variables.rst

Variables and Derivatives

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 ~numpy.ndarray with only one element:

>>> x_data = np.array([5], dtype=np.float32) >>> x = Variable(x_data)

A Variable object supports basic arithmetic operators. In order to compute y = x² − 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 ~chainer.Variable.array attribute:

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

Note

~chainer.Variable has two attributes to represent the underlying array: ~chainer.Variable.array and ~chainer.Variable.data. There is no difference between the two; both refer to exactly the same object. However it is not recommended to 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 ~Variable.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 ~chainer.Variable.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 >>> y.backward(retain_grad=True) >>> z.grad array([-1.], dtype=float32)

All these computations can be generalized to a multi-element array input. While single-element arrays are automatically initialized to [1], 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 ~chainer.Variable.grad attribute of the output variable:

>>> x = Variable(np.array([[1, 2, 3], [4, 5, 6]], dtype=np.float32)) >>> y = x*2 - 2x + 1 >>> y.grad = np.ones((2, 3), dtype=np.float32) >>> y.backward() >>> x.grad array([[ 0., 2., 4.], [ 6., 8., 10.]], dtype=float32)

Note

Many functions taking ~chainer.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 ~chainer.Variable.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

~chainer.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) >>> x.grad_var variable([[ 0., 12., 27.], [ 48., 75., 108.]]) >>> assert x.grad_var.array is x.grad >>> assert (x.grad == (3 * x**2).array).all()

chainer.Variable.grad_var is a ~chainer.Variable for chainer.Variable.grad (which is an ~numpy.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 >>> x.cleargrad() >>> gx.grad = np.ones((2, 3), dtype=np.float32) >>> gx.backward() >>> x.grad array([[ 0., 12., 18.], [24., 30., 36.]], dtype=float32) >>> assert (x.grad == (6 * x).array).all()