# Flux Overview: Fitting a Straight Line

Flux is a pure Julia ML stack that allows you to build predictive models. Here are the steps for a typical Flux program:

1. Provide training and test data
2. Build a model with configurable parameters to make predictions
3. Iteratively train the model by tweaking the parameters to improve predictions

Under the hood, Flux uses a technique called automatic differentiation to take gradients that help improve predictions. Flux is also fully written in Julia so you can easily replace any layer of Flux with your own code to improve your understanding or satisfy special requirements.

Here's how you'd use Flux to build and train the most basic of models, step by step.

### A Trivial Prediction

This example will predict the output of the function 4x + 2. Making such predictions is called "linear regression", and is really too simple to need a neural network. But it's a nice toy example.

First, import Flux and define the function we want to simulate:

julia> using Flux

julia> actual(x) = 4x + 2
actual (generic function with 1 method)

This example will build a model to approximate the actual function.

## 1. Provide Training and Test Data

Use the actual function to build sets of data for training and verification:

julia> x_train, x_test = hcat(0:5...), hcat(6:10...)
([0 1 … 4 5], [6 7 … 9 10])

julia> y_train, y_test = actual.(x_train), actual.(x_test)
([2 6 … 18 22], [26 30 … 38 42])

Normally, your training and test data come from real world observations, but here we simulate them.

## 2. Build a Model to Make Predictions

Now, build a model to make predictions with 1 input and 1 output:

julia> model = Dense(1 => 1)
Dense(1 => 1)       # 2 parameters

julia> model.weight
1×1 Matrix{Float32}:
0.95041317

julia> model.bias
1-element Vector{Float32}:
0.0

Under the hood, a dense layer is a struct with fields weight and bias. weight represents a weights' matrix and bias represents a bias vector. There's another way to think about a model. In Flux, models are conceptually predictive functions:

julia> predict = Dense(1 => 1)
Dense(1 => 1)       # 2 parameters

Dense(1 => 1) also implements the function σ(Wx+b) where W and b are the weights and biases. σ is an activation function (more on activations later). Our model has one weight and one bias, but typical models will have many more. Think of weights and biases as knobs and levers Flux can use to tune predictions. Activation functions are transformations that tailor models to your needs.

This model will already make predictions, though not accurate ones yet:

julia> predict(x_train)
1×6 Matrix{Float32}:
0.0  0.906654  1.81331  2.71996  3.62662  4.53327

In order to make better predictions, you'll need to provide a loss function to tell Flux how to objectively evaluate the quality of a prediction. Loss functions compute the cumulative distance between actual values and predictions.

julia> using Statistics

julia> loss(model, x, y) = mean(abs2.(model(x) .- y));

julia> loss(predict, x_train, y_train)
122.64734f0

More accurate predictions will yield a lower loss. You can write your own loss functions or rely on those already provided by Flux. This loss function is called mean squared error (and built-in as mse). Flux works by iteratively reducing the loss through training.

## 3. Improve the Prediction

Under the hood, the Flux Flux.train! function uses a loss function and training data to improve the parameters of your model based on a pluggable optimiser:

julia> using Flux: train!

julia> opt = Descent()
Descent(0.1)

julia> data = [(x_train, y_train)]
1-element Vector{Tuple{Matrix{Int64}, Matrix{Int64}}}:
([0 1 … 4 5], [2 6 … 18 22])

Now, we have the optimiser and data we'll pass to train!. All that remains are the parameters of the model. Remember, each model is a Julia struct with a function and configurable parameters. Remember, the dense layer has weights and biases that depend on the dimensions of the inputs and outputs:

julia> predict.weight
1×1 Matrix{Float32}:
0.9066542

julia> predict.bias
1-element Vector{Float32}:
0.0

The dimensions of these model parameters depend on the number of inputs and outputs.

Flux will adjust predictions by iteratively changing these parameters according to the optimizer.

This optimiser implements the classic gradient descent strategy. Now improve the parameters of the model with a call to Flux.train! like this:

julia> train!(loss, predict, data, opt)

And check the loss:

julia> loss(predict, x_train, y_train)
116.38745f0

It went down. Why?

julia> predict.weight, predict.bias
(Float32[7.5777884], Float32[1.9466728])

The parameters have changed. This single step is the essence of machine learning.

## 3+. Iteratively Train the Model

In the previous section, we made a single call to train! which iterates over the data we passed in just once. An epoch refers to one pass over the dataset. Typically, we will run the training for multiple epochs to drive the loss down even further. Let's run it a few more times:

julia> for epoch in 1:200
train!(loss, predict, data, opt)
end

julia> loss(predict, x_train, y_train)
0.00339581f0

julia> predict.weight, predict.bias
(Float32[4.0178537], Float32[2.0050256])

After 200 training steps, the loss went down, and the parameters are getting close to those in the function the model is built to predict.

## 4. Verify the Results

Now, let's verify the predictions:

julia> predict(x_test)
1×5 Matrix{Float32}:
26.1121  30.13  34.1479  38.1657  42.1836

julia> y_test
1×5 Matrix{Int64}:
26  30  34  38  42

The predictions are good. Here's how we got there.

First, we gathered real-world data into the variables x_train, y_train, x_test, and y_test. The x_* data defines inputs, and the y_* data defines outputs. The *_train data is for training the model, and the *_test data is for verifying the model. Our data was based on the function 4x + 2.

Then, we built a single input, single output predictive model, predict = Dense(1 => 1). The initial predictions weren't accurate, because we had not trained the model yet.

After building the model, we trained it with train!(loss, predict, data, opt). The loss function is first, followed by the model itself, the training data, and the Descent optimizer provided by Flux. We ran the training step once, and observed that the parameters changed and the loss went down. Then, we ran the train! many times to finish the training process.

After we trained the model, we verified it with the test data to verify the results.

This overall flow represents how Flux works. Let's drill down a bit to understand what's going on inside the individual layers of Flux.