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@CarloLucibello
CarloLucibello committed Oct 22, 2024
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Expand Up @@ -6,7 +6,7 @@ To introduce Flux's recurrence functionalities, we will consider the following v

![](../../assets/rnn-basic.png)

In the above, we have a sequence of length 3, where `x1` to `x3` represent the input at each step (could be a timestamp or a word in a sentence), and `y1` to `y3` are their respective outputs.
In the above, we have a sequence of length 3, where `x1` to `x3` represent the input at each step. It could be a timestamp or a word in a sentence encoded as vectors. `y1` to `y3` are their respective outputs.

An aspect to recognise is that in such a model, the recurrent cells `A` all refer to the same structure. What distinguishes it from a simple dense layer is that the cell `A` is fed, in addition to an input `x`, with information from the previous state of the model (hidden state denoted as `h1` & `h2` in the diagram).

Expand All @@ -17,213 +17,152 @@ output_size = 5
input_size = 2
Wxh = randn(Float32, output_size, input_size)
Whh = randn(Float32, output_size, output_size)
b = randn(Float32, output_size)
b = zeros(Float32, output_size)

function rnn_cell(h, x)
function rnn_cell(x, h)
h = tanh.(Wxh * x .+ Whh * h .+ b)
return h, h
return h
end

x = rand(Float32, input_size) # dummy input data
h = rand(Float32, output_size) # random initial hidden state

h, y = rnn_cell(h, x)
seq_len = 3
# dummy input data
x = [rand(Float32, input_size) for i = 1:seq_len]
# random initial hidden state
h0 = zeros(Float32, output_size)

y = []
ht = h0
for xt in x
ht = rnn_cell(xt, ht)
y = [y; [ht]] # concatenate in non-mutating (AD friendly) way
end
```

Notice how the above is essentially a `Dense` layer that acts on two inputs, `h` and `x`.
Notice how the above is essentially a `Dense` layer that acts on two inputs, `xt` and `ht`.

If you run the last line a few times, you'll notice the output `y` changing slightly even though the input `x` is the same.
The output at each time step, called the hidden state, is used as the input to the next time step and is also the output of the model.

There are various recurrent cells available in Flux, notably `RNNCell`, `LSTMCell` and `GRUCell`, which are documented in the [layer reference](../../reference/models/layers.md). The hand-written example above can be replaced with:

```julia
using Flux

rnn = Flux.RNNCell(2, 5)
output_size = 5
input_size = 2
seq_len = 3
x = [rand(Float32, input_size) for i = 1:seq_len]
h0 = zeros(Float32, output_size)

x = rand(Float32, 2) # dummy data
h = rand(Float32, 5) # initial hidden state
rnn_cell = Flux.RNNCell(input_size => output_size)

h, y = rnn(h, x)
y = []
ht = h0
for xt in x
ht = rnn_cell(xt, ht)
y = [y; [ht]]
end
```
The entire output `y` or just the last output `y[end]` can be used for further processing, such as classification or regression.

## Stateful Models
## Using a cell as part of a model

For the most part, we don't want to manage hidden states ourselves, but to treat our models as being stateful. Flux provides the `Recur` wrapper to do this.
Let's consider a simple model that is trained to predict a scalar quantity for each time step in a sequence. The model will have a single RNN cell, followed by a dense layer to produce the output.
Since the [`RNNCell`](@ref) can deal with batches of data, we can define the model to accept an input where
at each time step, the input is a matrix of size `(input_size, batch_size)`.

```julia
x = rand(Float32, 2)
h = rand(Float32, 5)

m = Flux.Recur(rnn, h)

y = m(x)
```

The `Recur` wrapper stores the state between runs in the `m.state` field.

If we use the `RNN(2, 5)` constructor – as opposed to `RNNCell` – you'll see that it's simply a wrapped cell.

```jldoctest recurrence
julia> using Flux
julia> RNN(2, 5) # or equivalently RNN(2 => 5)
Recur(
RNNCell(2 => 5, tanh), # 45 parameters
) # Total: 4 trainable arrays, 45 parameters,
# plus 1 non-trainable, 5 parameters, summarysize 404 bytes.
```

Equivalent to the `RNN` stateful constructor, `LSTM` and `GRU` are also available.

Using these tools, we can now build the model shown in the above diagram with:

```jldoctest recurrence
julia> m = Chain(RNN(2 => 5), Dense(5 => 1))
Chain(
Recur(
RNNCell(2 => 5, tanh), # 45 parameters
),
Dense(5 => 1), # 6 parameters
) # Total: 6 trainable arrays, 51 parameters,
# plus 1 non-trainable, 5 parameters, summarysize 540 bytes.
```
In this example, each output has only one component.

## Working with sequences

Using the previously defined `m` recurrent model, we can now apply it to a single step from our sequence:

