prompt: We are currently solving the following issue within our repository. Here is the issue text:
  --- BEGIN ISSUE ---
  `study.trials_dataframe` call fails for multi-objective studies
  When using the new multi-objective study framework in v2.4.0 (`optuna.create_study(..., directions=[...])`), calling the `trials_dataframe` message raises an exception caused by some incorrect column spec type comparison in the _dataframe framework. It appears the column aggregation for contains a bug when used with multiple objectives, which adds an extra `values` Tuple to the `values` Set during the initial mapping of `'value' -> 'values'` for a multi-objective study.
  
  ## Expected behavior
  
  Accessing the `trials_dataframe` method of a study works the same for both single and multi-objective studies.
  
  ## Environment
  
  - Optuna version: 2.3.0 -> 2.4.0
  - Python version: 3.8
  - OS: Ubuntu 20.04.1, Linux Kernel v5.4.0-56
  
  ## Error messages, stack traces, or logs
  
  ```
  Traceback (most recent call last):
    File "optuna_vis.py", line 12, in <module>
      study_df = study.trials_dataframe()
    File "/home/user.name/anaconda3/lib/python3.8/site-packages/optuna/study.py", line 499, in trials_dataframe
      return _trials_dataframe(self, attrs, multi_index)
    File "/home/user.name/anaconda3/lib/python3.8/site-packages/optuna/_dataframe.py", line 80, in _trials_dataframe
      columns: List[Tuple[str, str]] = sum(
    File "/home/user.name/anaconda3/lib/python3.8/site-packages/optuna/_dataframe.py", line 81, in <genexpr>
      (sorted(column_agg[k]) for k in attrs if k in column_agg), []
  TypeError: '<' not supported between instances of 'str' and 'int'
  ```
  
  By inspecting the `attr` field in the `_trials_dataframe` method (line 23, _dataframe.py), reveals an extra `values` item:
  
  ```
  {'number': {('number', '')}, 'values': {('values', 1), ('values', ''), ('values', 0), ('values', 3), ('values', 2)}, 'datetime_start': {('datetime_start', '')}, 'datetime_complete': {('datetime_complete', '')}, 'duration': {('duration', '')}, 'params': {(<param_tuples>), ...}, 'user_attrs': {('user_attrs', 'total_x'), ('user_attrs', 'total_y'), ('user_attrs', '_time_'), ('user_attrs', 'value_total')}, 'system_attrs': {('system_attrs', 'nsga2:generation')}, 'state': {('state', '')}}
  ```
  
  For context, I have defined 4 objectives in this study. After the first trial, there appears this `('values', '')` anomaly which causes the above exception.
  
  ## Steps to reproduce
  
  1. Create a MPO study with 4 objectives.
  2. Start your study and after at least 2 trials, try to call the `study.trials_dataframe()` method.
  3. The above exception will be raised.
  
