Neural Networks - The Math of Intelligence #4

**Understanding Neural Networks: A Comprehensive Guide**

Neural networks have revolutionized the field of machine learning, offering unparalleled capabilities in pattern recognition and function approximation. This article delves into the foundational concepts of neural networks, exploring their structure, functionality, and various types, all derived from the provided video transcription.

### Introduction to Neural Networks

Neural networks are inspired by the human brain's ability to model complex relationships. These networks are universal function approximators, capable of learning any function given sufficient data and proper training. The idea that emotions like love can be mathematically described through functions highlights the network's versatility in capturing intricate patterns.

At their core, neural networks aim to find a mathematical relationship between input features and output labels. This is achieved by initializing random weights, which are adjusted through a process known as gradient descent, minimizing prediction errors.

### Feedforward Neural Networks

The simplest form of neural network is the feedforward network. Here's how it works:

1. **Input Data**: Represented as a matrix where each row is a data point and columns represent features.

2. **Weight Matrix Initialization**: Weights are initialized randomly to break symmetry and enable learning.

3. **Forward Propagation**: The input is multiplied by the weight matrix, followed by an activation function (like sigmoid) to normalize outputs between 0 and 1.

4. **Error Calculation**: The difference between predicted and actual outputs is computed.

5. **Backpropagation**: Using gradient descent, gradients of errors with respect to weights are calculated and used to update weights iteratively.

This process allows the network to learn the optimal weight matrix, enabling accurate predictions.

### Recurrent Neural Networks (RNNs)

While feedforward networks handle static data well, they fall short when dealing with sequential information. Recurrent neural networks address this by incorporating previous time step information into current computations:

1. **Hidden State**: A mechanism that carries over information from the previous time step.

2. **Forward Propagation**: For each element in a sequence, the input is combined with the hidden state of the previous step using specific weight matrices.

3. **Backpropagation Through Time (BPTT)**: Extends backpropagation to handle temporal dependencies, allowing error gradients to be computed across time steps.

RNNs are ideal for tasks like predicting stock prices or generating musical notes, where sequence order matters.

### Self-Organizing Maps (SOMs)

Self-organizing maps are unsupervised neural networks used for clustering unlabeled data:

1. **Initialization**: A grid of neurons with weights initialized randomly.

2. **Best Matching Unit (BMU)**: For each input vector, the neuron with the closest weight vector is identified.

3. **Neighborhood Function**: Adjusts weights of BMU and its neighbors to move closer to the input vector in the weight space.

4. **Learning Process**: Repeatedly applies the above steps, causing the network to self-organize into clusters.

SOMs excel in visualizing data distributions and identifying patterns without prior labeling.

### Deep Learning and Beyond

Deep learning extends neural networks by adding multiple hidden layers, enhancing their ability to model complex functions. With adequate data and computational power, these networks achieve state-of-the-art performance across various domains, from image recognition to natural language processing.

### Conclusion

Neural networks are a powerful tool for approximating functions, whether linear or nonlinear. By understanding the nuances of different architectures—feedforward, recurrent, and self-organizing maps—you can select the appropriate model for your task. As technology advances, neural networks will continue to evolve, promising new insights and applications across diverse fields.

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This article provides a detailed exploration of neural networks, emphasizing their structure and functionality. By breaking down each concept and illustrating with examples, it serves as an accessible guide for anyone seeking to understand these foundational machine learning models.

