Function that computes the dot product between a vector v and the Hessian of a given scalar function at the point given by the inputs.

Parameters:
• func (function) – a Python function that takes Tensor inputs and returns a Tensor with a single element.

• inputs (tuple of Tensors or Tensor) – inputs to the function func.

• v (tuple of Tensors or Tensor) – The vector for which the vector Hessian product is computed. Must be the same size as the input of func. This argument is optional when func’s input contains a single element and (if it is not provided) will be set as a Tensor containing a single 1.

• create_graph (bool, optional) – If True, both the output and result will be computed in a differentiable way. Note that when strict is False, the result can not require gradients or be disconnected from the inputs. Defaults to False.

• strict (bool, optional) – If True, an error will be raised when we detect that there exists an input such that all the outputs are independent of it. If False, we return a Tensor of zeros as the vhp for said inputs, which is the expected mathematical value. Defaults to False.

Returns:

tuple with:

func_output (tuple of Tensors or Tensor): output of func(inputs)

vhp (tuple of Tensors or Tensor): result of the dot product with the same shape as the inputs.

Return type:

output (tuple)

Example

>>> def pow_reducer(x):
...   return x.pow(3).sum()
>>> inputs = torch.rand(2, 2)
>>> v = torch.ones(2, 2)
>>> vhp(pow_reducer, inputs, v)
(tensor(0.5591),
tensor([[1.0689, 1.2431],
[3.0989, 4.4456]]))
>>> vhp(pow_reducer, inputs, v, create_graph=True)