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Decorator to define a function with a custom gradient.

This decorator allows fine grained control over the gradients of a sequence for operations. This may be useful for multiple reasons, including providing a more efficient or numerically stable gradient for a sequence of operations.

For example, consider the following function that commonly occurs in the computation of cross entropy and log likelihoods:

def log1pexp(x):
  return tf.math.log(1 + tf.exp(x))

Due to numerical instability, the gradient of this function evaluated at x=100 is NaN. For example:

x = tf.constant(100.)
y = log1pexp(x)
dy = tf.gradients(y, x) # Will be NaN when evaluated.

The gradient expression can be analytically simplified to provide numerical stability:

def log1pexp(x):
  e = tf.exp(x)
  def grad(dy):
    return dy * (1 - 1 / (1 + e))
  return tf.math.log(1 + e), grad

With this definition, the gradient at x=100 will be correctly evaluated as 1.0.

Nesting custom gradients can lead to unintuitive results. The default behavior does not correspond to n-th order derivatives. For example

def op(x):
  y = op1(x)
  def grad_fn(dy):
    gdy = op2(x, y, dy)
    def grad_grad_fn(ddy):  # Not the 2nd order gradient of op w.r.t. x.
      return op3(x, y, dy, ddy)
    return gdy, grad_grad_fn
  return y, grad_fn

The function grad_grad_fn will be calculating the first order gradient of grad_fn with respect to dy, which is used to generate forward-mode gradient graphs from backward-mode gradient graphs, but is not the same as the second order gradient of op with respect to x.

Instead, wrap nested @tf.custom_gradients in another function:

def op_with_fused_backprop(x):
  y, x_grad = fused_op(x)
  def first_order_gradient(dy):
    def first_order_custom(unused_x):
      def second_order_and_transpose(ddy):
        return second_order_for_x(...), gradient_wrt_dy(...)
      return x_grad, second_order_and_transpose
    return dy * first_order_custom(x)
  return y, first_order_gradient

Additional arguments to the inner @tf.custom_gradient-decorated function control the expected return values of the innermost function.

See also tf.RegisterGradient which registers a gradient function for a primitive TensorFlow operation. tf.custom_gradient on the other hand allows for fine grained control over the gradient computation of a sequence of operations.

Note that if the decorated function uses Variables, the enclosing variable scope must be using ResourceVariables.

f function f(*x) that returns a tuple (y, grad_fn) where:

  • x is a sequence of (nested structures of) Tensor inputs to the function.
  • y is a (nested structure of) Tensor outputs of applying TensorFlow operations in f to x.
  • grad_fn is a function with the signature g(*grad_ys) which returns a list of Tensors the same size as (flattened) x - the derivatives of Tensors in y with respect to the Tensors in x. grad_ys is a sequence of Tensors the same size as (flattened) y holding the initial value gradients for each Tensor in y.

In a pure mathematical sense, a vector-argument vector-valued function f's derivatives should be its Jacobian matrix J. Here we are expressing the Jacobian J as a function grad_fn which defines how J will transform a vector grad_ys when left-multiplied with it (grad_ys * J, the vector-Jacobian product, or VJP). This functional representation of a matrix is convenient to use for chain-rule calculation (in e.g. the back-propagation algorithm).

If f uses Variables (that are not part of the inputs), i.e. through get_variable, then grad_fn should have signature g(*grad_ys, variables=None), where variables is a list of the Variables, and return a 2-tuple (grad_xs, grad_vars), where grad_xs is the same as above, and grad_vars is a list<Tensor> with the derivatives of Tensors in y with respect to the variables (that is, grad_vars has one Tensor per variable in variables).

A function h(x) which returns the same value as f(x)[0] and whose gradient (as calculated by tf.gradients) is determined by f(x)[1].