Automatic differentiation and gradient tape

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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


Gradient tapes

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.

For example:

x = tf.ones((2, 2))
with tf.GradientTape() as t:
  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" tf.GradientTape context.

x = tf.ones((2, 2))
with tf.GradientTape() as t:
  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:
  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 ifs and 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:
    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

Higher-order gradients

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
WARNING:tensorflow:From /usr/local/lib/python3.5/dist-packages/tensorflow/python/ops/ colocate_with (from tensorflow.python.framework.ops) is deprecated and will be removed in a future version.
Instructions for updating:
Colocations handled automatically by placer.

Next Steps

In this tutorial we covered gradient computation in TensorFlow. With that we have enough of the primitives required to build and train neural networks.