  
  --- END ISSUE ---
  
  Below are some code segments, each from a relevant file. One or more of these files may contain bugs.
  --- BEGIN FILES ---
  Path: `optuna/_dataframe.py`
  Content:
  ```
  1 import collections
  2 from typing import Any
  3 from typing import DefaultDict
  4 from typing import Dict
  5 from typing import List
  6 from typing import Set
  7 from typing import Tuple
  8 
  9 import optuna
  10 from optuna._imports import try_import
  11 from optuna.trial._state import TrialState
  12 
  13 
  14 with try_import() as _imports:
  15     # `Study.trials_dataframe` is disabled if pandas is not available.
  16     import pandas as pd
  17 
  18 # Required for type annotation in `Study.trials_dataframe`.
  19 if not _imports.is_successful():
  20     pd = object  # type: ignore # NOQA
  21 
  22 
  23 def _trials_dataframe(
  24     study: "optuna.Study", attrs: Tuple[str, ...], multi_index: bool
  25 ) -> "pd.DataFrame":
  26     _imports.check()
  27 
  28     trials = study.get_trials(deepcopy=False)
  29 
  30     # If no trials, return an empty dataframe.
  31     if not len(trials):
  32         return pd.DataFrame()
  33 
  34     if "value" in attrs and study._is_multi_objective():
  35         attrs = tuple("values" if attr == "value" else attr for attr in attrs)
  36 
  37     attrs_to_df_columns: Dict[str, str] = collections.OrderedDict()
  38     for attr in attrs:
  39         if attr.startswith("_"):
  40             # Python conventional underscores are omitted in the dataframe.
  41             df_column = attr[1:]
  42         else:
  43             df_column = attr
  44         attrs_to_df_columns[attr] = df_column
  45 
  46     # column_agg is an aggregator of column names.
  47     # Keys of column agg are attributes of `FrozenTrial` such as 'trial_id' and 'params'.
  48     # Values are dataframe columns such as ('trial_id', '') and ('params', 'n_layers').
  49     column_agg: DefaultDict[str, Set] = collections.defaultdict(set)
  50     non_nested_attr = ""
  51 
  52     def _create_record_and_aggregate_column(
  53         trial: "optuna.trial.FrozenTrial",
  54     ) -> Dict[Tuple[str, str], Any]:
  55 
  56         record = {}
  57         for attr, df_column in attrs_to_df_columns.items():
  58             value = getattr(trial, attr)
  59             if isinstance(value, TrialState):
  60                 # Convert TrialState to str and remove the common prefix.
  61                 value = str(value).split(".")[-1]
  62             if isinstance(value, dict):
  63                 for nested_attr, nested_value in value.items():
  64                     record[(df_column, nested_attr)] = nested_value
  65                     column_agg[attr].add((df_column, nested_attr))
  66             elif isinstance(value, list):
  67                 # Expand trial.values.
  68                 for nested_attr, nested_value in enumerate(value):
  69                     record[(df_column, nested_attr)] = nested_value
  70                     column_agg[attr].add((df_column, nested_attr))
  71             else:
  72                 record[(df_column, non_nested_attr)] = value
  73                 column_agg[attr].add((df_column, non_nested_attr))
  74         return record
  75 
  76     records = list([_create_record_and_aggregate_column(trial) for trial in trials])
  77 
  78     columns: List[Tuple[str, str]] = sum(
  79         (sorted(column_agg[k]) for k in attrs if k in column_agg), []
  80     )
  81 
  82     df = pd.DataFrame(records, columns=pd.MultiIndex.from_tuples(columns))
  83 
  84     if not multi_index:
  85         # Flatten the `MultiIndex` columns where names are concatenated with underscores.
  86         # Filtering is required to omit non-nested columns avoiding unwanted trailing
  87         # underscores.
  88         df.columns = ["_".join(filter(lambda c: c, map(lambda c: str(c), col))) for col in columns]
  89 
  90     return df
  91
  ```
  --- END FILES ---
  
  Please first localize the bug based on the issue statement, and then generate a patch according to the `git diff` format fenced by three backticks.
  
  Here is an example:
  
  ```diff
  diff --git a/examples/server_async.py b/examples/server_async.py
  --- a/examples/server_async.py
  +++ b/examples/server_async.py
  @@ -313,4 +313,4 @@
   
   
   if __name__ == "__main__":
  -    asyncio.run(run_async_server("."), debug=True)
  +    asyncio.run(run_async_server(), debug=True)
  diff --git a/examples/server_sync.py b/examples/server_sync.py
  --- a/examples/server_sync.py
  +++ b/examples/server_sync.py
  @@ -313,5 +313,5 @@
   
   
   if __name__ == "__main__":
  -    server = run_sync_server(".")
  +    server = run_sync_server()
       server.shutdown()
  
  ```
  
