# tf.custom_gradient
|---|
| [ View source on GitHub](https://github.com/tensorflow/tensorflow/blob/v2.13.1/tensorflow/python/ops/custom_gradient.py#L45-L301) |
Decorator to define a function with a custom gradient.
#### View aliases
**Compat aliases for migration**
See
[Migration guide](https://www.tensorflow.org/guide/migrate) for
more details.
[`tf.compat.v1.custom_gradient`](https://www.tensorflow.org/api_docs/python/tf/custom_gradient)
tf.custom_gradient(
f=None
)
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:
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:
@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
tape.gradient(z, x)
tape.gradient(z, y)
Nesting custom gradients can lead to unintuitive results. The default
behavior does not correspond to n-th order derivatives. For example
@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:
@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`](https://www.tensorflow.org/versions/r2.13/api_docs/python/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
tape.gradient(poly, x) # conventional scalar gradient dy/dx
tape.gradient(poly, weights)
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`](https://www.tensorflow.org/versions/r2.13/api_docs/python/tf/GradientTape) is still watching the forward pass of a
[`tf.custom_gradient`](https://www.tensorflow.org/versions/r2.13/api_docs/python/tf/custom_gradient), and will use the ops it watches. As a consequence,
calling [`tf.function`](https://www.tensorflow.org/versions/r2.13/api_docs/python/tf/function) while the tape is still watching leads
to a gradient graph being built. If an op is used in [`tf.function`](https://www.tensorflow.org/versions/r2.13/api_docs/python/tf/function) without
registered gradient, a `LookupError` will be raised.
Users can insert [`tf.stop_gradient`](https://www.tensorflow.org/versions/r2.13/api_docs/python/tf/stop_gradient) to customize this behavior. This
is demonstrated in the example below. [`tf.random.shuffle`](https://www.tensorflow.org/versions/r2.13/api_docs/python/tf/random/shuffle) does not have a
registered gradient. As a result [`tf.stop_gradient`](https://www.tensorflow.org/versions/r2.13/api_docs/python/tf/stop_gradient) is used to avoid the
`LookupError`.
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`](https://www.tensorflow.org/versions/r2.13/api_docs/python/tf/RegisterGradient) which registers a gradient function for a
primitive TensorFlow operation. [`tf.custom_gradient`](https://www.tensorflow.org/versions/r2.13/api_docs/python/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` 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`](https://www.tensorflow.org/versions/r2.13/api_docs/python/tf/gradients)) is determined by `f(x)[1]`. ||