code: def test_multiple_update_parameters(model):
      """
      The loss function (with respect to the weights of the model w) is defined as
          f(w) = 0.5 * (1 * w_1 + 2 * w_2 + 3 * w_3) ** 2
      with w = [2, 3, 5].
  
      The gradient of f with respect to w is:
          df / dw_1 = 1 * (1 * w_1 + 2 * w_2 + 3 * w_3) = 23
          df / dw_2 = 2 * (1 * w_1 + 2 * w_2 + 3 * w_3) = 46
          df / dw_3 = 3 * (1 * w_1 + 2 * w_2 + 3 * w_3) = 69
  
      The updated parameters are given by:
          w'_1 = w_1 - 1. * df / dw_1 = 2 - 1. * 23 = -21
          w'_2 = w_2 - 1. * df / dw_2 = 3 - 1. * 46 = -43
          w'_3 = w_3 - 1. * df / dw_3 = 5 - 1. * 69 = -64
      """
      train_inputs = torch.tensor([[1.0, 2.0, 3.0]])
  
      train_loss_1 = 0.5 * (model(train_inputs) ** 2)
      params_1 = gradient_update_parameters(
          model,
          train_loss_1,
          params=None,
          step_size=1.0,
          first_order=False,
      )
  
      assert train_loss_1.item() == 264.5
      assert list(params_1.keys()) == ["weight"]
      assert torch.all(
          params_1["weight"].data
          == torch.tensor([[-21.0, -43.0, -64.0]])
      )
  
      """
      The new loss function is defined as
          g(w') = 0.5 * (1 * w'_1 + 2 * w'_2 + 3 * w'_3) ** 2
      with w' = [-21, -43, -64].
  
      The gradient of g with respect to w' is:
          dg / dw'_1 = 1 * (1 * w'_1 + 2 * w'_2 + 3 * w'_3) = -299
          dg / dw'_2 = 2 * (1 * w'_1 + 2 * w'_2 + 3 * w'_3) = -598
          dg / dw'_3 = 3 * (1 * w'_1 + 2 * w'_2 + 3 * w'_3) = -897
  
      The updated parameters are given by:
          w''_1 = w'_1 - 1. * dg / dw'_1 = -21 - 1. * -299 = 278
          w''_2 = w'_2 - 1. * dg / dw'_2 = -43 - 1. * -598 = 555
          w''_3 = w'_3 - 1. * dg / dw'_3 = -64 - 1. * -897 = 833
      """
      train_loss_2 = 0.5 * (model(train_inputs, params=params_1) ** 2)
      params_2 = gradient_update_parameters(
          model,
          train_loss_2,
          params=params_1,
          step_size=1.0,
          first_order=False,
      )
  
      assert train_loss_2.item() == 44700.5
      assert list(params_2.keys()) == ["weight"]
      assert torch.all(
          params_2["weight"].data
          == torch.tensor([[278.0, 555.0, 833.0]])
      )
  
      """
      The new loss function is defined as
          h(w'') = 0.5 * (1 * w''_1 + 2 * w''_2 + 3 * w''_3) ** 2
      with w'' = [278, 555, 833].
  
      The gradient of h with respect to w'' is:
          dh / dw''_1 = 1 * (1 * w''_1 + 2 * w''_2 + 3 * w''_3) =  3887
          dh / dw''_2 = 2 * (1 * w''_1 + 2 * w''_2 + 3 * w''_3) =  7774
          dh / dw''_3 = 3 * (1 * w''_1 + 2 * w''_2 + 3 * w''_3) = 11661
  
      The updated parameters are given by:
          w'''_1 = w''_1 - 1. * dh / dw''_1 = 278 - 1. *  3887 =  -3609
          w'''_2 = w''_2 - 1. * dh / dw''_2 = 555 - 1. *  7774 =  -7219
          w'''_3 = w''_3 - 1. * dh / dw''_3 = 833 - 1. * 11661 = -10828
      """
      train_loss_3 = 0.5 * (model(train_inputs, params=params_2) ** 2)
      params_3 = gradient_update_parameters(
          model,
          train_loss_3,
          params=params_2,
          step_size=1.0,
          first_order=False,
      )
  
      assert train_loss_3.item() == 7554384.5
      assert list(params_3.keys()) == ["weight"]
      assert torch.all(
          params_3["weight"].data
          == torch.tensor([[-3609.0, -7219.0, -10828.0]])
      )
  
      """
      The new loss function is defined as
          l(w) = 4 * w'''_1 + 5 * w'''_2 + 6 * w'''_3
      with w = [2, 3, 5] and w''' = [-3609, -7219, -10828].
  
      The gradient of l with respect to w is:
          dl / dw_1 = 4 * dw'''_1 / dw_1 + 5 * dw'''_2 / dw_1 + 6 * dw'''_3 / dw_1
                    = ... =  -5020
          dl / dw_2 = 4 * dw'''_1 / dw_2 + 5 * dw'''_2 / dw_2 + 6 * dw'''_3 / dw_2
                    = ... = -10043
          dl / dw_3 = 4 * dw'''_1 / dw_3 + 5 * dw'''_2 / dw_3 + 6 * dw'''_3 / dw_3
                    = ... = -15066
      """
      test_inputs = torch.tensor([[4.0, 5.0, 6.0]])
      test_loss = model(test_inputs, params=params_3)
      grads = torch.autograd.grad(test_loss, model.parameters())
  
      assert test_loss.item() == -115499.0
      assert len(grads) == 1
      assert torch.all(
          grads[0].data == torch.tensor([[-5020.0, -10043.0, -15066.0]])
      )
  
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 4 for calling a global function with example usage: test_multiple_update_parameters(model) without return types