golden_diff: diff --git a/optuna/_dataframe.py b/optuna/_dataframe.py
  --- a/optuna/_dataframe.py
  +++ b/optuna/_dataframe.py
  @@ -68,6 +68,12 @@
                   for nested_attr, nested_value in enumerate(value):
                       record[(df_column, nested_attr)] = nested_value
                       column_agg[attr].add((df_column, nested_attr))
  +            elif attr == "values":
  +                # trial.values should be None when the trial's state is FAIL or PRUNED.
  +                assert value is None
  +                for nested_attr in range(len(study.directions)):
  +                    record[(df_column, nested_attr)] = None
  +                    column_agg[attr].add((df_column, nested_attr))
               else:
                   record[(df_column, non_nested_attr)] = value
                   column_agg[attr].add((df_column, non_nested_attr))
  
verification_info: {"golden_diff": "diff --git a/optuna/_dataframe.py b/optuna/_dataframe.py\n--- a/optuna/_dataframe.py\n+++ b/optuna/_dataframe.py\n@@ -68,6 +68,12 @@\n                 for nested_attr, nested_value in enumerate(value):\n                     record[(df_column, nested_attr)] = nested_value\n                     column_agg[attr].add((df_column, nested_attr))\n+            elif attr == \"values\":\n+                # trial.values should be None when the trial's state is FAIL or PRUNED.\n+                assert value is None\n+                for nested_attr in range(len(study.directions)):\n+                    record[(df_column, nested_attr)] = None\n+                    column_agg[attr].add((df_column, nested_attr))\n             else:\n                 record[(df_column, non_nested_attr)] = value\n                 column_agg[attr].add((df_column, non_nested_attr))\n", "issue": "`study.trials_dataframe` call fails for multi-objective studies\nWhen using the new multi-objective study framework in v2.4.0 (`optuna.create_study(..., directions=[...])`), calling the `trials_dataframe` message raises an exception caused by some incorrect column spec type comparison in the _dataframe framework. It appears the column aggregation for contains a bug when used with multiple objectives, which adds an extra `values` Tuple to the `values` Set during the initial mapping of `'value' -> 'values'` for a multi-objective study.\r\n\r\n## Expected behavior\r\n\r\nAccessing the `trials_dataframe` method of a study works the same for both single and multi-objective studies.\r\n\r\n## Environment\r\n\r\n- Optuna version: 2.3.0 -> 2.4.0\r\n- Python version: 3.8\r\n- OS: Ubuntu 20.04.1, Linux Kernel v5.4.0-56\r\n\r\n## Error messages, stack traces, or logs\r\n\r\n```\r\nTraceback (most recent call last):\r\n  File \"optuna_vis.py\", line 12, in <module>\r\n    study_df = study.trials_dataframe()\r\n  File \"/home/user.name/anaconda3/lib/python3.8/site-packages/optuna/study.py\", line 499, in trials_dataframe\r\n    return _trials_dataframe(self, attrs, multi_index)\r\n  File \"/home/user.name/anaconda3/lib/python3.8/site-packages/optuna/_dataframe.py\", line 80, in _trials_dataframe\r\n    columns: List[Tuple[str, str]] = sum(\r\n  File \"/home/user.name/anaconda3/lib/python3.8/site-packages/optuna/_dataframe.py\", line 81, in <genexpr>\r\n    (sorted(column_agg[k]) for k in attrs if k in column_agg), []\r\nTypeError: '<' not supported between instances of 'str' and 'int'\r\n```\r\n\r\nBy inspecting the `attr` field in the `_trials_dataframe` method (line 23, _dataframe.py), reveals an extra `values` item:\r\n\r\n```\r\n{'number': {('number', '')}, 'values': {('values', 1), ('values', ''), ('values', 0), ('values', 3), ('values', 2)}, 'datetime_start': {('datetime_start', '')}, 'datetime_complete': {('datetime_complete', '')}, 'duration': {('duration', '')}, 'params': {(<param_tuples>), ...