"WEBVTTKind: captionsLanguage: enHello Word! It's Siraj!By the end of this video you will understand the basic math behind neural networks since we’ll build 4 types of them.You know that feeling you get when you’re in love and you see that special someone?That magical, ethereal sensation that just can’t be described by words? Well, it actually can be described.By math. The features of a face, word associations, the tone of their voice, these are all related variables.And we can represent this relationship using a function.We have different models for approximating different functions.But humans can approximate almost any function.So wouldn't it make sense to model our own capability?Neural Networks, inspired by the brain, are universal function approximators. That means they can learn any function.Although silicon is a very different medium than the chemical soup in our head, they’re still able to replicate much of what we do like nothing else we’ve created.This was proven in 1989 when he stated that “we show that arbitrary decision regions can be arbitrarily well approximated by continuous feedforward neural networks with only a single internal,hidden layer and any continuous sigmoidal nonlinearity”.Wait, what? Are you trying to throw hands here, I'm, ok...Look, let’s just build a simple af neural network. Alright so given some input data X and some associated output labels Y, there exists a function that represents the mapping between them.Our goal is to learn that function so we can then input some random X value and it will predict its associated Y value.This input data is represented as a matrix where each row is a different datapoint and each column is one of its features, just arbitrary 1s and 0s in our case.So how do we learn this mapping? Imagine there existed a matrix such that every time we multiplied our input data by that matrix, the result would give us the correct output every time.That’d be pretty awesome right? Thats what we’re trying to do. Find that matrix, that coefficient of the function we’re trying to learn. We’ll call it our weight value. So we’ll initialize it as a matrix of random values.Eventually, we want this matrix to have the ideal values. Values such that when we compute the dot product between our input data and this matrix it will give us the correct output.But wait! We’ve gotta add something else here.It's not enough to just say the product of these two matrices is our output.No, we’ll need to pass that output into a sigmoid function to normalize or adjust the result to probability values between 0 and 1.So multiplying our input by this matrix and passing the result to this activation function gives us an output value, 4 guesses, one for each data point !But the guesses are wrong! We know what our actual output should be, so lets compute the difference between our prediction and the actual Y values. We’re pretty far off.We know that gradient descent works well for linear regression, so lets try it here too! We’ll compute the partial derivative of the error with respect to our weight and that gives us a gradient value.Gradient descent is all about finding the minimum of an error function, so lets adjust the values in our matrix by adding our gradient value to it. Now our predicted output is slightly better.Let's just do this over and over again. Our error will decrease every iteration or time step and eventually our network will be able to predict the right label.A neural network is really just one big composite function.That means its a function that consists of other functions.Each layer is just a function that takes as its input the result of the previous functions output. That’s it.*music*The dataset we used had a linear relationship , there was a direct one-to-one relationship between the input and output values.But what if we used a dataset that instead had a one-to-one relationship between a combination of inputs? A nonlinear relationship?Our simple af neural network wouldn’t be able to learn it, In order to first combine the input data into a representation that can then have a one-to-one relationship with the output, we need to add another layer.Our first layer will combine the inputs, and our second layer will then map them to the output using the output of the first layer as input.So we’ll multiply the input by the first weight matrix, activate it with our sigmoid and pass the result to the next layer as its input. It’s just does the exact same process.Input times weight, activate. It rhymes! When we compute our prediction value, since we have multiple weight values to updatewe’ll recursively compute our error gradient for each layer we have in the opposite direction.Then we can update each weight value individually. That's why when gradient descent is applied to neural networks specifically, its called back propagation.Because after we forward propagate, our input data, we backward propagate our error gradient to update our weight values.What we just built was called a feedforward network.They’re great for modeling relationships between any set of input variables and one or more output variables.What if time mattered in the input sequence?By that I mean, what if the ordering mattered. Like if we’re trying to predict the next stock price or musical note in a sequence?Well, well need to modify our feedforward network to become a recurrent network, which will allow it to learn a sequential mappingWe could apply a linear transformation to the matrix.Totally I'm all about the projections.So this time we’ve got sequential input dataWe still initialize our weights randomly like before.We still multiply our input by our weight matrix and apply an activation function to the result for every layer.The difference in forward propagation this time though,is that for each element in the sequence, we don’t just us it alone as the input we use the hidden state from the previous time step as well.So a hidden state at a given time step is a function of the input at the same time step modified by a weight matrix added to the hidden state of the previous time step multipliedby its own hidden state to hidden state matrix.Back-propagation in this case works the same way as in a feedforward network, we calculate the gradients of our error with respect to to the weights and use them to update the weights.Because recurrent nets are a temporal model, we call it back-propagation through time.Doc! Let's back-propagation through time.But can a neural network still learn a function if we have unlabeled data?You Betcha!=n=and one neural net that can is called a self organizing map. It works pretty differently, lets look at it.We still start off by initializing the weights of the network randomly.Think of it as a 2 dimensional array of neurons.Each neuron has a specific position topologically and contains a vector of weights of the same dimensions as the input vectors. The lines connecting nodes just represent adjacency.They don't signify a connection as normally indicated when talking about the previous networks.Then, we’ll pick a random data point from our training set to calculate the Euclidean distance between that vector and each weight.The one thats closest to it is the most similar, its the best matching unit.We iterate through all the other nodes to determine whether or not they are within the radius of it, then adjust the weights of its neighborsThis whole process is what we repeat iteratively and is apart of the training procedure.This map of nodes self organizes into clusters for each learned label. nearby locations in the map represent inputs with similar properties.We can even visualize it and this acts as a great tool to observe these clusters.and when give these networks more aka deeper layers, lots of data, and lots compute we call that field deep learning.It’s the hottest subset of machine learning.Although they’e not a cure all solution, they perform really well if you have those two things.To summarize, Neural Networks are just a series of matrix operations, no matter the type they are just one big composite function.Because they use nonlinearities in this series of operations like the sigmoid, they can approximate any function both linear and nonlinear.And If we don’t have labels for our data, we can use a type of neural network called a self organizing map to learn the label clusters.The Wizard of the Week is Hammad Shaikh.I was so, so impressed by his notebook :0It illustrated how using L2 regularization reduces overfitting for high degree polynomials in the context of linear regression(specifically the relationship between movie sales and movie ratings)You make me smile ♪♫And the runner up is Ong Ja Rui.He used a regularized linear regression to predict the impact of climate change on temperatures in LA.What a dope use case. This weeks coding challenge is to build your own self organized map.See the description for all the details and post your GitHub link in the comments section.Please Subscribe for more programming videos and for now I’ve gotta practice my Dutch so thanks for watching :)Subtitles by David Resendiz (Pinkie The Smiling Cat)\n"