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In the previous tutorial we introduced
Tensors and operations on them. In this tutorial we will cover automatic differentiation, a key technique for optimizing machine learning models.
import tensorflow as tf tf.enable_eager_execution()
TensorFlow provides the tf.GradientTape API for automatic differentiation - computing the gradient of a computation with respect to its input variables. Tensorflow "records" all operations executed inside the context of a
tf.GradientTape onto a "tape". Tensorflow then uses that tape and the gradients associated with each recorded operation to compute the gradients of a "recorded" computation using reverse mode differentiation.
x = tf.ones((2, 2)) with tf.GradientTape() as t: t.watch(x) y = tf.reduce_sum(x) z = tf.multiply(y, y) # Derivative of z with respect to the original input tensor x dz_dx = t.gradient(z, x) for i in [0, 1]: for j in [0, 1]: assert dz_dx[i][j].numpy() == 8.0
You can also request gradients of the output with respect to intermediate values computed during a "recorded"
x = tf.ones((2, 2)) with tf.GradientTape() as t: t.watch(x) y = tf.reduce_sum(x) z = tf.multiply(y, y) # Use the tape to compute the derivative of z with respect to the # intermediate value y. dz_dy = t.gradient(z, y) assert dz_dy.numpy() == 8.0
By default, the resources held by a GradientTape are released as soon as GradientTape.gradient() method is called. To compute multiple gradients over the same computation, create a
persistent gradient tape. This allows multiple calls to the
gradient() method. as resources are released when the tape object is garbage collected. For example:
x = tf.constant(3.0) with tf.GradientTape(persistent=True) as t: t.watch(x) y = x * x z = y * y dz_dx = t.gradient(z, x) # 108.0 (4*x^3 at x = 3) dy_dx = t.gradient(y, x) # 6.0 del t # Drop the reference to the tape
Recording control flow
Because tapes record operations as they are executed, Python control flow (using
whiles for example) is naturally handled:
def f(x, y): output = 1.0 for i in range(y): if i > 1 and i < 5: output = tf.multiply(output, x) return output def grad(x, y): with tf.GradientTape() as t: t.watch(x) out = f(x, y) return t.gradient(out, x) x = tf.convert_to_tensor(2.0) assert grad(x, 6).numpy() == 12.0 assert grad(x, 5).numpy() == 12.0 assert grad(x, 4).numpy() == 4.0
Operations inside of the
GradientTape context manager are recorded for automatic differentiation. If gradients are computed in that context, then the gradient computation is recorded as well. As a result, the exact same API works for higher-order gradients as well. For example:
x = tf.Variable(1.0) # Create a Tensorflow variable initialized to 1.0 with tf.GradientTape() as t: with tf.GradientTape() as t2: y = x * x * x # Compute the gradient inside the 't' context manager # which means the gradient computation is differentiable as well. dy_dx = t2.gradient(y, x) d2y_dx2 = t.gradient(dy_dx, x) assert dy_dx.numpy() == 3.0 assert d2y_dx2.numpy() == 6.0
In this tutorial we covered gradient computation in TensorFlow. With that we have enough of the primitives required to build and train neural networks.