(air-lightgbm-example-ref)=
In this example we will train a model in Ray AIR using distributed LightGBM.
Let's start with installing our dependencies:
!pip install -qU "ray[tune]" lightgbm_ray
Then we need some imports:
from typing import Tuple
import ray
from ray.train.batch_predictor import BatchPredictor
from ray.train.lightgbm import LightGBMPredictor
from ray.data.preprocessors.chain import Chain
from ray.data.preprocessors.encoder import Categorizer
from ray.train.lightgbm import LightGBMTrainer
from ray.air.config import ScalingConfig
from ray.data import Dataset
from ray.air.result import Result
from ray.data.preprocessors import StandardScaler
Next we define a function to load our train, validation, and test datasets.
def prepare_data() -> Tuple[Dataset, Dataset, Dataset]:
dataset = ray.data.read_csv("s3://anonymous@air-example-data/breast_cancer_with_categorical.csv")
train_dataset, valid_dataset = dataset.train_test_split(test_size=0.3)
test_dataset = valid_dataset.drop_columns(cols=["target"])
return train_dataset, valid_dataset, test_dataset
The following function will create a LightGBM trainer, train it, and return the result.
def train_lightgbm(num_workers: int, use_gpu: bool = False) -> Result:
train_dataset, valid_dataset, _ = prepare_data()
# Scale some random columns, and categorify the categorical_column,
# allowing LightGBM to use its built-in categorical feature support
preprocessor = Chain(
Categorizer(["categorical_column"]),
StandardScaler(columns=["mean radius", "mean texture"])
)
# LightGBM specific params
params = {
"objective": "binary",
"metric": ["binary_logloss", "binary_error"],
}
trainer = LightGBMTrainer(
scaling_config=ScalingConfig(num_workers=num_workers, use_gpu=use_gpu),
label_column="target",
params=params,
datasets={"train": train_dataset, "valid": valid_dataset},
preprocessor=preprocessor,
num_boost_round=100,
)
result = trainer.fit()
print(result.metrics)
return result
Once we have the result, we can do batch inference on the obtained model. Let's define a utility function for this.
def predict_lightgbm(result: Result):
_, _, test_dataset = prepare_data()
batch_predictor = BatchPredictor.from_checkpoint(
result.checkpoint, LightGBMPredictor
)
predicted_labels = (
batch_predictor.predict(test_dataset)
.map_batches(lambda df: (df > 0.5).astype(int), batch_format="pandas")
)
print(f"PREDICTED LABELS")
predicted_labels.show()
shap_values = batch_predictor.predict(test_dataset, pred_contrib=True)
print(f"SHAP VALUES")
shap_values.show()
Now we can run the training:
result = train_lightgbm(num_workers=2, use_gpu=False)
2022-06-22 17:26:41,346 WARNING read_api.py:260 -- The number of blocks in this dataset (1) limits its parallelism to 1 concurrent tasks. This is much less than the number of available CPU slots in the cluster. Use `.repartition(n)` to increase the number of dataset blocks. Map_Batches: 100%|██████████| 1/1 [00:00<00:00, 46.26it/s]
Trial name | status | loc | iter | total time (s) | train-binary_logloss | train-binary_error | valid-binary_logloss |
---|---|---|---|---|---|---|---|
LightGBMTrainer_7b049_00000 | TERMINATED | 172.31.43.110:1491578 | 100 | 10.9726 | 0.000574522 | 0 | 0.171898 |
UserWarning: cpus_per_actor is set to less than 2. Distributed LightGBM needs at least 2 CPUs per actor to train efficiently. This may lead to a degradation of performance during training. (pid=1491578) /home/ubuntu/ray/venv/lib/python3.8/site-packages/dask/dataframe/backends.py:181: FutureWarning: pandas.Int64Index is deprecated and will be removed from pandas in a future version. Use pandas.Index with the appropriate dtype instead. (pid=1491578) _numeric_index_types = (pd.Int64Index, pd.Float64Index, pd.UInt64Index) (pid=1491578) /home/ubuntu/ray/venv/lib/python3.8/site-packages/dask/dataframe/backends.py:181: