Now th # Update the parameter and reset the gradient update!(W, -0.1Δ) -loss(x, y) # ~ 2.5
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.
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 (σ) in between them. In the above style we could write this as:
W1 = param(rand(3, 5))
+loss(x, y) # ~ 2.5The 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.
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 (σ) in between them. In the above style we could write this as:
using Flux
+
+W1 = param(rand(3, 5))
b1 = param(rand(3))
layer1(x) = W1 * x .+ b1
diff --git a/dev/models/layers/index.html b/dev/models/layers/index.html
index 6e4bfd32..3bc4ca05 100644
--- a/dev/models/layers/index.html
+++ b/dev/models/layers/index.html
@@ -11,28 +11,28 @@ m(5) == 26
m = Chain(Dense(10, 5), Dense(5, 2))
x = rand(10)
-m(x) == m[2](m[1](x))Chain also supports indexing and slicing, e.g. m[2] or m[1:end-1]. m[1:3](x) will calculate the output of the first three layers.
Flux.Dense — Type.Dense(in::Integer, out::Integer, σ = identity)Creates a traditional Dense layer with parameters W and b.
y = σ.(W * x .+ b)The input x must be a vector of length in, or a batch of vectors represented as an in × N matrix. The out y will be a vector or batch of length out.
julia> d = Dense(5, 2)
+m(x) == m[2](m[1](x))Chain also supports indexing and slicing, e.g. m[2] or m[1:end-1]. m[1:3](x) will calculate the output of the first three layers.
Flux.Dense — Type.Dense(in::Integer, out::Integer, σ = identity)Creates a traditional Dense layer with parameters W and b.
y = σ.(W * x .+ b)The input x must be a vector of length in, or a batch of vectors represented as an in × N matrix. The out y will be a vector or batch of length out.
julia> d = Dense(5, 2)
Dense(5, 2)
julia> d(rand(5))
Tracked 2-element Array{Float64,1}:
0.00257447
- -0.00449443Flux.Conv — Type.Conv(size, in=>out)
-Conv(size, in=>out, relu)Standard convolutional layer. size should be a tuple like (2, 2). in and out specify the number of input and output channels respectively.
Data should be stored in WHCN order. In other words, a 100×100 RGB image would be a 100×100×3 array, and a batch of 50 would be a 100×100×3×50 array.
Takes the keyword arguments pad, stride and dilation.
Flux.MaxPool — Type.MaxPool(k)Max pooling layer. k stands for the size of the window for each dimension of the input.
Takes the keyword arguments pad and stride.
Flux.MeanPool — Type.MeanPool(k)Mean pooling layer. k stands for the size of the window for each dimension of the input.
Takes the keyword arguments pad and stride.
Flux.DepthwiseConv — Type.DepthwiseConv(size, in)
+ -0.00449443Flux.Conv — Type.Conv(size, in=>out)
+Conv(size, in=>out, relu)Standard convolutional layer. size should be a tuple like (2, 2). in and out specify the number of input and output channels respectively.
Data should be stored in WHCN order. In other words, a 100×100 RGB image would be a 100×100×3 array, and a batch of 50 would be a 100×100×3×50 array.
Takes the keyword arguments pad, stride and dilation.
Flux.MaxPool — Type.MaxPool(k)Max pooling layer. k stands for the size of the window for each dimension of the input.
Takes the keyword arguments pad and stride.
Flux.MeanPool — Type.MeanPool(k)Mean pooling layer. k stands for the size of the window for each dimension of the input.
Takes the keyword arguments pad and stride.
Flux.DepthwiseConv — Type.DepthwiseConv(size, in)
DepthwiseConv(size, in=>mul)
-DepthwiseConv(size, in=>mul, relu)Depthwise convolutional layer. size should be a tuple like (2, 2). in and mul specify the number of input channels and channel multiplier respectively. In case the mul is not specified it is taken as 1.
Data should be stored in WHCN order. In other words, a 100×100 RGB image would be a 100×100×3 array, and a batch of 50 would be a 100×100×3×50 array.
Takes the keyword arguments pad and stride.
Much like the core layers above, but can be used to process sequence data (as well as other kinds of structured data).
Flux.RNN — Function.RNN(in::Integer, out::Integer, σ = tanh)The most basic recurrent layer; essentially acts as a Dense layer, but with the output fed back into the input each time step.
Flux.LSTM — Function.LSTM(in::Integer, out::Integer)Long Short Term Memory recurrent layer. Behaves like an RNN but generally exhibits a longer memory span over sequences.
See this article for a good overview of the internals.
Flux.GRU — Function.GRU(in::Integer, out::Integer)Gated Recurrent Unit layer. Behaves like an RNN but generally exhibits a longer memory span over sequences.
See this article for a good overview of the internals.
