www.3blue1brown.com/lessons/neural-networks

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This number inside the neuron is called the “activation” of that neuron, and the image you might have in your mind is that each neuron is lit up when its activation is a high number

This is meant to indicate how the activation of each neuron in one layer, the little number inside it, has some influence on the activation of each neuron in the next layer.

But what if we could write a program that mimics the structure of your brain? That’s the idea behind neural networks. The hope is that by writing brain-inspired software, we might be able to create programs that tackle the kinds of fuzzy and difficult-to-reason-about problems that your mind is so good at solving

Moreover, just as you learn by seeing many examples, the “learning” part of machine learning comes from the fact that we never give the program any specific instructions for how to identify digits. Instead, we’ll show it many examples of hand-drawn digits together with labels for what they should be, and leave it up to the computer to adapt the network based on each new example.

There are many variants of neural networks, such as convolutional neural networks (CNN), recurrent neural networks (RNN), transformers, and countless others

In this network I have 2 hidden layers, each with 16 neurons, which is admittedly kind of an arbitrary choice. To be honest, I chose 2 layers based on how I want to motivate the structure in just a moment, and 16 was simply a nice number to fit on the screen. In practice, there’s a lot of room to experiment with the specific structure.

However, not all these connections are equal. Some will be stronger than others, and as you’ll see shortly, determining how strong these connections are is really the heart of how a neural network operates, as an information processing mechanism

And beyond image recognition, there are all sorts of intelligent tasks that you can break down into layers of abstraction. Parsing speech, for example, involves parsing raw audio into distinct sounds, which combine to make certain syllables, which combine to form words, which combine to make up phrases and more abstract thoughts, etc

Every neuron has an activation between 0.0 and 1.0, sort of analogous to how neurons in the brain can be active or inactive.

So to actually compute the value of this second-layer neuron, you take all the activations from the neurons in the first layer, and compute their weighted sum.

One common function that does this is called the “sigmoid” function, also known as a logistic curve, which we represent using the symbol � σ. Very negative inputs end up close to 0, very positive inputs end up close to 1, and it steadily increases around 0. So the activation of the neuron here will basically be a measure of how positive the weighted sum is.

But maybe it’s not that we want the neuron to light up when this weighted sum is bigger than 0. Maybe we only want it to be meaningfully active when that sum is bigger than, say, 10. That is, we want some bias for it to be inactive. What we’ll do then is add some number, like -10, to the weighted sum before plugging it into the sigmoid function that squishes everything into the range between 0 and 1. We call this additional number a bias.

When we talk about learning, which we’ll do in the next lesson, we mean getting the computer to find an optimal setting for all these many, many numbers that will solve the problem at hand.

So it’s actually more accurate to think of each neuron as a function. It takes in the activations of all neurons in the previous layer, and spits out a number between 0 and 1. And really, the entire network is just a function! It takes in 784 numbers as its input, and spits out 10 numbers as its output.

0.0 and 1.0.

So we need to represent the inputs and outputs of our network (the images and digit predictions) in terms of these neuron values between 0.0 and 1.0.

When we want to feed the network an image, we’ll set each input neuron’s activation to the brightness of its corresponding pixel.

The last layer of our network will have 10 neurons, each representing one of the possible digits. The activation in these neurons, again some number between 0.0 and 1.0, will represent how much the system thinks an image corresponds to a given digit.

In a perfect world, we might hope that each neuron in the second-to-last layer corresponds to one of these subcomponents.

The hope would be that any generally loopy pattern toward the top of the image sets off this neuron. That way, going from this third layer to the last one would only require learning which combinations of subcomponents correspond to which digits.

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