def hessians(ys,
xs,
name="hessians",
colocate_gradients_with_ops=False,
gate_gradients=False,
aggregation_method=None):
"""Constructs the Hessian of sum of `ys` with respect to `x` in `xs`.
`hessians()` adds ops to the graph to output the Hessian matrix of `ys`
with respect to `xs`. It returns a list of `Tensor` of length `len(xs)`
where each tensor is the Hessian of `sum(ys)`.
The Hessian is a matrix of second-order partial derivatives of a scalar
tensor (see https://en.wikipedia.org/wiki/Hessian_matrix for more details).
Args:
ys: A `Tensor` or list of tensors to be differentiated.
xs: A `Tensor` or list of tensors to be used for differentiation.
name: Optional name to use for grouping all the gradient ops together.
defaults to 'hessians'.
colocate_gradients_with_ops: See `gradients()` documentation for details.
gate_gradients: See `gradients()` documentation for details.
aggregation_method: See `gradients()` documentation for details.
Returns:
A list of Hessian matrices of `sum(ys)` for each `x` in `xs`.
Raises:
LookupError: if one of the operations between `xs` and `ys` does not
have a registered gradient function.
"""
xs = gradients_util._AsList(xs) # pylint: disable=protected-access
kwargs = {
"colocate_gradients_with_ops": colocate_gradients_with_ops,
"gate_gradients": gate_gradients,
"aggregation_method": aggregation_method
}
# Compute first-order derivatives and iterate for each x in xs.
hessians = []
_gradients = gradients(ys, xs, **kwargs)
for gradient, x in zip(_gradients, xs):
# change shape to one-dimension without graph branching
gradient = array_ops.reshape(gradient, [-1])
# Declare an iterator and tensor array loop variables for the gradients.
n = array_ops.size(x)
loop_vars = [
array_ops.constant(0, dtypes.int32),
tensor_array_ops.TensorArray(x.dtype, n)
]
# Iterate over all elements of the gradient and compute second order
# derivatives.
_, hessian = control_flow_ops.while_loop(
lambda j, _: j < n, lambda j, result:
(j + 1, result.write(j,
gradients(gradient[j], x)[0])), loop_vars)
_shape = array_ops.shape(x)
_reshaped_hessian = array_ops.reshape(
hessian.stack(), array_ops.concat((_shape, _shape), 0))
hessians.append(_reshaped_hessian)
return hessians
def hessians(ys,
xs,
name="hessians",
colocate_gradients_with_ops=False,
gate_gradients=False,
aggregation_method=None):
"""Constructs the Hessian of sum of `ys` with respect to `x` in `xs`.
`hessians()` adds ops to the graph to output the Hessian matrix of `ys`
with respect to `xs`. It returns a list of `Tensor` of length `len(xs)`
where each tensor is the Hessian of `sum(ys)`.
The Hessian is a matrix of second-order partial derivatives of a scalar
tensor (see https://en.wikipedia.org/wiki/Hessian_matrix for more details).
Args:
ys: A `Tensor` or list of tensors to be differentiated.
xs: A `Tensor` or list of tensors to be used for differentiation.
name: Optional name to use for grouping all the gradient ops together.
defaults to 'hessians'.
colocate_gradients_with_ops: See `gradients()` documentation for details.
gate_gradients: See `gradients()` documentation for details.
aggregation_method: See `gradients()` documentation for details.
Returns:
A list of Hessian matrices of `sum(ys)` for each `x` in `xs`.
Raises:
LookupError: if one of the operations between `xs` and `ys` does not
have a registered gradient function.
"""
xs = gradients_util._AsList(xs) # pylint: disable=protected-access
kwargs = {
"colocate_gradients_with_ops": colocate_gradients_with_ops,
"gate_gradients": gate_gradients,
"aggregation_method": aggregation_method
}
# Compute first-order derivatives and iterate for each x in xs.
hessians = []
_gradients = gradients(ys, xs, **kwargs)
for gradient, x in zip(_gradients, xs):
# change shape to one-dimension without graph branching
gradient = array_ops.reshape(gradient, [-1])
# Declare an iterator and tensor array loop variables for the gradients.
n = array_ops.size(x)
loop_vars = [
array_ops.constant(0, dtypes.int32),
tensor_array_ops.TensorArray(x.dtype, n)
]
# Iterate over all elements of the gradient and compute second order
# derivatives.
_, hessian = control_flow_ops.while_loop(
lambda j, _: j < n,
lambda j, result: (j + 1,
result.write(j, gradients(gradient[j], x)[0])),
loop_vars
)
_shape = array_ops.shape(x)
_reshaped_hessian = array_ops.reshape(hessian.stack(),
array_ops.concat((_shape, _shape), 0))
hessians.append(_reshaped_hessian)
return hessians