Flux.Recur — Type.Recur(cell)Recur takes a recurrent cell and makes it stateful, managing the hidden state in the background. cell should be a model of the form:
h, y = cell(h, x...)For example, here's a recurrent network that keeps a running total of its inputs.
accum(h, x) = (h+x, x)
+DepthwiseConv(size, in=>mul, relu)Depthwise convolutional layer. size should be a tuple like (2, 2). in and mul specify the number of input channels and channel multiplier respectively. In case the mul is not specified it is taken as 1.
Data should be stored in WHCN order. In other words, a 100×100 RGB image would be a 100×100×3 array, and a batch of 50 would be a 100×100×3×50 array.
Takes the keyword arguments pad and stride.
Much like the core layers above, but can be used to process sequence data (as well as other kinds of structured data).
Flux.RNN — Function.RNN(in::Integer, out::Integer, σ = tanh)The most basic recurrent layer; essentially acts as a Dense layer, but with the output fed back into the input each time step.
Flux.LSTM — Function.LSTM(in::Integer, out::Integer)Long Short Term Memory recurrent layer. Behaves like an RNN but generally exhibits a longer memory span over sequences.
See this article for a good overview of the internals.
Flux.GRU — Function.GRU(in::Integer, out::Integer)Gated Recurrent Unit layer. Behaves like an RNN but generally exhibits a longer memory span over sequences.
See this article for a good overview of the internals.
Flux.Recur — Type.Recur(cell)Recur takes a recurrent cell and makes it stateful, managing the hidden state in the background. cell should be a model of the form:
h, y = cell(h, x...)For example, here's a recurrent network that keeps a running total of its inputs.
accum(h, x) = (h+x, x)
rnn = Flux.Recur(accum, 0)
rnn(2) # 2
rnn(3) # 3
rnn.state # 5
rnn.(1:10) # apply to a sequence
-rnn.state # 60Non-linearities that go between layers of your model. Most of these functions are defined in NNlib but are available by default in Flux.
Note that, unless otherwise stated, activation functions operate on scalars. To apply them to an array you can call σ.(xs), relu.(xs) and so on.
NNlib.σ — Function.σ(x) = 1 / (1 + exp(-x))Classic sigmoid activation function.
NNlib.relu — Function.relu(x) = max(0, x)Rectified Linear Unit activation function.
NNlib.leakyrelu — Function.leakyrelu(x) = max(0.01x, x)Leaky Rectified Linear Unit activation function. You can also specify the coefficient explicitly, e.g. leakyrelu(x, 0.01).
NNlib.elu — Function.elu(x, α = 1) =
+rnn.state # 60Non-linearities that go between layers of your model. Most of these functions are defined in NNlib but are available by default in Flux.
Note that, unless otherwise stated, activation functions operate on scalars. To apply them to an array you can call σ.(xs), relu.(xs) and so on.
NNlib.σ — Function.σ(x) = 1 / (1 + exp(-x))Classic sigmoid activation function.
NNlib.relu — Function.relu(x) = max(0, x)Rectified Linear Unit activation function.
NNlib.leakyrelu — Function.leakyrelu(x) = max(0.01x, x)Leaky Rectified Linear Unit activation function. You can also specify the coefficient explicitly, e.g. leakyrelu(x, 0.01).
NNlib.elu — Function.elu(x, α = 1) =
x > 0 ? x : α * (exp(x) - 1)Exponential Linear Unit activation function. See Fast and Accurate Deep Network Learning by Exponential Linear Units. You can also specify the coefficient explicitly, e.g. elu(x, 1).
NNlib.swish — Function.swish(x) = x * σ(x)Self-gated actvation function. See Swish: a Self-Gated Activation Function.
These layers don't affect the structure of the network but may improve training times or reduce overfitting.
Flux.testmode! — Function.Flux.BatchNorm — Type.Flux.BatchNorm — Type.BatchNorm(channels::Integer, σ = identity;
initβ = zeros, initγ = ones,
ϵ = 1e-8, momentum = .1)Batch Normalization layer. The channels input should be the size of the channel dimension in your data (see below).
Given an array with N dimensions, call the N-1th the channel dimension. (For a batch of feature vectors this is just the data dimension, for WHCN images it's the usual channel dimension.)
BatchNorm computes the mean and variance for each each W×H×1×N slice and shifts them to have a new mean and variance (corresponding to the learnable, per-channel bias and scale parameters).
See Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift.
Example:
m = Chain(
Dense(28^2, 64),
BatchNorm(64, relu),
Dense(64, 10),
BatchNorm(10),
- softmax)Flux.Dropout — Type.Dropout(p)A Dropout layer. For each input, either sets that input to 0 (with probability p) or scales it by 1/(1-p). This is used as a regularisation, i.e. it reduces overfitting during training.
Does nothing to the input once in testmode!.
Flux.LayerNorm — Type.LayerNorm(h::Integer)A normalisation layer designed to be used with recurrent hidden states of size h. Normalises the mean/stddev of each input before applying a per-neuron gain/bias.