FutureWarning: pandas.Float64Index is deprecated and will be removed from pandas in a future version. Use pandas.Index with the appropriate dtype instead. (pid=1491578) _numeric_index_types = (pd.Int64Index, pd.Float64Index, pd.UInt64Index) (pid=1491578) /home/ubuntu/ray/venv/lib/python3.8/site-packages/dask/dataframe/backends.py:181: FutureWarning: pandas.UInt64Index is deprecated and will be removed from pandas in a future version. Use pandas.Index with the appropriate dtype instead. (pid=1491578) _numeric_index_types = (pd.Int64Index, pd.Float64Index, pd.UInt64Index) (pid=1491578) /home/ubuntu/ray/venv/lib/python3.8/site-packages/xgboost/compat.py:31: FutureWarning: pandas.Int64Index is deprecated and will be removed from pandas in a future version. Use pandas.Index with the appropriate dtype instead. (pid=1491578) from pandas import MultiIndex, Int64Index (LightGBMTrainer pid=1491578) UserWarning: Dataset 'train' has 1 blocks, which is less than the `num_workers` 2. This dataset will be automatically repartitioned to 2 blocks. (LightGBMTrainer pid=1491578) UserWarning: Dataset 'valid' has 1 blocks, which is less than the `num_workers` 2. This dataset will be automatically repartitioned to 2 blocks. (LightGBMTrainer pid=1491578) UserWarning: cpus_per_actor is set to less than 2. Distributed LightGBM needs at least 2 CPUs per actor to train efficiently. This may lead to a degradation of performance during training. (pid=1491651) /home/ubuntu/ray/venv/lib/python3.8/site-packages/xgboost/compat.py:31: FutureWarning: pandas.Int64Index is deprecated and will be removed from pandas in a future version. Use pandas.Index with the appropriate dtype instead. (pid=1491651) from pandas import MultiIndex, Int64Index (pid=1491651) FutureWarning: pandas.Int64Index is deprecated and will be removed from pandas in a future version. Use pandas.Index with the appropriate dtype instead. (pid=1491651) FutureWarning: pandas.Float64Index is deprecated and will be removed from pandas in a future version. Use pandas.Index with the appropriate dtype instead. (pid=1491651) FutureWarning: pandas.UInt64Index is deprecated and will be removed from pandas in a future version. Use pandas.Index with the appropriate dtype instead. (pid=1491653) /home/ubuntu/ray/venv/lib/python3.8/site-packages/dask/dataframe/backends.py:181: FutureWarning: pandas.Int64Index is deprecated and will be removed from pandas in a future version. Use pandas.Index with the appropriate dtype instead. (pid=1491653) _numeric_index_types = (pd.Int64Index, pd.Float64Index, pd.UInt64Index) (pid=1491653) /home/ubuntu/ray/venv/lib/python3.8/site-packages/dask/dataframe/backends.py:181: FutureWarning: pandas.Float64Index is deprecated and will be removed from pandas in a future version. Use pandas.Index with the appropriate dtype instead. (pid=1491653) _numeric_index_types = (pd.Int64Index, pd.Float64Index, pd.UInt64Index) (pid=1491653) /home/ubuntu/ray/venv/lib/python3.8/site-packages/dask/dataframe/backends.py:181: FutureWarning: pandas.UInt64Index is deprecated and will be removed from pandas in a future version. Use pandas.Index with the appropriate dtype instead. (pid=1491653) _numeric_index_types = (pd.Int64Index, pd.Float64Index, pd.UInt64Index) (pid=1491653) /home/ubuntu/ray/venv/lib/python3.8/site-packages/xgboost/compat.py:31: FutureWarning: pandas.Int64Index is deprecated and will be removed from pandas in a future version. Use pandas.Index with the appropriate dtype instead. (pid=1491653) from pandas import MultiIndex, Int64Index (pid=1491652) /home/ubuntu/ray/venv/lib/python3.8/site-packages/dask/dataframe/backends.py:181: FutureWarning: pandas.Int64Index is deprecated and will be removed from