code: def rcal(
      base_loss_fn: BCLossWithoutAux,
      discount: float,
      alpha: float,
      num_bins: Optional[int] = None,
  ) -> BCLossWithoutAux:
      """https://www.cristal.univ-lille.fr/~pietquin/pdf/AAMAS_2014_BPMGOP.pdf.
  
      Args:
        base_loss_fn: the base loss to add RCAL on top of.
        discount: the gamma discount used in RCAL.
        alpha: the regularization parameter.
        num_bins: how many bins were used for discretization. If None the
          environment was originally discrete already.
      Returns:
        The loss function.
      """
  
      def loss(
          networks: bc_networks.BCNetworks,
          params: networks_lib.Params,
          key: jax_types.PRNGKey,
          transitions: types.Transition,
      ) -> jnp.ndarray:
          def logits_fn(
              key: jax_types.PRNGKey,
              observations: networks_lib.Observation,
              actions: Optional[networks_lib.Action] = None,
          ):
              logits = networks.policy_network.apply(
                  params, observations, key=key, is_training=True
              )
              if num_bins:
                  logits = jnp.reshape(
                      logits, list(logits.shape[:-1]) + [-1, num_bins]
                  )
              if actions is None:
                  actions = jnp.argmax(logits, axis=-1)
              logits_actions = jnp.sum(
                  jax.nn.one_hot(actions, logits.shape[-1]) * logits,
                  axis=-1,
              )
              return logits_actions
  
          key, key1, key2 = jax.random.split(key, 3)
  
          logits_a_tm1 = logits_fn(
              key1, transitions.observation, transitions.action
          )
          logits_a_t = logits_fn(key2, transitions.next_observation)
  
          # RCAL, by making a parallel between the logits of BC and Q-values,
          # defines a regularization loss that encourages the implicit reward
          # (inferred by inversing the Bellman Equation) to be sparse.
          # NOTE: In case of discretized envs jnp.mean goes over batch and num_bins
          # dimensions.
          regularization_loss = jnp.mean(
              jnp.abs(logits_a_tm1 - discount * logits_a_t)
          )
  
          loss = base_loss_fn(networks, params, key, transitions)
          return loss + alpha * regularization_loss
  
      return loss
  
file_path: acme/acme/agents/jax/bc/losses.py
imports: from typing import Callable, Optional, Tuple, Union
  from acme import types
  from acme.agents.jax.bc import networks
  from acme.jax import networks
  from acme.jax import types
  from acme.utils import loggers
  import jax
  import jax.numpy
label_desc: Use image node_id 4 for calling a global function with example usage: rcal(base_loss_fn, discount, alpha, num_bins) and returns: loss, unknown, logits_actions