}, 'user_attrs': {('user_attrs', 'total_x'), ('user_attrs', 'total_y'), ('user_attrs', '_time_'), ('user_attrs', 'value_total')}, 'system_attrs': {('system_attrs', 'nsga2:generation')}, 'state': {('state', '')}}\r\n```\r\n\r\nFor context, I have defined 4 objectives in this study. After the first trial, there appears this `('values', '')` anomaly which causes the above exception.\r\n\r\n## Steps to reproduce\r\n\r\n1. Create a MPO study with 4 objectives.\r\n2. Start your study and after at least 2 trials, try to call the `study.trials_dataframe()` method.\r\n3. The above exception will be raised.\r\n\n", "before_files": [{"content": "import collections\nfrom typing import Any\nfrom typing import DefaultDict\nfrom typing import Dict\nfrom typing import List\nfrom typing import Set\nfrom typing import Tuple\n\nimport optuna\nfrom optuna._imports import try_import\nfrom optuna.trial._state import TrialState\n\n\nwith try_import() as _imports:\n    # `Study.trials_dataframe` is disabled if pandas is not available.\n    import pandas as pd\n\n# Required for type annotation in `Study.trials_dataframe`.\nif not _imports.is_successful():\n    pd = object  # type: ignore # NOQA\n\n\ndef _trials_dataframe(\n    study: \"optuna.Study\", attrs: Tuple[str, ...], multi_index: bool\n) -> \"pd.DataFrame\":\n    _imports.check()\n\n    trials = study.get_trials(deepcopy=False)\n\n    # If no trials, return an empty dataframe.\n    if not len(trials):\n        return pd.DataFrame()\n\n    if \"value\" in attrs and study._is_multi_objective():\n        attrs = tuple(\"values\" if attr == \"value\" else attr for attr in attrs)\n\n    attrs_to_df_columns: Dict[str, str] = collections.OrderedDict()\n    for attr in attrs:\n        if attr.startswith(\"_\"):\n            # Python conventional underscores are omitted in the dataframe.\n            df_column = attr[1:]\n        else:\n            df_column = attr\n        attrs_to_df_columns[attr] = df_column\n\n    # column_agg is an aggregator of column names.\n    # Keys of column agg are attributes of `FrozenTrial` such as 'trial_id' and 'params'.\n    # Values are dataframe columns such as ('trial_id', '') and ('params', 'n_layers').\n    column_agg: DefaultDict[str, Set] = collections.defaultdict(set)\n    non_nested_attr = \"\"\n\n    def _create_record_and_aggregate_column(\n        trial: \"optuna.trial.FrozenTrial\",\n    ) -> Dict[Tuple[str, str], Any]:\n\n        record = {}\n        for attr, df_column in attrs_to_df_columns.items():\n            value = getattr(trial, attr)\n            if isinstance(value, TrialState):\n                # Convert TrialState to str and remove the common prefix.\n                value = str(value).split(\".\")[-1]\n            if isinstance(value, dict):\n                for nested_attr, nested_value in value.items():\n                    record[(df_column, nested_attr)] = nested_value\n                    column_agg[attr].add((df_column, nested_attr))\n            elif isinstance(value, list):\n                # Expand trial.values.\n                for nested_attr, nested_value in enumerate(value):\n                    record[(df_column, nested_attr)] = nested_value\n                    column_agg[attr].add((df_column, nested_attr))\n            else:\n                record[(df_column, non_nested_attr)] = value\n                column_agg[attr].add((df_column, non_nested_attr))\n        return record\n\n    records = list([_create_record_and_aggregate_column(trial) for trial in trials])\n\n    columns: List[Tuple[str, str]] = sum(\n        (sorted(column_agg[k]) for k in attrs if k in column_agg), []\n    )\n\n    df = pd.DataFrame(records, columns=pd.MultiIndex.from_tuples(columns))\n\n    if not multi_index:\n        # Flatten the `MultiIndex` columns where names are concatenated with underscores.\n        # Filtering is required to omit non-nested columns avoiding unwanted trailing\n        # underscores.