Flux's core feature is taking gradients of Julia code. The `gradient` function takes another Julia function `f` and a set of arguments, and returns the gradient with respect to each argument. (It's a good idea to try pasting these examples in the Julia terminal.)
```julia
using Flux.Tracker
f(x) = 3x^2 + 2x + 1
# df/dx = 6x + 2
f′(x) = Tracker.gradient(f, x)[1]
f′(2) # 14.0 (tracked)
# d²f/dx² = 6
f′′(x) = Tracker.gradient(f′, x)[1]
f′′(2) # 6.0 (tracked)
```
(We'll learn more about why these numbers show up as `(tracked)` below.)
When a function has many parameters, we can pass them all in explicitly:
```julia
f(W, b, x) = W * x + b
Tracker.gradient(f, 2, 3, 4)
(4.0 (tracked), 1.0, 2.0 (tracked))
```
But machine learning models can have *hundreds* of parameters! Flux offers a nice way to handle this. We can tell Flux to treat something as a parameter via `param`. Then we can collect these together and tell `gradient` to collect the gradients of all of them at once.
```julia
W = param(2) # 2.0 (tracked)
b = param(3) # 3.0 (tracked)
f(x) = W * x + b
params = Params([W, b])
grads = Tracker.gradient(() -> f(4), params)
grads[W] # 4.0
grads[b] # 1.0
```
There are a few things to notice here. Firstly, `W` and `b` now show up as *tracked*. Tracked things behave like normal numbers or arrays, but keep records of everything you do with them, allowing Flux to calculate their gradients. `gradient` takes a zero-argument function; no arguments are necessary because the `Params` tell it what to differentiate.
This will come in really handy when dealing with big, complicated models. For now, though, let's start with something simple.
## Simple Models
Consider a simple linear regression, which tries to predict an output array `y` from an input `x`.
To improve the prediction we can take the gradients of `W` and `b` with respect to the loss and perform gradient descent. Let's tell Flux that `W` and `b` are parameters, just like we did above.
Now that we have gradients, we can pull them out and update `W` to train the model. The `update!(W, Δ)` function applies `W = W + Δ`, which we can use for gradient descent.
The loss has decreased a little, meaning that our prediction `x` is closer to the target `y`. If we have some data we can already try [training the model](../training/training.md).
All deep learning in Flux, however complex, is a simple generalisation of this example. Of course, models can *look* very different – they might have millions of parameters or complex control flow. Let's see how Flux handles more complex models.
It's common to create more complex models than the linear regression above. For example, we might want to have two linear layers with a nonlinearity like [sigmoid](https://en.wikipedia.org/wiki/Sigmoid_function) (`σ`) in between them. In the above style we could write this as:
This works but is fairly unwieldy, with a lot of repetition – especially as we add more layers. One way to factor this out is to create a function that returns linear layers.
```julia
function linear(in, out)
W = param(randn(out, in))
b = param(randn(out))
x -> W * x .+ b
end
linear1 = linear(5, 3) # we can access linear1.W etc
Another (equivalent) way is to create a struct that explicitly represents the affine layer.
```julia
struct Affine
W
b
end
Affine(in::Integer, out::Integer) =
Affine(param(randn(out, in)), param(randn(out)))
# Overload call, so the object can be used as a function
(m::Affine)(x) = m.W * x .+ m.b
a = Affine(10, 5)
a(rand(10)) # => 5-element vector
```
Congratulations! You just built the `Dense` layer that comes with Flux. Flux has many interesting layers available, but they're all things you could have built yourself very easily.
(There is one small difference with `Dense`– for convenience it also takes an activation function, like `Dense(10, 5, σ)`.)
## Stacking It Up
It's pretty common to write models that look something like:
```julia
layer1 = Dense(10, 5, σ)
# ...
model(x) = layer3(layer2(layer1(x)))
```
For long chains, it might be a bit more intuitive to have a list of layers, like this:
```julia
using Flux
layers = [Dense(10, 5, σ), Dense(5, 2), softmax]
model(x) = foldl((x, m) -> m(x), x, layers)
model(rand(10)) # => 2-element vector
```
Handily, this is also provided for in Flux:
```julia
model2 = Chain(
Dense(10, 5, σ),
Dense(5, 2),
softmax)
model2(rand(10)) # => 2-element vector
```
This quickly starts to look like a high-level deep learning library; yet you can see how it falls out of simple abstractions, and we lose none of the power of Julia code.
A nice property of this approach is that because "models" are just functions (possibly with trainable parameters), you can also see this as simple function composition.
```julia
m = Dense(5, 2) ∘ Dense(10, 5, σ)
m(rand(10))
```
Likewise, `Chain` will happily work with any Julia function.
This enables a useful extra set of functionality for our `Affine` layer, such as [collecting its parameters](../training/optimisers.md) or [moving it to the GPU](../gpu.md).
