pass is performed, and the model generates its prediction.

University of Michigan - Ann Arbor

Deep feedforward networks, also called feedforward neural networks, or multilayer perceptrons (MLPs), are the quintessential deep learning models.

It was the first and simplest type of artificial neural network devised. In this network, the information moves in only one direction, forward, from the input nodes, through the hidden nodes (if any) and to the output nodes. There are no cycles or loops in the network.

Deep Feedforward Networks (MLP = Multi-Layer Perceptrons)

A helpful website that introduces neural networks:
https://missinglink.ai/guides/neural-network-concepts/

Neural Network Reference

Medium

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

Stages of Feed Forward Neural Network Learning

A hyperparameter is a setting that affects the structure or operation of the neural network. In real deep learning projects, tuning hyperparameters is the primary way to build a network that provides accurate predictions for a certain problem. 

Hyperparameters of Feedforward Neural Network

There are a few different types of cost functions, but for the most part the same type of cost function is used for neural networks. We usually use the principle of maximum likelihood for most cases, such as when our parametric model uses a distribution p(y | x;θ), where the relationship between the input (training data) and output (model's predictions) of the model as the cost function. A sometimes simpler approach used, where we are predicting some statistic of y conditioned on x, instead of predicting a complete probability distribution over y. 

Deep Feedforward Network Cost Functions

-The chain rule that underlies the back-propagation algorithm (one of the common algorithms used to train neural networks) was invented in the seventeenth century.

-While algebra and advanced mathematics have always been implemented to solve computer optimization problems, gradient descent was not introduced as a technique for iteratively approximating the solution to optimization problems until the nineteenth century.

-Non-linear functions could only be properly studied once a multilayer perceptron and a means of computing the gradient were developed. One of the biggest advancements for this development occurred once eﬃcient applications of the chain rule based on dynamic programming were implemented between the 1960s and 70s. 


Deep Feedfoward Network as time passed.

Approaches to back-propagation taking a computational graph and a set ofnumerical values for the inputs to the graph and returning a set of numerical valuesdescribing the gradient at those input values is called symbol-to-number diﬀerentiation.

Symbol-to-number Diﬀerentiation

Theano and Tensorflow Approach is to take a computational graph and add additional nodes to the graph that provide a symbolic description of the desired derivatives.

Theano and Tensorflow Approach

These algebraic and graph-based representations operating on symbols, orvariables that do not have speciﬁc values are called symbolic representations.

Symbolic Representations

Language modeling is one of the application of feedforward networks: predicting upcoming words from prior word context. Neural language modeling
is an important NLP task in itself, and it plays a role in many important algorithms for tasks like machine translation, summarization, speech recognition, grammar correction, and dialogue.



Feedforward Neural Language Modeling

Most Neural NLP classifications take word embeddings as their inputs.
$$ x = [e_{w1};e_{w2};e_{w3};...;e_{wn}]$$
These kind of embeddings are gained from the pertaining process relying on another algorithm.
Embedding can also be learned as part of the NLP task in the neural language modeling task.



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