# Variables and Derivatives¶

In the example code of this tutorial, we assume for simplicity that the following symbols are already imported.

import math
import numpy as np
import chainer
from chainer import backend
from chainer import backends
from chainer.backends import cuda
from chainer import Function, FunctionNode, gradient_check, report, training, utils, Variable
from chainer import datasets, initializers, iterators, optimizers, serializers
from chainer import Link, Chain, ChainList
import chainer.functions as F
from chainer.training import extensions


As described previously, Chainer uses the “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 a chainer.Variable object. Here we start with a simple ndarray with only one element:

>>> x_data = np.array(, dtype=np.float32)
>>> x = Variable(x_data)


A Variable object supports basic arithmetic operators. In order to compute $$y = x^2 - 2x + 1$$, just write:

>>> y = x**2 - 2 * x + 1


The resulting y is also a Variable object, whose value can be extracted by accessing the array attribute:

>>> y.array
array([16.], dtype=float32)


Note

Variable has two attributes to represent the underlying array: array and data. There is no difference between the two; both refer to exactly the same object. However it is not recommended that you use .data because it might be confused with numpy.ndarray.data attribute.

What y holds is not only the result value. It also holds the history of computation (or computational graph), which enables us to compute its derivative. This is done by calling its backward() method:

>>> y.backward()


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:

>>> 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
array([-1.], dtype=float32)


All these computations can be generalized to a multi-element array input. While single-element arrays are automatically initialized to , to start backward computation from a variable holding a multi-element array, we must set the initial error manually. This is done simply 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()
array([[ 0.,  2.,  4.],
[ 6.,  8., 10.]], dtype=float32)


Note

Many functions taking Variable object(s) are defined in the chainer.functions module. You can combine them to realize complicated functions with automatic backward computation.

Note

Instead of using backward(), you can also calculate gradients of any variables in a computational graph w.r.t. any other variables in the graph using the chainer.grad() function.

## Higher-Order Derivatives¶

Variable also supports higher-order derivatives (a.k.a. double backpropagation).

Let’s see a simple example. First calculate the first-order derivative. Note that enable_double_backprop=True is passed to y.backward().

>>> x = chainer.Variable(np.array([[0, 2, 3], [4, 5, 6]], dtype=np.float32))
>>> y = x ** 3
>>> y.grad = np.ones((2, 3), dtype=np.float32)
>>> y.backward(enable_double_backprop=True)
variable([[  0.,  12.,  27.],
[ 48.,  75., 108.]])
>>> assert (x.grad == (3 * x**2).array).all()


chainer.Variable.grad_var is a Variable for chainer.Variable.grad (which is an ndarray). By passing enable_double_backprop=True to backward(), a computational graph for the backward calculation is recorded. So, you can start backpropagation from x.grad_var to calculate the second-order derivative.

>>> gx = x.grad_var