code: def test_init(self):
      mean = torch.zeros(1, 1)
      variance = torch.ones(1, 1)
      mm = MockModel(MockPosterior(mean=mean, variance=variance))
      # basic test w/ invalid model.
      sampler = IIDNormalSampler(sample_shape=torch.Size([1]))
      with self.assertWarns(DeprecationWarning):
          DeprecatedCachedCholeskyAcqf(model=mm, sampler=sampler)
  
      constructors = [
          DeprecatedCachedCholeskyAcqf,
          DummyCachedCholeskyAcqf,
      ]
      for constructor in constructors:
          acqf = constructor(model=mm, sampler=sampler)
          self.assertFalse(acqf._cache_root)  # no cache by default
          with self.assertWarnsRegex(RuntimeWarning, "cache_root"):
              acqf = constructor(
                  model=mm, sampler=sampler, cache_root=True
              )
          self.assertFalse(acqf._cache_root)  # gets turned to False
          # Unsupported outcome transform.
          stgp = SingleTaskGP(
              torch.zeros(1, 1),
              torch.zeros(1, 1),
              outcome_transform=Log(),
          )
          with self.assertWarnsRegex(RuntimeWarning, "cache_root"):
              acqf = constructor(model=stgp, cache_root=True)
          self.assertFalse(acqf._cache_root)
          # ModelList is not supported.
          model_list = ModelList(
              SingleTaskGP(torch.zeros(1, 1), torch.zeros(1, 1))
          )
          with self.assertWarnsRegex(RuntimeWarning, "cache_root"):
              acqf = constructor(model=model_list, cache_root=True)
          self.assertFalse(acqf._cache_root)
  
          # basic test w/ supported model.
          stgp = SingleTaskGP(torch.zeros(1, 1), torch.zeros(1, 1))
          acqf = constructor(
              model=stgp, sampler=sampler, cache_root=True
          )
          self.assertTrue(acqf._cache_root)
          self.assertEqual(acqf.sampler, sampler)
  
          # test the base_samples are set to None
          self.assertIsNone(acqf.sampler.base_samples)
          # test model that uses matheron's rule and sampler.batch_range != (0, -1)
          hogp = HigherOrderGP(
              torch.zeros(1, 1), torch.zeros(1, 1, 1)
          ).eval()
          with self.assertWarnsRegex(RuntimeWarning, "cache_root"):
              acqf = constructor(
                  model=hogp, sampler=sampler, cache_root=True
              )
          self.assertFalse(acqf._cache_root)
  
          # test deterministic model
          model = GenericDeterministicModel(f=lambda X: X)
          with self.assertWarnsRegex(RuntimeWarning, "cache_root"):
              acqf = constructor(
                  model=model, sampler=sampler, cache_root=True
              )
          self.assertFalse(acqf._cache_root)
  
imports: import warnings
  from typing import Optional
  from unittest import mock
  import torch
  from botorch import settings
  from botorch.acquisition.cached_cholesky import CachedCholeskyMCAcquisitionFunction, CachedCholeskyMCSamplerMixin
  from botorch.acquisition.monte_carlo import MCAcquisitionFunction
  from botorch.acquisition.objective import GenericMCObjective, MCAcquisitionObjective
  from botorch.exceptions.warnings import BotorchWarning
  from botorch.models import SingleTaskGP
  from botorch.models.deterministic import GenericDeterministicModel
  from botorch.models.higher_order_gp import HigherOrderGP
  from botorch.models.model import Model, ModelList
  from botorch.models.transforms.outcome import Log
  from botorch.sampling.normal import IIDNormalSampler, MCSampler
  from botorch.utils.low_rank import extract_batch_covar
  from botorch.utils.testing import BotorchTestCase, MockModel, MockPosterior
  from linear_operator.utils.errors import NanError, NotPSDError
label_desc: Use image node_id 1 for calling the TestCachedCholeskyMCSamplerMixin obj's underlying member method code with example usage: obj.test_init() without return types