pandas in a future version. Use pandas.Index with the appropriate dtype instead. (pid=1491652) _numeric_index_types = (pd.Int64Index, pd.Float64Index, pd.UInt64Index) (pid=1491652) /home/ubuntu/ray/venv/lib/python3.8/site-packages/dask/dataframe/backends.py:181: FutureWarning: pandas.Float64Index is deprecated and will be removed from pandas in a future version. Use pandas.Index with the appropriate dtype instead. (pid=1491652) _numeric_index_types = (pd.Int64Index, pd.Float64Index, pd.UInt64Index) (pid=1491652) /home/ubuntu/ray/venv/lib/python3.8/site-packages/dask/dataframe/backends.py:181: FutureWarning: pandas.UInt64Index is deprecated and will be removed from pandas in a future version. Use pandas.Index with the appropriate dtype instead. (pid=1491652) _numeric_index_types = (pd.Int64Index, pd.Float64Index, pd.UInt64Index) (pid=1491652) /home/ubuntu/ray/venv/lib/python3.8/site-packages/xgboost/compat.py:31: FutureWarning: pandas.Int64Index is deprecated and will be removed from pandas in a future version. Use pandas.Index with the appropriate dtype instead. (pid=1491652) from pandas import MultiIndex, Int64Index (_RemoteRayLightGBMActor pid=1491653) 2022-06-22 17:26:50,509 WARNING __init__.py:190 -- DeprecationWarning: `ray.worker.get_resource_ids` is a private attribute and access will be removed in a future Ray version. (_RemoteRayLightGBMActor pid=1491652) 2022-06-22 17:26:50,658 WARNING __init__.py:190 -- DeprecationWarning: `ray.worker.get_resource_ids` is a private attribute and access will be removed in a future Ray version.
(_RemoteRayLightGBMActor pid=1491653) [LightGBM] [Info] Trying to bind port 59039... (_RemoteRayLightGBMActor pid=1491653) [LightGBM] [Info] Binding port 59039 succeeded (_RemoteRayLightGBMActor pid=1491653) [LightGBM] [Info] Listening... (_RemoteRayLightGBMActor pid=1491652) [LightGBM] [Info] Trying to bind port 46955... (_RemoteRayLightGBMActor pid=1491652) [LightGBM] [Info] Binding port 46955 succeeded (_RemoteRayLightGBMActor pid=1491652) [LightGBM] [Info] Listening... (_RemoteRayLightGBMActor pid=1491652) [LightGBM] [Warning] Connecting to rank 1 failed, waiting for 200 milliseconds
(_RemoteRayLightGBMActor pid=1491653) UserWarning: Overriding the parameters from Reference Dataset. (_RemoteRayLightGBMActor pid=1491653) UserWarning: categorical_column in param dict is overridden. (_RemoteRayLightGBMActor pid=1491652) UserWarning: Overriding the parameters from Reference Dataset. (_RemoteRayLightGBMActor pid=1491652) UserWarning: categorical_column in param dict is overridden.
(_RemoteRayLightGBMActor pid=1491653) [LightGBM] [Info] Connected to rank 0 (_RemoteRayLightGBMActor pid=1491653) [LightGBM] [Info] Local rank: 1, total number of machines: 2 (_RemoteRayLightGBMActor pid=1491653) [LightGBM] [Warning] num_threads is set=1, n_jobs=-1 will be ignored. Current value: num_threads=1 (_RemoteRayLightGBMActor pid=1491652) [LightGBM] [Info] Connected to rank 1 (_RemoteRayLightGBMActor pid=1491652) [LightGBM] [Info] Local rank: 0, total number of machines: 2 (_RemoteRayLightGBMActor pid=1491652) [LightGBM] [Warning] num_threads is set=1, n_jobs=-1 will be ignored. Current value: num_threads=1
(_QueueActor pid=1491650) /home/ubuntu/ray/venv/lib/python3.8/site-packages/dask/dataframe/backends.py:181: FutureWarning: pandas.Int64Index is deprecated and will be removed from pandas in a future version. Use pandas.Index with the appropriate dtype instead. (_QueueActor pid=1491650) _numeric_index_types = (pd.Int64Index, pd.Float64Index, pd.UInt64Index) (_QueueActor pid=1491650) /home/ubuntu/ray/venv/lib/python3.8/site-packages/dask/dataframe/backends.py:181: FutureWarning: pandas.Float64Index is deprecated and will be removed from pandas in a future version. Use pandas.Index