```jldoctest recurrence; filter = r"[+-]?([0-9]*[.])?[0-9]+(f[+-]*[0-9])?"
julia> x = rand(Float32, 2);
julia> m(x)
1-element Vector{Float32}:
0.45860028
```

The `m(x)` operation would be represented by `x1 -> A -> y1` in our diagram.
If we perform this operation a second time, it will be equivalent to `x2 -> A -> y2`
since the model `m` has stored the state resulting from the `x1` step.

Now, instead of computing a single step at a time, we can get the full `y1` to `y3` sequence in a single pass by
iterating the model on a sequence of data.

To do so, we'll need to structure the input data as a `Vector` of observations at each time step. This `Vector` will therefore be of `length = seq_length` and each of its elements will represent the input features for a given step. In our example, this translates into a `Vector` of length 3, where each element is a `Matrix` of size `(features, batch_size)`, or just a `Vector` of length `features` if dealing with a single observation.

```jldoctest recurrence; filter = r"[+-]?([0-9]*[.])?[0-9]+(f[+-]*[0-9])?"
julia> x = [rand(Float32, 2) for i = 1:3];
julia> [m(xi) for xi in x]
3-element Vector{Vector{Float32}}:
[0.36080405]
[-0.13914406]
[0.9310162]
```

!!! warning "Use of map and broadcast"
Mapping and broadcasting operations with stateful layers such are discouraged,
since the julia language doesn't guarantee a specific execution order.
Therefore, avoid
```julia
y = m.(x)
# or
y = map(m, x)
```
and use explicit loops
```julia
y = [m(x) for x in x]
```

If for some reason one wants to exclude the first step of the RNN chain for the computation of the loss, that can be handled with:
struct RecurrentCellModel{H,C,D}
h0::H
cell::C
dense::D
end

```julia
using Flux.Losses: mse
# we choose to not train the initial hidden state
Flux.@layer RecurrentCellModel trainable=(cell,dense)

function loss(x, y)
m(x[1]) # ignores the output but updates the hidden states
sum(mse(m(xi), yi) for (xi, yi) in zip(x[2:end], y))
function RecurrentCellModel(input_size::Int, hidden_size::Int)
return RecurrentCellModel(
zeros(Float32, hidden_size),
RNNCell(input_size => hidden_size),
Dense(hidden_size => 1))
end

y = [rand(Float32, 1) for i=1:2]
loss(x, y)
function (m::RecurrentCellModel)(x)
z = []
ht = m.h0
for xt in x
ht = m.cell(xt, ht)
z = [z; [ht]]
end
z = stack(z, dims=2) # [hidden_size, seq_len, batch_size] or [hidden_size, seq_len]
ŷ = m.dense(z) # [1, seq_len, batch_size] or [1, seq_len]
return ŷ
end
```

In such a model, only the last two outputs are used to compute the loss, hence the target `y` being of length 2. This is a strategy that can be used to easily handle a `seq-to-one` kind of structure, compared to the `seq-to-seq` assumed so far.
Notice that we stack the hidden states `z` to form a tensor of size `(hidden_size, seq_len, batch_size)`. This can speedup the final classification, since we then process all the outputs at once with a single forward pass of the dense layer.

Alternatively, if one wants to perform some warmup of the sequence, it could be performed once, followed with a regular training where all the steps of the sequence would be considered for the gradient update:
Let's now define the training loop for this model:

```julia
function loss(m, x, y)
sum(mse(m(xi), yi) for (xi, yi) in zip(x, y))
end

seq_init = [rand(Float32, 2)]
seq_1 = [rand(Float32, 2) for i = 1:3]
seq_2 = [rand(Float32, 2) for i = 1:3]
using Optimisers: AdamW

y1 = [rand(Float32, 1) for i = 1:3]
y2 = [rand(Float32, 1) for i = 1:3]
function loss(model, x, y)
ŷ = model(x)
y = stack(y, dims=2)
return Flux.mse(ŷ, y)
end

X = [seq_1, seq_2]
Y = [y1, y2]
data = zip(X,Y)
# create dummy data
seq_len, batch_size, input_size = 3, 4, 2
x = [rand(Float32, input_size, batch_size) for _ = 1:seq_len]
y = [rand(Float32, 1, batch_size) for _ = 1:seq_len]

