import tensorflow as tf

def optimizer_archt(method: str = 'sgd', learning_rate: float = 0.01, momentum: float = 0.0, nesterov: bool = False, batch_size: int = 32) -> tuple:
    """
    Returns the optimizer, training method, and batch size based on the specified method.

    Args:
        method (str, optional): The optimization method to use. Default is 'sgd'.
                                Supported methods are:
                                'sgd-standard', 'sgd-stochastic', 'sgd-mini-batch', 'sgd-momentum', 
                                'nesterov', 'adagrad', 'adadelta', 'rmsprop', 'adam', 'adamax', 'nadam'.
        learning_rate (float, optional): Learning rate for the optimizer. Default is 0.01.
        momentum (float, optional): Momentum for optimizers that support it. Default is 0.0.
        nesterov (bool, optional): Whether to use Nesterov momentum. Default is False.
        batch_size (int, optional): Batch size for mini-batch optimization. Default is 32.

    Returns:
        tuple: A tuple containing the optimizer instance, the training method, and the batch size.
               The batch size is None for methods that do not use mini-batch training.

    Raises:
        ValueError: If the specified method is not supported.
    """
    if method == 'sgd-standard':
        return tf.optimizers.SGD(learning_rate=learning_rate), 'batch', None
    elif method == 'sgd-stochastic':
        return tf.optimizers.SGD(learning_rate=learning_rate), 'stochastic', None
    elif method == 'sgd-mini-batch':
        return tf.optimizers.SGD(learning_rate=learning_rate), 'mini-batch', batch_size
    elif method == 'sgd-momentum':
        return tf.optimizers.SGD(learning_rate=learning_rate, momentum=momentum), 'batch', None
    elif method == 'nesterov':
        return tf.optimizers.SGD(learning_rate=learning_rate, momentum=momentum, nesterov=True), 'batch', None
    elif method == 'adagrad':
        return tf.optimizers.Adagrad(learning_rate=learning_rate), 'batch', None
    elif method == 'adadelta':
        return tf.optimizers.Adadelta(learning_rate=learning_rate), 'batch', None
    elif method == 'rmsprop':
        return tf.optimizers.RMSprop(learning_rate=learning_rate), 'batch', None
    elif method == 'adam':
        return tf.optimizers.Adam(learning_rate=learning_rate), 'batch', None
    elif method == 'adamax':
        return tf.optimizers.Adamax(learning_rate=learning_rate), 'batch', None
    elif method == 'nadam':
        return tf.optimizers.Nadam(learning_rate=learning_rate), 'batch', None
    else:
        raise ValueError(f'{method} is not a valid optimizer.')