code: def custom_gradient(f=None):
      """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:
  
      ```python
      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:
  
      ```python
      with tf.GradientTape() as tape:
        tape.watch(x)
        y=log1pexp(x)
      dy_dx = tape.gradient(y, x) # Will be NaN when evaluated.
      ```
  
      The gradient expression can be analytically simplified to provide numerical
      stability:
  
      ```python
      @tf.custom_gradient
      def log1pexp(x):
        e = tf.exp(x)
        def grad(upstream):
          return upstream * (1 - 1 / (1 + e))
        return tf.math.log(1 + e), grad
      ```
  
      With this definition, the gradient `dy_dx` at `x = 100` will be correctly
      evaluated as 1.0.
  
      The variable `upstream` is defined as the upstream gradient. i.e. the gradient
      from all the layers or functions originating from this layer. The above
      example has no upstream functions, therefore `upstream = dy/dy = 1.0`.
  
      Assume that `x_i` is `log1pexp` in the forward pass `x_1 = x_1(x_0)`,
      `x_2 = x_2(x_1)`, ..., `x_i = x_i(x_i-1)`, ..., `x_n = x_n(x_n-1)`. By
      chain rule we know that `dx_n/dx_0 = dx_n/dx_n-1 * dx_n-1/dx_n-2 * ... *
      dx_i/dx_i-1 * ... * dx_1/dx_0`.
  
      In this case the gradient of our current function defined as
      `dx_i/dx_i-1 = (1 - 1 / (1 + e))`. The upstream gradient `upstream` would be
      `dx_n/dx_n-1 * dx_n-1/dx_n-2 * ... * dx_i+1/dx_i`. The upstream gradient
      multiplied by the current gradient is then passed downstream.
  
      In case the function takes multiple variables as input, the `grad`
      function must also return  the same number of variables.
      We take the function `z = x * y` as an example.
  
      >>> @tf.custom_gradient
      ... def bar(x, y):
      ...   def grad(upstream):
      ...     dz_dx = y
      ...     dz_dy = x
      ...     return upstream * dz_dx, upstream * dz_dy
      ...   z = x * y
      ...   return z, grad
      >>> x = tf.constant(2.0, dtype=tf.float32)
      >>> y = tf.constant(3.0, dtype=tf.float32)
      >>> with tf.GradientTape(persistent=True) as tape:
      ...   tape.watch(x)
      ...   tape.watch(y)
      ...   z = bar(x, y)
      >>> z
      <tf.Tensor: shape=(), dtype=float32, numpy=6.0>
      >>> tape.gradient(z, x)
      <tf.Tensor: shape=(), dtype=float32, numpy=3.0>
      >>> tape.gradient(z, y)
      <tf.Tensor: shape=(), dtype=float32, numpy=2.0>
  
      Nesting custom gradients can lead to unintuitive results. The default
      behavior does not correspond to n-th order derivatives. For example
  
      ```python
      @tf.custom_gradient
      def op(x):
        y = op1(x)
        @tf.custom_gradient
        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:
  
      ```python
      @tf.custom_gradient
      def op_with_fused_backprop(x):
        y, x_grad = fused_op(x)
        def first_order_gradient(dy):
          @tf.custom_gradient
          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.
  
      The examples above illustrate how to specify custom gradients for functions
      which do not read from variables. The following example uses variables, which
      require special handling because they are effectively inputs of the forward
      function.
  
      >>> weights = tf.Variable(tf.ones([2]))  # Trainable variable weights
      >>> @tf.custom_gradient
      ... def linear_poly(x):
      ...   # Creating polynomial
      ...   poly = weights[1] * x + weights[0]
      ...
      ...   def grad_fn(dpoly, variables):
      ...     # dy/dx = weights[1] and we need to left multiply dpoly
      ...     grad_xs = dpoly * weights[1]  # Scalar gradient
      ...
      ...     grad_vars = []  # To store gradients of passed variables
      ...     assert variables is not None
      ...     assert len(variables) == 1
      ...     assert variables[0] is weights
      ...     # Manually computing dy/dweights
      ...     dy_dw = dpoly * tf.stack([x ** 1, x ** 0])
      ...     grad_vars.append(
      ...         tf.reduce_sum(tf.reshape(dy_dw, [2, -1]), axis=1)
      ...     )
      ...     return grad_xs, grad_vars
      ...   return poly, grad_fn
      >>> x = tf.constant([1., 2., 3.])
      >>> with tf.GradientTape(persistent=True) as tape:
      ...   tape.watch(x)
      ...   poly = linear_poly(x)
      >>> poly # poly = x + 1
      <tf.Tensor: shape=(3,),
        dtype=float32,
        numpy=array([2., 3., 4.], dtype=float32)>
      >>> tape.gradient(poly, x)  # conventional scalar gradient dy/dx
      <tf.Tensor: shape=(3,),
        dtype=float32,
        numpy=array([1., 1., 1.], dtype=float32)>
      >>> tape.gradient(poly, weights)
      <tf.Tensor: shape=(2,), dtype=float32, numpy=array([6., 3.], dtype=float32)>
  
