Introduction to Chainer¶
This is the first section of the Chainer Tutorial. In this section, you will learn about the following things:
- Pros and cons of existing frameworks and why we are developing Chainer
- Simple example of forward and backward computation
- Usage of parameterized functions and their gradient computation
- Management of a set of parameterized functions (a.k.a. “model” in most frameworks)
- Parameter optimization
After reading this section, you will be able to:
- Compute gradients of some arithmetics
- Write a multi-layer perceptron with Chainer
As mentioned on the front page, Chainer is a flexible framework for neural networks. One major goal is flexibility, so it must enable us to write complex architectures simply and intuitively.
Most existing deep learning frameworks are based on the “Define-and-Run” scheme. That is, first a network is defined and fixed, and then the user periodically feeds it with minibatches. Since the network is statically defined before any forward/backward computation, all the logic must be embedded into the network architecture as data. Consequently, defining a network architecture in such systems (e.g. Caffe) follows a declarative approach. Note that one can still produce such a static network definition using imperative languages (e.g. Torch7 and Theano-based frameworks).
In contrast, Chainer adopts a “Define-by-Run” scheme, i.e., the network is defined on-the-fly via the actual forward computation. More precisely, Chainer stores the history of computation instead of programming logic. This strategy enables to fully leverage the power of programming logic in Python. For example, Chainer does not need any magic to introduce conditionals and loops into the network definitions. The Define-by-Run scheme is the core concept of Chainer. We will show in this tutorial how to define networks dynamically.
This strategy also makes it easy to write multi-GPU parallelization, since logic comes closer to network manipulation. We will review such amenities in later sections of this tutorial.
In example codes of this tutorial, we assume for simplicity that the following symbols are already imported:
import numpy as np from chainer import cuda, Function, FunctionSet, gradient_check, Variable, optimizers import chainer.functions as F
These imports appear widely in Chainer’s codes and examples. For simplicity, we omit this idiom in this tutorial.
As described above, Chainer uses “Define-by-Run” scheme, so forward computation itself defines the network.
In order to start forward computation, we have to set the input array to
Here we start with simple
ndarray with only one element:
>>> x_data = np.array(, dtype=np.float32) >>> x = Variable(x_data)
Chainer currently only supports 32-bit float for most computations.
A Variable object has basic arithmetic operators. In order to compute \(y = x^2 - 2x + 1\), just write
>>> y = x**2 - 2 * x + 1
y is also Variable object, whose value can be extracted by accessing the
>>> y.data array([ 16.], dtype=float32)
y holds is not only the result value.
It also holds the history of computation (or computational graph), which enables us to compute its differentiation.
This is done by calling its
This runs error backpropagation (a.k.a. backprop or reverse-mode automatic differentiation).
Then, the gradient is computed and stored in the
grad attribute of the input variable
>>> x.grad array([ 8.], dtype=float32)
Also we can compute gradients of intermediate variables.
Note that Chainer, by default, releases the gradient arrays of intermediate variables for memory efficiency.
In order to preserve gradient information, pass the
retain_grad argument to the backward method:
>>> z = 2*x >>> y = x**2 - z + 1 >>> y.backward(retain_grad=True) >>> z.grad array([-1.], dtype=float32)
All these computations are easily generalized to multi-element array input.
Note that if we want to start backward computation from a variable holding a multi-element array, we must set the initial error manually.
This is simply done by setting the
grad attribute of the output variable:
>>> x = Variable(np.array([[1, 2, 3], [4, 5, 6]], dtype=np.float32)) >>> y = x**2 - 2*x + 1 >>> y.grad = np.ones((2, 3), dtype=np.float32) >>> y.backward() >>> x.grad array([[ 0., 2., 4.], [ 6., 8., 10.]], dtype=float32)
Many functions taking
Variable object(s) are defined in the
You can combine them to realize complicated functions with automatic backward computation.
In order to write neural networks, we have to use some parameterized functions and optimize their parameters.
As noted above, functions are predefined in
functions module, which also includes parameterized functions.
One of the most fundamental parameterized functions is the
Linear function (a.k.a. fully-connected layer or affine transformation).
It represents a mathematical function \(f(x) = Wx + b\), where the matrix \(W\) and the vector \(b\) are parameters.
A linear function from three-dimensional space to two-dimensional space is defined by:
>>> f = F.Linear(3, 2)
Most functions only accept minibatch input, where the first dimension of input arrays is considered as the batch dimension. In the above Linear function case, input must has shape of (N, 3), where N is the minibatch size.
The parameters of Linear function are stored in
By default, the matrix W is initialized randomly, while the vector b is initialized with zeros.
>>> f.W array([[ 1.33545339, -0.01839679, 0.7662735 ], [-1.21562171, -0.44784674, -0.07128379]], dtype=float32) >>> f.b array([ 0., 0.], dtype=float32)
Instances of a parameterized function class act like usual functions:
>>> x = Variable(np.array([[1, 2, 3], [4, 5, 6]], dtype=np.float32)) >>> y = f(x) >>> y.data array([[ 3.5974803 , -2.3251667 ], [ 9.84747124, -7.52942371]], dtype=float32)
Gradients of parameters are computed by
Note that gradients are accumulated by the method rather than overwritten.
So first you must initialize gradients to zero to renew the computation.
Gradients of Linear function are stored in
>>> f.gW.fill(0) >>> f.gb.fill(0)
This procedure is simplified by FunctionSet and Optimizer, which we will see in the next seciton.
