code: def _parallel_update_step(
      self,
      grads,
      variables,
      learning_rate,
  ):
      keras_variables = variables
      variables = [v.value for v in variables]
  
      dtype = variables[0].dtype
      lr = ops.cast(learning_rate, dtype)
      rho = self.rho
  
      accumulated_grads = [
          self._accumulated_grads[
              self._get_variable_index(variable)
          ].value
          for variable in keras_variables
      ]
      accumulated_delta_vars = [
          self._accumulated_delta_vars[
              self._get_variable_index(variable)
          ].value
          for variable in keras_variables
      ]
      torch._foreach_mul_(accumulated_grads, rho)
      torch._foreach_add_(
          accumulated_grads,
          torch._foreach_mul(grads, grads),
          alpha=1 - rho,
      )
  
      def rms(x):
          return torch._foreach_sqrt(
              torch._foreach_add(x, self.epsilon)
          )
  
      delta_vars = torch._foreach_mul(
          torch._foreach_div(
              torch._foreach_mul(rms(accumulated_delta_vars), grads),
              rms(accumulated_grads),
          ),
          -1,
      )
      torch._foreach_mul_(accumulated_delta_vars, rho)
      torch._foreach_add_(
          accumulated_delta_vars,
          torch._foreach_mul(delta_vars, delta_vars),
          alpha=1 - rho,
      )
  
      torch._foreach_add_(variables, delta_vars, alpha=lr)
  
file_path: keras/keras/backend/torch/optimizers/torch_adadelta.py
imports: import torch
  from keras import ops
  from keras import optimizers
  from keras.backend.torch.optimizers import torch_parallel_optimizer
label_desc: Use image node_id 1 for calling the Adadelta obj's underlying member method code with example usage: obj._parallel_update_step(grads, variables, learning_rate) and returns: torch