with the appropriate dtype instead. (_QueueActor pid=1491650) _numeric_index_types = (pd.Int64Index, pd.Float64Index, pd.UInt64Index) (_QueueActor pid=1491650) /home/ubuntu/ray/venv/lib/python3.8/site-packages/dask/dataframe/backends.py:181: FutureWarning: pandas.UInt64Index is deprecated and will be removed from pandas in a future version. Use pandas.Index with the appropriate dtype instead. (_QueueActor pid=1491650) _numeric_index_types = (pd.Int64Index, pd.Float64Index, pd.UInt64Index) (_QueueActor pid=1491650) /home/ubuntu/ray/venv/lib/python3.8/site-packages/xgboost/compat.py:31: FutureWarning: pandas.Int64Index is deprecated and will be removed from pandas in a future version. Use pandas.Index with the appropriate dtype instead. (_QueueActor pid=1491650) from pandas import MultiIndex, Int64Index
Result for LightGBMTrainer_7b049_00000: date: 2022-06-22_17-26-53 done: false experiment_id: b4a87c26a7604a43baf895755d4f16b3 hostname: ip-172-31-43-110 iterations_since_restore: 1 node_ip: 172.31.43.110 pid: 1491578 should_checkpoint: true time_since_restore: 8.369545459747314 time_this_iter_s: 8.369545459747314 time_total_s: 8.369545459747314 timestamp: 1655918813 timesteps_since_restore: 0 train-binary_error: 0.5175879396984925 train-binary_logloss: 0.6302848981539763 training_iteration: 1 trial_id: 7b049_00000 valid-binary_error: 0.2 valid-binary_logloss: 0.558752017793943 warmup_time: 0.008721590042114258 Result for LightGBMTrainer_7b049_00000: date: 2022-06-22_17-26-56 done: true experiment_id: b4a87c26a7604a43baf895755d4f16b3 experiment_tag: '0' hostname: ip-172-31-43-110 iterations_since_restore: 100 node_ip: 172.31.43.110 pid: 1491578 should_checkpoint: true time_since_restore: 10.972588300704956 time_this_iter_s: 0.027977466583251953 time_total_s: 10.972588300704956 timestamp: 1655918816 timesteps_since_restore: 0 train-binary_error: 0.0 train-binary_logloss: 0.0005745220956391456 training_iteration: 100 trial_id: 7b049_00000 valid-binary_error: 0.058823529411764705 valid-binary_logloss: 0.17189847605331432 warmup_time: 0.008721590042114258
2022-06-22 17:26:56,406 INFO tune.py:734 -- Total run time: 14.73 seconds (14.06 seconds for the tuning loop).
{'train-binary_logloss': 0.0005745220956391456, 'train-binary_error': 0.0, 'valid-binary_logloss': 0.17189847605331432, 'valid-binary_error': 0.058823529411764705, 'time_this_iter_s': 0.027977466583251953, 'should_checkpoint': True, 'done': True, 'timesteps_total': None, 'episodes_total': None, 'training_iteration': 100, 'trial_id': '7b049_00000', 'experiment_id': 'b4a87c26a7604a43baf895755d4f16b3', 'date': '2022-06-22_17-26-56', 'timestamp': 1655918816, 'time_total_s': 10.972588300704956, 'pid': 1491578, 'hostname': 'ip-172-31-43-110', 'node_ip': '172.31.43.110', 'config': {}, 'time_since_restore': 10.972588300704956, 'timesteps_since_restore': 0, 'iterations_since_restore': 100, 'warmup_time': 0.008721590042114258, 'experiment_tag': '0'}
And perform inference on the obtained model:
predict_lightgbm(result)
2022-06-22 17:26:57,517 WARNING read_api.py:260 -- The number of blocks in this dataset (1) limits its parallelism to 1 concurrent tasks. This is much less than the number of available CPU slots in the cluster. Use `.repartition(n)` to increase the number of dataset blocks. Map_Batches: 100%|██████████| 1/1 [00:00<00:00, 50.96it/s] Map_Batches: 0%| | 0/1 [00:00<?, ?it/s](pid=1491998) /home/ubuntu/ray/venv/lib/python3.8/site-packages/dask/dataframe/backends.py:181: FutureWarning: pandas.Int64Index is deprecated and will be removed from pandas in a future version. Use pandas.Index with the appropriate dtype instead. (pid=1491998) _numeric_index_types = (pd.Int64Index, pd.Float64Index, pd.UInt64Index) (pid=1491998) /home/ubuntu/ray/venv/lib/python3.8/site-packages/dask/dataframe/backends.py:181: