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How to compute the Hessian of a given scalar function in PyTorch?
The hessian() function computes the Hessian of a given function. The hessian() function can be accessed from the torch.autograd.functional module. The function whose Hessian is being computed takes a tensor as the input and returns a tuple of tensors or a tensor. The hessian() function returns a tensor with the Hessian values computed for a function with the given input.
Syntax
torch.autograd.functional.hessian(func, input)
Parameters
func − It's a Python function for which the Hessian is computed.
input − It’s input to the function, func.
Steps
We could use the following steps to compute the Hessian of a given function −
Import the required library. In all the following examples, the required Python libraries are torch. Make sure you have already installed it.
import torch from torch.autograd.functional import hessian
Define a function func for which the Hessian is to be calculated. The input to this function is input.
def func(x): return x**3 + 4*x -10
Define the tensor input to the function func.
input = torch.tensor([2.,3.,4.])
Compute the Hessian of the function defined above for the given input input.
output = hessian(func, input)
Print the tensor containing the computed hessian.
print("Hessian Tensor:
", output)Example 1
# Import the required libraries
import torch
from torch.autograd.functional import hessian
# define a function
def func(x):
return x**3 + 4*x -10
input = torch.tensor([2.])
output = hessian(func, input)
print("Hessian Tensor:
",output)Output
Hessian Tensor: tensor([[12.]])
In the above example, we have computed the Hessian of a function for a given input. The function here we have used is a univariate function.
Example 2
# Import the required libraries import torch from torch.autograd.functional import hessian def func(x): return (x**3 + 4*x -10).sum() # apply an input having more than one elements input = torch.randn(2,2) output = hessian(func, input) print(output)
Output
tensor([[[[-1.4218, -0.0000], [ 0.0000, -0.0000]], [[-0.0000, -2.7878], [ 0.0000, -0.0000]]], [[[-0.0000, -0.0000], [ 1.9817, -0.0000]], [[-0.0000, -0.0000], [ 0.0000, -2.8517]]]])
In the above example, we have computed the Hessian of a function for a given 2×2 input tensor. The function here we have used is a univariate function.
Example 3
# Import the required libraries import torch from torch.autograd.functional import hessian # define a function def func(x, y): return x**3 + x*y input = (torch.tensor([2.]),torch.tensor([4.])) output = hessian(func, input) print(output) # to apply multi element tensor as input def func(x, y): return (x**3 + x*y).sum() input = (torch.tensor([2.,3.]),torch.tensor([4., 5.])) output = hessian(func, input) print(output)
Output
((tensor([[12.]]), tensor([[1.]])), (tensor([[1.]]), tensor([[0.]]))) ((tensor([[12., 0.], [ 0., 18.]]), tensor([[1., 0.], [0., 1.]])), (tensor([[1., 0.], [0., 1.]]), tensor([[0., 0.], [0., 0.]])))
In the above example, we have computed the Hessian of a function for a given input tensor. The function here we have used is a bivariate function. So, we have defined the input as a tuple of two tensors.
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