code: def test_update_parameters_first_order(model):
      """
      The loss function (with respect to the weights of the model w) is defined as
          f(w) = 0.5 * (4 * w_1 + 5 * w_2 + 6 * w_3) ** 2
      with w = [2, 3, 5].
  
      The gradient of the function f with respect to w, and evaluated
      at w = [2, 3, 5] is:
          df / dw_1 = 4 * (4 * w_1 + 5 * w_2 + 6 * w_3) = 212
          df / dw_2 = 5 * (4 * w_1 + 5 * w_2 + 6 * w_3) = 265
          df / dw_3 = 6 * (4 * w_1 + 5 * w_2 + 6 * w_3) = 318
  
      The updated parameter w' is then given by one step of gradient descent,
      with step size 0.5:
          w'_1 = w_1 - 0.5 * df / dw_1 = 2 - 0.5 * 212 = -104
          w'_2 = w_2 - 0.5 * df / dw_2 = 3 - 0.5 * 265 = -129.5
          w'_3 = w_3 - 0.5 * df / dw_3 = 5 - 0.5 * 318 = -154
      """
      train_inputs = torch.tensor([[4.0, 5.0, 6.0]])
      train_loss = 0.5 * (model(train_inputs) ** 2)
  
      params = gradient_update_parameters(
          model,
          train_loss,
          params=None,
          step_size=0.5,
          first_order=True,
      )
  
      assert train_loss.item() == 1404.5
      assert list(params.keys()) == ["weight"]
      assert torch.all(
          params["weight"].data
          == torch.tensor([[-104.0, -129.5, -154.0]])
      )
  
      """
      The new loss function (still with respect to the weights of the model w) is
      defined as:
          g(w) = 0.5 * (1 * w'_1 + 2 * w'_2 + 3 * w'_3) ** 2
  
      Since we computed w' with the first order approximation, the gradient of the
      function g with respect to w, and evaluated at w = [2, 3, 5], is:
          dg / dw_1 = 1 * (1 * w'_1 + 2 * w'_2 + 3 * w'_3) =  -825
          dg / dw_2 = 2 * (1 * w'_1 + 2 * w'_2 + 3 * w'_3) = -1650
          dg / dw_3 = 3 * (1 * w'_1 + 2 * w'_2 + 3 * w'_3) = -2475
      """
      test_inputs = torch.tensor([[1.0, 2.0, 3.0]])
      test_loss = 0.5 * (model(test_inputs, params=params) ** 2)
  
      grads = torch.autograd.grad(test_loss, model.parameters())
  
      assert test_loss.item() == 340312.5
      assert len(grads) == 1
      assert torch.all(
          grads[0].data == torch.tensor([[-825.0, -1650.0, -2475.0]])
      )
  
file_path: pytorch-meta/torchmeta/tests/utils/test_gradient_based.py
imports: import pytest
  import torch
  from torchmeta.modules import MetaLinear
  from torchmeta.utils.gradient_based import gradient_update_parameters
label_desc: Use image node_id 3 for calling a global function with example usage: test_update_parameters_first_order(model) without return types