2nd Order Cross-derivative Matrix-Vector Product (Similar to Hessian-Vector Product)

autograd
2

Hi Pkoch!

The short answer is that you when you use autograd to compute a second derivative,
you need to use create_graph = True so that the first derivative you compute has
a computation graph through which you can backpropagate.

Here is a script that illustrates this both with .backward() and autograd.grad():

import torch
print (torch.__version__)

def h(x, y):
    return x**2 @ y + y**2 @ x

x = torch.tensor([1., 2., 3.], requires_grad=True)
y = torch.tensor([2., 3., 4.], requires_grad=True)
b = torch.tensor([1., 10., 100.,])

# first version -- doesn't work

eval = h (x, y)
eval.backward()                       # doesn't create computation graph for x.grad
print ('x.grad.requires_grad:', x.grad.requires_grad)
eval_2 = (x.grad * b).sum()
print ('eval_2.requires_grad:', eval_2.requires_grad)

# eval_2.backward()                   # fails -- no computation graph for eval_2

x.grad = None
y.grad = None

# fixed version

eval = h (x, y)
eval.backward (create_graph = True)   # creates computation graph for x.grad
y.grad = None                         # zero out unwanted d_eval / d_y
eval_2 = (x.grad * b).sum()
eval_2.backward()
print ('y.grad:', y.grad)

x.grad = None
y.grad = None

# autograd.grad version -- a little cleaner

eval = h (x, y)
x_grad = torch.autograd.grad (eval, x, create_graph = True)[0]
y_grad = torch.autograd.grad ((x_grad * b).sum(), y)[0]

print ('y_grad:', y_grad)

And here is its output:

2.3.0
x.grad.requires_grad: False
eval_2.requires_grad: False
<path_to_pytorch_install>\torch\autograd\graph.py:744: UserWarning: Using backward() with create_graph=True will create a reference cycle between the parameter and its gradient which can cause a memory leak. We recommend using autograd.grad when creating the graph to avoid this. If you have to use this function, make sure to reset the .grad fields of your parameters to None after use to break the cycle and avoid the leak. (Triggered internally at C:\cb\pytorch_1000000000000\work\torch\csrc\autograd\engine.cpp:1208.)
  return Variable._execution_engine.run_backward(  # Calls into the C++ engine to run the backward pass
y.grad: tensor([   6.,  100., 1400.])
y_grad: tensor([   6.,  100., 1400.])

Best.

K. Frank

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