      Above example illustrates usage of trainable variable `weights`.
      In the example, the inner `grad_fn` accepts an extra `variables` input
      parameter and also returns an extra `grad_vars` output. That extra argument
      is passed if the forward function reads any variables. You need to
      compute the gradient w.r.t. each of those `variables` and output it as a list
      of `grad_vars`. Note here that default value of `variables` is set to `None`
      when no variables are used in the forward function.
  
      It should be noted `tf.GradientTape` is still watching the forward pass of a
      `tf.custom_gradient`, and will use the ops it watches. As a consequence,
      calling `tf.function` while the tape is still watching leads
      to a gradient graph being built. If an op is used in `tf.function` without
      registered gradient, a `LookupError` will be raised.
  
      Users can insert `tf.stop_gradient` to customize this behavior. This
      is demonstrated in the example below. `tf.random.shuffle` does not have a
      registered gradient. As a result `tf.stop_gradient` is used to avoid the
      `LookupError`.
  
      ```python
      x = tf.constant([0.3, 0.5], dtype=tf.float32)
  
      @tf.custom_gradient
      def test_func_with_stop_grad(x):
        @tf.function
        def _inner_func():
          # Avoid exception during the forward pass
          return tf.stop_gradient(tf.random.shuffle(x))
          # return tf.random.shuffle(x)  # This will raise
  
        res = _inner_func()
        def grad(upstream):
          return upstream  # Arbitrarily defined custom gradient
        return res, grad
  
      with tf.GradientTape() as g:
        g.watch(x)
        res = test_func_with_stop_grad(x)
  
      g.gradient(res, x)
      ```
  
      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 `Variable`s, the enclosing variable
      scope must be using
      [ResourceVariables](https://www.tensorflow.org/guide/migrate/tf1_vs_tf2#resourcevariables_instead_of_referencevariables).
  
      Args:
        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 `Tensor`s the same size as (flattened) `x` - the derivatives
             of `Tensor`s in `y` with respect to the `Tensor`s in `x`.  `grad_ys` is
             a sequence of `Tensor`s 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 `Variable`s (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 `Variable`s, 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 `Tensor`s in `y` with respect to the variables
             (that is, grad_vars has one Tensor per variable in variables).
  
      Returns:
        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]`.
      """
  
      if f is None:
          return lambda f: custom_gradient(f=f)
  
      @Bind.decorator
      def decorated(wrapped, args, kwargs):
          """Decorated function with custom gradient."""
          if context.executing_eagerly():
              return _eager_mode_decorator(wrapped, args, kwargs)
          else:
              return _graph_mode_decorator(wrapped, args, kwargs)
  
      return tf_decorator.make_decorator(f, decorated(f))
  
imports: from tensorflow.python.eager import backprop
  from tensorflow.python.eager import context
  from tensorflow.python.eager import record
  from tensorflow.python.framework import composite_tensor_gradient
  from tensorflow.python.framework import dtypes
  from tensorflow.python.framework import ops
  from tensorflow.python.ops import array_ops
  from tensorflow.python.ops import gen_array_ops
  from tensorflow.python.ops import handle_data_util
  from tensorflow.python.ops import math_ops
  from tensorflow.python.ops import op_selector
  from tensorflow.python.ops import resource_variable_ops
  from tensorflow.python.ops import variable_scope
  from tensorflow.python.ops.unconnected_gradients import UnconnectedGradients
  from tensorflow.python.platform import tf_logging
  from tensorflow.python.util import nest
  from tensorflow.python.util import tf_decorator
  from tensorflow.python.util import tf_inspect
  from tensorflow.python.util import variable_utils
  from tensorflow.python.util.tf_export import tf_export
label_desc: Use image node_id 1 for calling a global function with example usage: custom_gradient(f) and returns: tf_decorator, function, _eager_mode_decorator, _graph_mode_decorator