python - 如何计算损失 w.r.t. 的 Hessian PyTorch 中使用 autograd.grad 的参数

Question

我知道在 pytorch 中有很多关于“计算 Hessian”的内容，但据我所知，我没有找到任何对我有用的东西。因此，尽量准确地说，我想要的 Hessian 是相对于网络参数的损失梯度的雅可比行列式。也称为关于参数的二阶导数矩阵。
我发现了一些以直观方式工作的代码，虽然不应该很快。它显然只是计算损失 w.r.t. 的梯度的梯度。 params w.r.t params，它一次只做一个元素(梯度的)。我认为逻辑绝对是正确的，但我收到了一个错误，与 requires_grad 有关。 .我是 pytorch 初学者，所以这可能是一件简单的事情，但错误似乎是说它不能采用 env_grads 的梯度。变量，它是前一个 grad 函数调用的输出。
对此的任何帮助将不胜感激。这是代码后跟错误消息。我还打印了 env_grads[0]变量所以我们可以看到它实际上是一个张量，这是前面 grad 的正确输出称呼。

env_loss = loss_fn(env_outputs, env_targets)
total_loss += env_loss
env_grads = torch.autograd.grad(env_loss, params,retain_graph=True)

print( env_grads[0] )
hess_params = torch.zeros_like(env_grads[0])
for i in range(env_grads[0].size(0)):
    for j in range(env_grads[0].size(1)):
        hess_params[i, j] = torch.autograd.grad(env_grads[0][i][j], params, retain_graph=True)[0][i, j] #  <--- error here
print( hess_params )
exit()

输出:

tensor([[-6.4064e-03, -3.1738e-03,  1.7128e-02,  8.0391e-03],
        [ 7.1698e-03, -2.4640e-03, -2.2769e-03, -1.0687e-03],
        [-3.0390e-04, -2.4273e-03, -4.0799e-02, -1.9149e-02],
        ...,
        [ 1.1258e-02, -2.5911e-05, -9.8133e-02, -4.6059e-02],
        [ 8.1502e-04, -2.5814e-03,  4.1772e-02,  1.9606e-02],
        [-1.0075e-02,  6.6072e-03,  8.3118e-04,  3.9011e-04]], device='cuda:0')

错误:

Traceback (most recent call last):
  File "/home/jefferythewind/anaconda3/envs/rapids3/lib/python3.7/runpy.py", line 193, in _run_module_as_main
    "__main__", mod_spec)
  File "/home/jefferythewind/anaconda3/envs/rapids3/lib/python3.7/runpy.py", line 85, in _run_code
    exec(code, run_globals)
  File "/home/jefferythewind/Projects/Irina/learning-explanations-hard-to-vary/and_mask/run_synthetic.py", line 258, in <module>
    main(args)
  File "/home/jefferythewind/Projects/Irina/learning-explanations-hard-to-vary/and_mask/run_synthetic.py", line 245, in main
    deep_mask=args.deep_mask
  File "/home/jefferythewind/Projects/Irina/learning-explanations-hard-to-vary/and_mask/run_synthetic.py", line 103, in train
    scale_grad_inverse_sparsity=scale_grad_inverse_sparsity
  File "/home/jefferythewind/Projects/Irina/learning-explanations-hard-to-vary/and_mask/and_mask_utils.py", line 154, in get_grads_deep
    hess_params[i, j] = torch.autograd.grad(env_grads[0][i][j], params, retain_graph=True)[0][i, j]
  File "/home/jefferythewind/anaconda3/envs/rapids3/lib/python3.7/site-packages/torch/autograd/__init__.py", line 157, in grad
    inputs, allow_unused)
RuntimeError: element 0 of tensors does not require grad and does not have a grad_fn

Answer 1

PyTorch 最近添加了一个 functional higher level API至 torch.autograd提供 torch.autograd.functional.hessian(func, inputs, ... )直接计算标量函数的 Hessian func 关于 它的参数 在 由 inputs 指定的位置，对应于 func 的参数的张量元组. hessian我相信它本身不支持自动微分。
但请注意，截至 2021 年 3 月，它仍处于测试阶段。

使用 torch.autograd.functional.hessian 的完整示例创建一个 score-test对于非零均值(作为一个样本 t 检验的(坏)替代方案):

import numpy as np
import torch, torchvision
from torch.autograd import Variable, grad
import torch.distributions as td
import math
from torch.optim import Adam
import scipy.stats


x_data = torch.randn(100)+0.0 # observed data (here sampled under H0)

N = x_data.shape[0] # number of observations

mu_null = torch.zeros(1)
sigma_null_hat = Variable(torch.ones(1), requires_grad=True)

def log_lik(mu, sigma):
  return td.Normal(loc=mu, scale=sigma).log_prob(x_data).sum()

# Find theta_null_hat by some gradient descent algorithm (in this case an closed-form expression would be trivial to obtain (see below)):
opt = Adam([sigma_null_hat], lr=0.01)
for epoch in range(2000):
    opt.zero_grad() # reset gradient accumulator or optimizer
    loss = - log_lik(mu_null, sigma_null_hat) # compute log likelihood with current value of sigma_null_hat  (= Forward pass)
    loss.backward() # compute gradients (= Backward pass)
    opt.step()      # update sigma_null_hat
    
print(f'parameter fitted under null: sigma: {sigma_null_hat}, expected: {torch.sqrt((x_data**2).mean())}')

theta_null_hat = (mu_null, sigma_null_hat)

U = torch.tensor(torch.autograd.functional.jacobian(log_lik, theta_null_hat)) # Jacobian (= vector of partial derivatives of log likelihood w.r.t. the parameters (of the full/alternative model)) = score
I = -torch.tensor(torch.autograd.functional.hessian(log_lik, theta_null_hat)) / N # estimate of the Fisher information matrix
S = torch.t(U) @ torch.inverse(I) @ U / N # test statistic, often named "LM" (as in Lagrange multiplier), would be zero at the maximum likelihood estimate

pval_score_test = 1 - scipy.stats.chi2(df = 1).cdf(S) # S asymptocially follows a chi^2 distribution with degrees of freedom equal to the number of parameters fixed under H0
print(f'p-value Chi^2-based score test: {pval_score_test}')
#parameter fitted under null: sigma: tensor([0.9260], requires_grad=True), expected: 0.9259940385818481

# comparison with Student's t-test:
pval_t_test = scipy.stats.ttest_1samp(x_data, popmean = 0).pvalue
#> p-value Chi^2-based score test: 0.9203232752568568
print(f'p-value Student\'s t-test: {pval_t_test}')
#> p-value Student's t-test: 0.9209265268946605