[m(x) for x in seq_init]
# initialize the model and optimizer
model = RecurrentCellModel(input_size, 5)
opt_state = Flux.setup(AdamW(1e-3), model)

opt = Flux.setup(Adam(1e-3), m)
Flux.train!(loss, m, data, opt)
# compute the gradient and update the model
g = gradient(m -> loss(m, x, y),model)[1]
Flux.update!(opt_state, model, g)
```

Then, there's a warmup that is performed over a sequence of length 1 by feeding it with `seq_init`, resulting in a warmup state. The model can then be trained for 1 epoch, where 2 batches are provided (`seq_1` and `seq_2`) and all the timesteps outputs are considered for the loss.
## Handling the whole sequence at once

In this scenario, it is important to note that a single continuous sequence is considered. Since the model state is not reset between the 2 batches, the state of the model flows through the batches, which only makes sense in the context where `seq_1` is the continuation of `seq_init` and so on.
In the above example, we processed the sequence one time step at a time using a recurrent cell. However, it is possible to process the entire sequence at once. This can be done by stacking the input data `x` to form a tensor of size `(input_size, seq_len)` or `(input_size, seq_len, batch_size)`.
One can then use the [`RNN`](@ref), [`LSTM`](@ref) or [`GRU`](@ref) layers to process the entire input tensor.

Batch size would be 1 here as there's only a single sequence within each batch. If the model was to be trained on multiple independent sequences, then these sequences could be added to the input data as a second dimension. For example, in a language model, each batch would contain multiple independent sentences. In such scenario, if we set the batch size to 4, a single batch would be of the shape:
Let's consider the same example as above, but this time we use an `RNN` layer instead of an `RNNCell`:

```julia
x = [rand(Float32, 2, 4) for i = 1:3]
y = [rand(Float32, 1, 4) for i = 1:3]
```

That would mean that we have 4 sentences (or samples), each with 2 features (let's say a very small embedding!) and each with a length of 3 (3 words per sentence). Computing `m(batch[1])`, would still represent `x1 -> y1` in our diagram and returns the first word output, but now for each of the 4 independent sentences (second dimension of the input matrix). Each sentence in the batch will output in its own "column", and the outputs of the different sentences won't mix.

To illustrate, we go through an example of batching with our implementation of `rnn_cell`. The implementation doesn't need to change; the batching comes for "free" from the way Julia does broadcasting and the rules of matrix multiplication.

```julia
output_size = 5
input_size = 2
Wxh = randn(Float32, output_size, input_size)
Whh = randn(Float32, output_size, output_size)
b = randn(Float32, output_size)

function rnn_cell(h, x)
h = tanh.(Wxh * x .+ Whh * h .+ b)
return h, h
struct RecurrentModel{H,C,D}
h0::H
rnn::C
dense::D
end
```

Here, we use the last dimension of the input and the hidden state as the batch dimension. I.e., `h[:, n]` would be the hidden state of the nth sentence in the batch.
Flux.@layer RecurrentModel trainable=(rnn, dense)

```julia
batch_size = 4
x = rand(Float32, input_size, batch_size) # dummy input data
h = rand(Float32, output_size, batch_size) # random initial hidden state
function RecurrentModel(input_size::Int, hidden_size::Int)
return RecurrentModel(
zeros(Float32, hidden_size),
RNN(input_size => hidden_size),
Dense(hidden_size => 1))
end

h, y = rnn_cell(h, x)
```
function (m::RecurrentModel)(x)
z = m.rnn(x, m.h0) # [hidden_size, seq_len, batch_size] or [hidden_size, seq_len]
ŷ = m.dense(z) # [1, seq_len, batch_size] or [1, seq_len]
return ŷ
end

```julia
julia> size(h) == size(y) == (output_size, batch_size)
true
```
seq_len, batch_size, input_size = 3, 4, 2
x = rand(Float32, input_size, seq_len, batch_size)
y = rand(Float32, 1, seq_len, batch_size)

In many situations, such as when dealing with a language model, the sentences in each batch are independent (i.e. the last item of the first sentence of the first batch is independent from the first item of the first sentence of the second batch), so we cannot handle the model as if each batch was the direct continuation of the previous one. To handle such situations, we need to reset the state of the model between each batch, which can be conveniently performed within the loss function:
model = RecurrentModel(input_size, 5)
opt_state = Flux.setup(AdamW(1e-3), model)

```julia
function loss(x, y)
sum(mse(m(xi), yi) for (xi, yi) in zip(x, y))
end
g = gradient(m -> Flux.mse(m(x), y), model)[1]
Flux.update!(opt_state, model, g)
```

A potential source of ambiguity with RNN in Flux can come from the different data layout compared to some common frameworks where data is typically a 3 dimensional array: `(features, seq length, samples)`. In Flux, those 3 dimensions are provided through a vector of seq length containing a matrix `(features, samples)`.

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