Model-Building Basics
Taking Gradients
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.)
using Flux.Tracker
+
Model-Building Basics
Taking Gradients
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
+julia> f(x) = 3x^2 + 2x + 1;
-# df/dx = 6x + 2
-df(x) = Tracker.gradient(f, x; nest = true)[1]
+julia> df(x) = Tracker.gradient(f, x; nest = true)[1]; # df/dx = 6x + 2
-df(2) # 14.0 (tracked)
+julia> df(2)
+14.0 (tracked)
-# d²f/dx² = 6
-d2f(x) = Tracker.gradient(df, x; nest = true)[1]
+julia> d2f(x) = Tracker.gradient(df, x; nest = true)[1]; # d²f/dx² = 6
-d2f(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:
f(W, b, x) = W * x + b
+julia> d2f(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 (tracked), 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 params at once.
W = param(2) # 2.0 (tracked)
-b = param(3) # 3.0 (tracked)
+julia> Tracker.gradient(f, 2, 3, 4)
+(4.0 (tracked), 1.0 (tracked), 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 params at once.
julia> using Flux
-f(x) = W * x + b
+julia> W = param(2)
+2.0 (tracked)
-grads = Tracker.gradient(() -> f(4), params(W, b))
+julia> b = param(3)
+3.0 (tracked)
-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.
W = rand(2, 5)
+julia> f(x) = W * x + b;
+
+julia> grads = Tracker.gradient(() -> f(4), params(W, b));
+
+julia> grads[W]
+4.0
+
+julia> 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.
W = rand(2, 5)
b = rand(2)
predict(x) = W*x .+ b
diff --git a/dev/models/layers/index.html b/dev/models/layers/index.html
index b832b765..dffbd228 100644
--- a/dev/models/layers/index.html
+++ b/dev/models/layers/index.html
@@ -11,29 +11,29 @@ 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.
sourceFlux.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.
sourceFlux.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
- -0.00449443
sourceFlux.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 100×100×3×1 array, and a batch of 50 would be a 100×100×3×50 array.
Takes the keyword arguments pad, stride and dilation.
sourceFlux.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.
sourceFlux.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.
sourceAdditional Convolution Layers
Flux.DepthwiseConv — Type.DepthwiseConv(size, in)
+ -0.00449443
sourceFlux.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 100×100×3×1 array, and a batch of 50 would be a 100×100×3×50 array.
Takes the keyword arguments pad, stride and dilation.
sourceFlux.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.
sourceFlux.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.
sourceAdditional Convolution Layers
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.
sourceFlux.ConvTranspose — Type.ConvTranspose(size, in=>out)
-ConvTranspose(size, in=>out, relu)
Standard convolutional transpose 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.
sourceRecurrent Layers
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.
sourceFlux.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.
sourceFlux.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.
sourceFlux.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.
sourceFlux.ConvTranspose — Type.ConvTranspose(size, in=>out)
+ConvTranspose(size, in=>out, relu)
Standard convolutional transpose 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.
sourceRecurrent Layers
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.
sourceFlux.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.
sourceFlux.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.
sourceFlux.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 # 60
sourceActivation Functions
Non-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 # 60
sourceActivation Functions
Non-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.
Normalisation & Regularisation
These layers don't affect the structure of the network but may improve training times or reduce overfitting.
Flux.testmode! — Function.sourceFlux.BatchNorm — Type.sourceFlux.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)
sourceFlux.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!.
sourceFlux.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.
source