\n        df.columns = [\"_\".join(filter(lambda c: c, map(lambda c: str(c), col))) for col in columns]\n\n    return df\n", "path": "optuna/_dataframe.py"}], "after_files": [{"content": "import collections\nfrom typing import Any\nfrom typing import DefaultDict\nfrom typing import Dict\nfrom typing import List\nfrom typing import Set\nfrom typing import Tuple\n\nimport optuna\nfrom optuna._imports import try_import\nfrom optuna.trial._state import TrialState\n\n\nwith try_import() as _imports:\n    # `Study.trials_dataframe` is disabled if pandas is not available.\n    import pandas as pd\n\n# Required for type annotation in `Study.trials_dataframe`.\nif not _imports.is_successful():\n    pd = object  # type: ignore # NOQA\n\n\ndef _trials_dataframe(\n    study: \"optuna.Study\", attrs: Tuple[str, ...], multi_index: bool\n) -> \"pd.DataFrame\":\n    _imports.check()\n\n    trials = study.get_trials(deepcopy=False)\n\n    # If no trials, return an empty dataframe.\n    if not len(trials):\n        return pd.DataFrame()\n\n    if \"value\" in attrs and study._is_multi_objective():\n        attrs = tuple(\"values\" if attr == \"value\" else attr for attr in attrs)\n\n    attrs_to_df_columns: Dict[str, str] = collections.OrderedDict()\n    for attr in attrs:\n        if attr.startswith(\"_\"):\n            # Python conventional underscores are omitted in the dataframe.\n            df_column = attr[1:]\n        else:\n            df_column = attr\n        attrs_to_df_columns[attr] = df_column\n\n    # column_agg is an aggregator of column names.\n    # Keys of column agg are attributes of `FrozenTrial` such as 'trial_id' and 'params'.\n    # Values are dataframe columns such as ('trial_id', '') and ('params', 'n_layers').\n    column_agg: DefaultDict[str, Set] = collections.defaultdict(set)\n    non_nested_attr = \"\"\n\n    def _create_record_and_aggregate_column(\n        trial: \"optuna.trial.FrozenTrial\",\n    ) -> Dict[Tuple[str, str], Any]:\n\n        record = {}\n        for attr, df_column in attrs_to_df_columns.items():\n            value = getattr(trial, attr)\n            if isinstance(value, TrialState):\n                # Convert TrialState to str and remove the common prefix.\n                value = str(value).split(\".\")[-1]\n            if isinstance(value, dict):\n                for nested_attr, nested_value in value.items():\n                    record[(df_column, nested_attr)] = nested_value\n                    column_agg[attr].add((df_column, nested_attr))\n            elif isinstance(value, list):\n                # Expand trial.values.\n                for nested_attr, nested_value in enumerate(value):\n                    record[(df_column, nested_attr)] = nested_value\n                    column_agg[attr].add((df_column, nested_attr))\n            elif attr == \"values\":\n                # trial.values should be None when the trial's state is FAIL or PRUNED.\n                assert value is None\n                for nested_attr in range(len(study.directions)):\n                    record[(df_column, nested_attr)] = None\n                    column_agg[attr].add((df_column, nested_attr))\n            else:\n                record[(df_column, non_nested_attr)] = value\n                column_agg[attr].add((df_column, non_nested_attr))\n        return record\n\n    records = list([_create_record_and_aggregate_column(trial) for trial in trials])\n\n    columns: List[Tuple[str, str]] = sum(\n        (sorted(column_agg[k]) for k in attrs if k in column_agg), []\n    )\n\n    df = pd.DataFrame(records, columns=pd.MultiIndex.from_tuples(columns))\n\n    if not multi_index:\n        # Flatten the `MultiIndex` columns where names are concatenated with underscores.\n        # Filtering is required to omit non-nested columns avoiding unwanted trailing\n        # underscores.\n        df.columns = [\"_\".join(filter(lambda c: c, map(lambda c: str(c), col))) for col in columns]\n\n    return df\n", "path": "optuna/_dataframe.py"}]}