Now we can compute the gradients of parameters by simply calling backward method:
>>> y.grad = np.ones((2, 2), dtype=np.float32) >>> y.backward() >>> >>> f.gW array([[ 5., 7., 9.], [ 5., 7., 9.]], dtype=float32) >>> f.gb array([ 2., 2.], dtype=float32)
Most neural network architectures contain multiple parameterized functions.
FunctionSet makes it easy to manage them.
This class acts like a simple object, with attributes initialized by keyword arguments of the initializer:
>>> model = FunctionSet( ... l1 = F.Linear(4, 3), ... l2 = F.Linear(3, 2), ... ) >>> model.l1 <chainer.functions.linear.Linear object at 0x7f7f03e4f350> >>> model.l2 <chainer.functions.linear.Linear object at 0x7f7f03e4f590>
You can also add additional functions later by setting attributes:
>>> model.l3 = F.Linear(2, 2)
model is just an object with functions stored as its attributes, we can use these functions in forward computation:
>>> x = Variable(np.array([[1, 2, 3, 4], [5, 6, 7, 8]], dtype=np.float32)) >>> h1 = model.l1(x) >>> h2 = model.l2(h1) >>> h3 = model.l3(h2)
One of the features of FunctionSet is the ability to collect parameters and gradients.
A tuple of all parameters and a tuple of all gradients are extracted by
FunctionSet.gradients properties, respectively.
Optimizer is the last core feature of Chainer described in this section.
It runs a numerical optimization algorithm given tuples of parameters and gradients.
Many algorithms are implemented in
Here we use the simplest one, called Stochastic Gradient Descent:
>>> optimizer = optimizers.SGD() >>> optimizer.setup(model.collect_parameters())
Since Optimizer does not know the functions that actually own the parameters and gradients, once parameters and gradients are given to Optimizer, functions must use same parameter and gradient array objects throughout all forward/backward computations.
In order to run optimization, you first have to compute gradients.
Zeroing the initial gradient arrays are simply done by calling
We have done the zeroing manually in the previous section. The line above is an equivalent and simpler way to initialize the gradients.
Then, after computing gradient of each parameter,
update() method runs one iteration of optimization:
>>> (compute gradient) >>> optimizer.update()
Optimizer also contains some features related to parameter and gradient manipulation, e.g. weight decay and gradient clipping.
Example: Multi-layer Perceptron on MNIST¶
Now you can solve a multiclass classification task using a multi-layer perceptron.
Here we use hand-written digits dataset called MNIST, which is the long-standing de-facto “hello world” of machine learning.
This MNIST example is also found in
examples/mnist directory of the official repository.
>>> from sklearn.datasets import fetch_mldata >>> mnist = fetch_mldata('MNIST original')
The mnist dataset consists of 70,000 grayscale images of size 28x28 (i.e. 784 pixels) and corresponding digit labels. First, we scale pixels to [0, 1] values, and divide the dataset into 60,000 training samples and 10,000 test samples.
>>> x_all = mnist.data.astype(np.float32) / 255 >>> y_all = mnist.target.astype(np.int32) >>> x_train, x_test = np.split(x_all, ) >>> y_train, y_test = np.split(y_all, )
Next, we want to define the architecture. We use a simple three-layer rectifier network with 100 units per layer as an example. Before defining the forward routine, we have to prepare our parameterized functions:
>>> model = FunctionSet( ... l1 = F.Linear(784, 100), ... l2 = F.Linear(100, 100), ... l3 = F.Linear(100, 10), ... ) >>> optimizer = optimizers.SGD() >>> optimizer.setup(model.collect_parameters())
model.l3 is the final linear layer whose output corresponds to the ten digits.
We also set up the optimizer here.
Now we can define the forward routine using these Linear functions. Typically it is defined as a simple python function given input arrays:
>>> def forward(x_data, y_data): ... x = Variable(x_data) ... t = Variable(y_data) ... h1 = F.relu(model.l1(x)) ... h2 = F.relu(model.l2(h1)) ... y = model.l3(h2) ... return F.softmax_cross_entropy(y, t), F.accuracy(y, t)
This function uses
functions.relu() as an activation function.
Since ReLU does not have parameters to optimize, it does not need to be included in model.
functions.softmax_cross_entropy() computes the loss function of softmax regression.
functions.accuracy() computes the classification accuracy of this minibatch.
Finally, we can write a learning loop as following:
>>> batchsize = 100 >>> for epoch in xrange(20): ... print 'epoch', epoch ... indexes = np.random.permutation(60000) ... for i in xrange(0, 60000, batchsize): ... x_batch = x_train[indexes[i : i + batchsize]] ... y_batch = y_train[indexes[i : i + batchsize]] ... ... optimizer.zero_grads() ... loss, accuracy = forward(x_batch, y_batch) ... loss.backward() ... optimizer.update()
Only the last four lines are the code related to Chainer, which are already described above.
Here you find that, at each iteration, the network is defined by forward computation, used for backprop, and then disposed. By leveraging this “Define-by-Run” scheme, you can imagine that recurrent nets with variable length input are simply handled by just using loop over different length input for each iteration.
After or during optimization, we want to evaluate the model on the test set. It can be achieved simply by calling forward function:
>>> sum_loss, sum_accuracy = 0, 0 >>> for i in xrange(0, 10000, batchsize): ... x_batch = x_test[i : i + batchsize] ... y_batch = y_test[i : i + batchsize] ... loss, accuracy = forward(x_batch, y_batch) ... sum_loss += loss.data * batchsize ... sum_accuracy += accuracy.data * batchsize ... >>> mean_loss = sum_loss / 10000 >>> mean_accuracy = sum_accuracy / 10000
The example code contains GPU support, though the essential part is same as the code in this tutorial. We will review in later sections how to use GPU(s).