def fit(
        self,
        x: np.ndarray,
        y: np.ndarray,
        validation_data: tuple[np.ndarray, np.ndarray] | None = None,
        batch_size: int = 128,
        nb_epochs: int = 20,
        scheduler: "torch.optim.lr_scheduler._LRScheduler" | None = None,
        **kwargs,
    ):
        """
        Train a model adversarially with TRADES protocol.
        See class documentation for more information on the exact procedure.

        :param x: Training set.
        :param y: Labels for the training set.
        :param validation_data: Tuple consisting of validation data, (x_val, y_val)
        :param batch_size: Size of batches.
        :param nb_epochs: Number of epochs to use for trainings.
        :param scheduler: Learning rate scheduler to run at the end of every epoch.
        :param kwargs: Dictionary of framework-specific arguments. These will be passed as such to the `fit` function of
                                  the target classifier.
        """
        import torch

        logger.info("Performing adversarial training with TRADES protocol")

        if (scheduler is not None) and (
            not isinstance(scheduler, torch.optim.lr_scheduler._LRScheduler)
        ):  # pylint: enable=protected-access
            raise ValueError("Invalid Pytorch scheduler is provided for adversarial training.")

        nb_batches = int(np.ceil(len(x) / batch_size))
        ind = np.arange(len(x))

        logger.info("Adversarial Training TRADES")
        y = check_and_transform_label_format(y, nb_classes=self.classifier.nb_classes)

        if validation_data is not None:
            (x_test, y_test) = validation_data
            y_test = check_and_transform_label_format(y_test, nb_classes=self.classifier.nb_classes)

            x_preprocessed_test, y_preprocessed_test = self._classifier._apply_preprocessing(x_test, y_test, fit=True)

        for i_epoch in trange(nb_epochs, desc="Adversarial Training TRADES - Epochs"):
            # Shuffle the examples
            np.random.shuffle(ind)
            start_time = time.time()
            train_loss = 0.0
            train_acc = 0.0
            train_n = 0.0

            for batch_id in range(nb_batches):
                # Create batch data
                x_batch = x[ind[batch_id * batch_size : min((batch_id + 1) * batch_size, x.shape[0])]].copy()
                y_batch = y[ind[batch_id * batch_size : min((batch_id + 1) * batch_size, x.shape[0])]]

                _train_loss, _train_acc, _train_n = self._batch_process(x_batch, y_batch)

                train_loss += _train_loss
                train_acc += _train_acc
                train_n += _train_n

            if scheduler:
                scheduler.step()

            train_time = time.time()

            # compute accuracy
            if validation_data is not None:
                output = np.argmax(self.predict(x_preprocessed_test), axis=1)
                nb_correct_pred = np.sum(output == np.argmax(y_preprocessed_test, axis=1))

                logger.info(
                    "epoch: %s time(s): %.1f loss: %.4f acc(tr): %.4f acc(val): %.4f",
                    i_epoch,
                    train_time - start_time,
                    train_loss / train_n,
                    train_acc / train_n,
                    nb_correct_pred / x_test.shape[0],
                )
            else:
                logger.info(
                    "epoch: %s time(s): %.1f loss: %.4f acc: %.4f",
                    i_epoch,
                    train_time - start_time,
                    train_loss / train_n,
                    train_acc / train_n,
                )

        Train a model adversarially with TRADES protocol.
        See class documentation for more information on the exact procedure.

        :param x: Training set.
        :param y: Labels for the training set.
        :param validation_data: Tuple consisting of validation data, (x_val, y_val)
        :param batch_size: Size of batches.
        :param nb_epochs: Number of epochs to use for trainings.
        :param scheduler: Learning rate scheduler to run at the end of every epoch.
        :param kwargs: Dictionary of framework-specific arguments. These will be passed as such to the `fit` function of
                                  the target classifier.
        
art/defences/trainer/adversarial_trainer_trades_pytorch.py