FutureWarning: pandas.Float64Index is deprecated and will be removed from pandas in a future version. Use pandas.Index with the appropriate dtype instead. (pid=1491998) _numeric_index_types = (pd.Int64Index, pd.Float64Index, pd.UInt64Index) (pid=1491998) /home/ubuntu/ray/venv/lib/python3.8/site-packages/dask/dataframe/backends.py:181: FutureWarning: pandas.UInt64Index is deprecated and will be removed from pandas in a future version. Use pandas.Index with the appropriate dtype instead. (pid=1491998) _numeric_index_types = (pd.Int64Index, pd.Float64Index, pd.UInt64Index) (pid=1491998) /home/ubuntu/ray/venv/lib/python3.8/site-packages/xgboost/compat.py:31: FutureWarning: pandas.Int64Index is deprecated and will be removed from pandas in a future version. Use pandas.Index with the appropriate dtype instead. (pid=1491998) from pandas import MultiIndex, Int64Index Map Progress (1 actors 1 pending): 100%|██████████| 1/1 [00:02<00:00, 2.05s/it] Map_Batches: 100%|██████████| 1/1 [00:00<00:00, 75.07it/s]
PREDICTED LABELS {'predictions': 1} {'predictions': 1} {'predictions': 0} {'predictions': 1} {'predictions': 1} {'predictions': 1} {'predictions': 1} {'predictions': 1} {'predictions': 1} {'predictions': 1} {'predictions': 0} {'predictions': 1} {'predictions': 1} {'predictions': 1} {'predictions': 1} {'predictions': 0} {'predictions': 1} {'predictions': 1} {'predictions': 1} {'predictions': 0}
Map_Batches: 0%| | 0/1 [00:00<?, ?it/s](pid=1492031) /home/ubuntu/ray/venv/lib/python3.8/site-packages/dask/dataframe/backends.py:181: FutureWarning: pandas.Int64Index is deprecated and will be removed from pandas in a future version. Use pandas.Index with the appropriate dtype instead. (pid=1492031) _numeric_index_types = (pd.Int64Index, pd.Float64Index, pd.UInt64Index) (pid=1492031) /home/ubuntu/ray/venv/lib/python3.8/site-packages/dask/dataframe/backends.py:181: FutureWarning: pandas.Float64Index is deprecated and will be removed from pandas in a future version. Use pandas.Index with the appropriate dtype instead. (pid=1492031) _numeric_index_types = (pd.Int64Index, pd.Float64Index, pd.UInt64Index) (pid=1492031) /home/ubuntu/ray/venv/lib/python3.8/site-packages/dask/dataframe/backends.py:181: FutureWarning: pandas.UInt64Index is deprecated and will be removed from pandas in a future version. Use pandas.Index with the appropriate dtype instead. (pid=1492031) _numeric_index_types = (pd.Int64Index, pd.Float64Index, pd.UInt64Index) (pid=1492031) /home/ubuntu/ray/venv/lib/python3.8/site-packages/xgboost/compat.py:31: FutureWarning: pandas.Int64Index is deprecated and will be removed from pandas in a future version. Use pandas.Index with the appropriate dtype instead. (pid=1492031) from pandas import MultiIndex, Int64Index (pid=1492033) /home/ubuntu/ray/venv/lib/python3.8/site-packages/dask/dataframe/backends.py:181: FutureWarning: pandas.Int64Index is deprecated and will be removed from pandas in a future version. Use pandas.Index with the appropriate dtype instead. (pid=1492033) _numeric_index_types = (pd.Int64Index, pd.Float64Index, pd.UInt64Index) (pid=1492033) /home/ubuntu/ray/venv/lib/python3.8/site-packages/dask/dataframe/backends.py:181: FutureWarning: pandas.Float64Index is deprecated and will be removed from pandas in a future version. Use pandas.Index with the appropriate dtype instead. (pid=1492033) _numeric_index_types = (pd.Int64Index, pd.Float64Index, pd.UInt64Index) (pid=1492033) /home/ubuntu/ray/venv/lib/python3.8/site-packages/dask/dataframe/backends.py:181: FutureWarning: pandas.UInt64Index is deprecated and will be removed from pandas in a future version. Use pandas.Index with the appropriate dtype instead. (pid=1492033) _numeric_index_types = (pd.Int64Index, pd.Float64Index, pd.UInt64Index) (pid=1492033) /home/ubuntu/ray/venv/lib/python3.8/site-packages/xgboost/compat.py:31: FutureWarning: pandas.Int64Index is deprecated and will be removed from pandas in a future version. Use pandas.Index with the appropriate dtype instead. (pid=1492033) from pandas import MultiIndex, Int64Index Map Progress (1 actors 1 pending): 100%|██████████| 1/1 [00:02<00:00, 2.09s/it]
