Pytorch autograd.grad how to write the parameters for multiple outputs?

Question

In the documentation of torch.autograd.grad, it is stated that, for parameters,

parameters:

outputs (sequence of Tensor) – outputs of the differentiated function.

inputs (sequence of Tensor) – Inputs w.r.t. which the gradient will be returned (and not accumulated into .grad).

I try the following:

a = torch.rand(2, requires_grad=True)
b = torch.rand(2, requires_grad=True)
c = a+b
d = a-b

torch.autograd.grad([c, d], [a, b]) #ValueError: only one element tensors can be converted to Python scalars
torch.autograd.grad(torch.tensor([c, d]), torch.tensor([a, b])) #RuntimeError: grad can be implicitly created only for scalar outputs

I would like to get gradients of a list of tensors w.r.t another list of tensors. What is the correct way to feed the parameters?

Answer 1

As the torch.autograd.grad mentioned, torch.autograd.grad computes and returns the sum of gradients of outputs w.r.t. the inputs. Since your c and d are not scalar values, grad_outputs are required.

import torch

a = torch.rand(2,requires_grad=True)
b = torch.rand(2, requires_grad=True)

a
# tensor([0.2308, 0.2388], requires_grad=True)

b
# tensor([0.6314, 0.7867], requires_grad=True)

c = a*a + b*b
d = 2*a+4*b

torch.autograd.grad([c,d], inputs=[a,b], grad_outputs=[torch.Tensor([1.,1.]), torch.Tensor([1.,1.])])
# (tensor([2.4616, 2.4776]), tensor([5.2628, 5.5734]))

Explanation: dc/da = 2*a = [0.2308*2, 0.2388*2] dd/da = [2.,2.] So the first output is dc/da*grad_outputs[0]+dd/da*grad_outputs[1] = [2.4616, 2.4776]. Same calculation for the second output.

If you just want to get the gradient of c and d w.r.t. the inputs, probably you can do this:

a = torch.rand(2,requires_grad=True)
b = torch.rand(2, requires_grad=True)

a
# tensor([0.9566, 0.6066], requires_grad=True)
b
# tensor([0.5248, 0.4833], requires_grad=True)

c = a*a + b*b
d = 2*a+4*b

[torch.autograd.grad(t, inputs=[a,b], grad_outputs=[torch.Tensor([1.,1.])]) for t in [c,d]]
# [(tensor([1.9133, 1.2132]), tensor([1.0496, 0.9666])),
# (tensor([2., 2.]), tensor([4., 4.]))]

Answer 2

Here you go In the example you gave:

a = torch.rand(2, requires_grad=True)
b = torch.rand(2, requires_grad=True)
loss = a + b

As the loss is a vector with 2 elements, you can't perform the autograd operation at once.

typically,

loss = torch.sum(a + b)
torch.autograd.grad([loss], [a, b])

This would return the correct value of gradient for the loss tensor which contains one element. You can pass mutiple scalar tensors to outputs argument of the torch.autograd.grad method