code: def _test_bias_correction(base_policy, another_policy, action_value):
      x = torch.full((1, 1), 1, dtype=torch.float)
      pi = base_policy(x)
      mu = another_policy(x)
  
      def evaluate_action(action):
          return float(action_value.evaluate_actions(action))
  
      n = 1000
  
      pi_samples = [pi.sample() for _ in range(n)]
      mu_samples = [mu.sample() for _ in range(n)]
  
      onpolicy_gs = []
      for sample in pi_samples:
          base_policy.zero_grad()
          loss = -evaluate_action(sample) * pi.log_prob(sample)
          loss.squeeze().backward(retain_graph=True)
          onpolicy_gs.append(
              extract_gradients_as_single_vector(base_policy)
          )
      # on-policy
      onpolicy_gs_mean = np.mean(onpolicy_gs, axis=0)
      onpolicy_gs_var = np.var(onpolicy_gs, axis=0)
      print("on-policy")
      print("g mean", onpolicy_gs_mean)
      print("g var", onpolicy_gs_var)
  
      # off-policy without importance sampling
      offpolicy_gs = []
      for sample in mu_samples:
          base_policy.zero_grad()
          loss = -evaluate_action(sample) * pi.log_prob(sample)
          loss.squeeze().backward(retain_graph=True)
          offpolicy_gs.append(
              extract_gradients_as_single_vector(base_policy)
          )
      offpolicy_gs_mean = np.mean(offpolicy_gs, axis=0)
      offpolicy_gs_var = np.var(offpolicy_gs, axis=0)
      print("off-policy")
      print("g mean", offpolicy_gs_mean)
      print("g var", offpolicy_gs_var)
  
      # off-policy with importance sampling
      is_gs = []
      for sample in mu_samples:
          base_policy.zero_grad()
          rho = float((pi.log_prob(sample) - mu.log_prob(sample)).exp())
          loss = -rho * evaluate_action(sample) * pi.log_prob(sample)
          loss.squeeze().backward(retain_graph=True)
          is_gs.append(extract_gradients_as_single_vector(base_policy))
      is_gs_mean = np.mean(is_gs, axis=0)
      is_gs_var = np.var(is_gs, axis=0)
      print("importance sampling")
      print("g mean", is_gs_mean)
      print("g var", is_gs_var)
  
      # off-policy with truncated importance sampling + bias correction
      def bias_correction_policy_gradients(truncation_threshold):
          gs = []
          for sample in mu_samples:
              base_policy.zero_grad()
              loss = acer.compute_policy_gradient_loss(
                  action=sample,
                  advantage=evaluate_action(sample),
                  action_distrib=pi,
                  action_distrib_mu=mu,
                  action_value=action_value,
                  v=0,
                  truncation_threshold=truncation_threshold,
              )
              loss.squeeze().backward(retain_graph=True)
              gs.append(extract_gradients_as_single_vector(base_policy))
          return gs
  
      # c=0 means on-policy sampling
      print("truncated importance sampling + bias correction c=0")
      tis_c0_gs = bias_correction_policy_gradients(0)
      tis_c0_gs_mean = np.mean(tis_c0_gs, axis=0)
      tis_c0_gs_var = np.var(tis_c0_gs, axis=0)
      print("g mean", tis_c0_gs_mean)
      print("g var", tis_c0_gs_var)
      # c=0 must be low-bias compared to naive off-policy sampling
      assert np.linalg.norm(
          onpolicy_gs_mean - tis_c0_gs_mean
      ) <= np.linalg.norm(onpolicy_gs_mean - offpolicy_gs_mean)
  
      # c=1 means truncated importance sampling with bias correction
      print("truncated importance sampling + bias correction c=1")
      tis_c1_gs = bias_correction_policy_gradients(1)
      tis_c1_gs_mean = np.mean(tis_c1_gs, axis=0)
      tis_c1_gs_var = np.var(tis_c1_gs, axis=0)
      print("g mean", tis_c1_gs_mean)
      print("g var", tis_c1_gs_var)
      # c=1 must be low-variance compared to naive importance sampling
      assert tis_c1_gs_var.sum() <= is_gs_var.sum()
      # c=1 must be low-bias compared to naive off-policy sampling
      assert np.linalg.norm(
          onpolicy_gs_mean - tis_c1_gs_mean
      ) < np.linalg.norm(onpolicy_gs_mean - offpolicy_gs_mean)
  
      # c=inf means importance sampling no truncation
      print("truncated importance sampling + bias correction c=inf")
      tis_cinf_gs = bias_correction_policy_gradients(np.inf)
      tis_cinf_gs_mean = np.mean(tis_cinf_gs, axis=0)
      tis_cinf_gs_var = np.var(tis_cinf_gs, axis=0)
      print("g mean", tis_cinf_gs_mean)
      print("g var", tis_cinf_gs_var)
      np.testing.assert_allclose(
          tis_cinf_gs_mean, is_gs_mean, rtol=1e-3
      )
      np.testing.assert_allclose(tis_cinf_gs_var, is_gs_var, rtol=1e-3)
  
imports: import copy
  import logging
  import os
  import tempfile
  import warnings
  import numpy
  import pytest
  import torch
  from torch import nn
  import pfrl
  from pfrl.agents import acer
  from pfrl.envs.abc import ABC
  from pfrl.experiments.evaluator import run_evaluation_episodes
  from pfrl.experiments.train_agent_async import train_agent_async
  from pfrl.nn import ConcatObsAndAction
  from pfrl.policies import GaussianHeadWithDiagonalCovariance, SoftmaxCategoricalHead
  from pfrl.q_functions import DiscreteActionValueHead
  from pfrl.replay_buffers import EpisodicReplayBuffer
label_desc: Use image node_id 4 for calling a global function with example usage: _test_bias_correction(base_policy, another_policy, action_value) and returns: float, gs