SHAP VALUES {'predictions_0': 0.006121974664714535, 'predictions_1': 0.8940294162424869, 'predictions_2': -0.013623909529011522, 'predictions_3': -0.26580572803883, 'predictions_4': 0.2897686828261492, 'predictions_5': -0.03784232120648852, 'predictions_6': 0.021865334852359534, 'predictions_7': 1.1753326094382734, 'predictions_8': -0.02525466292349231, 'predictions_9': 0.0733463992354119, 'predictions_10': 0.09191922035401615, 'predictions_11': -0.0035196096494634313, 'predictions_12': 0.20211476104388482, 'predictions_13': 0.7813488658944929, 'predictions_14': 0.10000464816891827, 'predictions_15': 0.11543593649642907, 'predictions_16': -0.009732477634862284, 'predictions_17': 0.19117650484758314, 'predictions_18': -0.17600075102817322, 'predictions_19': 0.5829434737180024, 'predictions_20': 1.4220773445509465, 'predictions_21': 0.6086211783805069, 'predictions_22': 2.0031654232526925, 'predictions_23': 0.3090376110779834, 'predictions_24': -0.21156467772251453, 'predictions_25': 0.14122943819731193, 'predictions_26': -0.1324700025487787, 'predictions_27': 0.8280650504246968, 'predictions_28': 0.03147457104755769, 'predictions_29': 0.00029604737237433516, 'predictions_30': 0.024336487839325866, 'predictions_31': 1.5201632854544105} {'predictions_0': 0.00565090216762466, 'predictions_1': 0.7173247917145018, 'predictions_2': -0.01352989648419376, 'predictions_3': -0.204508963539279, 'predictions_4': -0.11703564338083555, 'predictions_5': 0.059858710083059874, 'predictions_6': 0.06974454296095976, 'predictions_7': 1.5952991804773315, 'predictions_8': 0.30494490847895245, 'predictions_9': 0.03770331660034111, 'predictions_10': 0.08779844216179675, 'predictions_11': 0.0001818669974550241, 'predictions_12': -0.10871732001356472, 'predictions_13': 0.49872871949407244, 'predictions_14': 0.16083030838859202, 'predictions_15': 0.4071487385487001, 'predictions_16': -0.00920287075428388, 'predictions_17': 0.21519060265555054, 'predictions_18': -0.24141319659570365, 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(pid=1492090) /home/ubuntu/ray/venv/lib/python3.8/site-packages/dask/dataframe/backends.py:181: FutureWarning: pandas.Int64Index is deprecated and will be removed from pandas in a future version. Use pandas.Index with the appropriate dtype instead. (pid=1492090) _numeric_index_types = (pd.Int64Index, pd.Float64Index, pd.UInt64Index) (pid=1492090) /home/ubuntu/ray/venv/lib/python3.8/site-packages/dask/dataframe/backends.py:181: FutureWarning: pandas.Float64Index is deprecated and will be removed from pandas in a future version. Use pandas.Index with the appropriate dtype instead. (pid=1492090) _numeric_index_types = (pd.Int64Index, pd.Float64Index, pd.UInt64Index) (pid=1492090) /home/ubuntu/ray/venv/lib/python3.8/site-packages/dask/dataframe/backends.py:181: FutureWarning: pandas.UInt64Index is deprecated and will be removed from pandas in a future version. Use pandas.Index with the appropriate dtype instead. (pid=1492090) _numeric_index_types = (pd.Int64Index, pd.Float64Index, pd.UInt64Index) (pid=1492090) /home/ubuntu/ray/venv/lib/python3.8/site-packages/xgboost/compat.py:31: FutureWarning: pandas.Int64Index is deprecated and will be removed from pandas in a future version. Use pandas.Index with the appropriate dtype instead. (pid=1492090) from pandas import MultiIndex, Int64Index