import copy
import functools
import warnings
from dataclasses import dataclass
from typing import (
    Any,
    cast,
    Dict,
    Iterable,
    Iterator,
    List,
    NamedTuple,
    Optional,
    Sequence,
    Set,
    Tuple,
    Union,
)

import torch
import torch.distributed as dist
import torch.distributed.fsdp._traversal_utils as traversal_utils
import torch.nn as nn
from torch.distributed._shard.sharded_tensor import ShardedTensor
from torch.distributed.fsdp._common_utils import (
    _apply_to_modules,
    _FSDPState,
    _get_module_fsdp_state_if_fully_sharded_module,
    _get_param_to_fqns,
    _module_handles,
    clean_tensor_name,
)
from torch.distributed.fsdp._fsdp_extensions import _ext_chunk_tensor
from torch.distributed.fsdp._runtime_utils import _clear_grads_if_needed, _lazy_init
from torch.distributed.fsdp._shard_utils import _gather_state_dict
from torch.distributed.fsdp.api import ShardingStrategy
from torch.distributed.fsdp.flat_param import FlatParameter, FlatParamHandle


@dataclass
class FSDPParamInfo:
    state: _FSDPState
    flat_param: FlatParameter
    param_indices: Dict[str, int]


def sorted_items(dictionary: Dict[str, Any]) -> Iterator[Tuple[str, Any]]:
    keys = sorted(dictionary.keys())
    for k in keys:
        yield k, dictionary[k]


class _ConsolidatedOptimState:
    """
    This holds the consolidated optimizer state on the target rank. Positive-
    dimension tensor state is communicated across ranks, while zero-dimension
    tensor state and non-tensor state is taken directly from the target rank.

    PyTorch version 1.12 moved to using zero-dimension tensors for scalar
    values, but user implemented optimizers may still use float (i.e. a
    non-tensor). Thus, we support both and handle them identically.

    Attributes:
        tensor_state (Dict[str, torch.Tensor]): Mapping from positive-dimension
            tensor state name to the unsharded flattened tensor representing
            the state.
        zero_dim_tensor_state (Dict[str, torch.Tensor]): Mapping from zero-
            dimension tensor state name to its value.
        non_tensor_state (Dict[str, Any]): Mapping from non-tensor state
            name to its value.
    """

    tensor_state: Dict[str, torch.Tensor] = {}
    zero_dim_tensor_state: Dict[str, torch.Tensor] = {}
    non_tensor_state: Dict[str, Any] = {}


class _PosDimTensorInfo(NamedTuple):
    """
    Meatadata for positive-dimension tensors used internally for
    :meth:`scatter_full_optim_state_dict`.

    Attributes:
        shape (torch.Size): Sharded tensor shape (which is equal to the
            unsharded tensor shape if the tensor is optimizer state for a
            non-FSDP parameter and is hence not sharded).
        dtype (torch.dtype): Data type of the tensor.
    """

    shape: torch.Size
    dtype: torch.dtype


class _OptimStateKey(NamedTuple):
    """
    This represents an optimizer state key that may be used commonly across
    ranks. It is based on the unflattened parameter names rather than parameter
    IDs to make it indepenendent of each rank's own optimizer construction.
    """

    unflat_param_names: Tuple[str, ...]
    is_fsdp_managed: bool


def _unflatten_optim_state(
    fsdp_param_info: FSDPParamInfo,
    flat_param_state: Dict[str, Any],
    to_save: bool,
    shard_state: bool,
) -> List[Dict[str, Any]]:
    """
    Unflattens the optimizer state, consisting of the "state" part and the
    "param_groups" part. Unflattening the "state" part involves consolidating
    the state on the target rank and remapping from flattened to unflattened
    parameter IDs, and the "param_groups" part only involves remapping from
    flattened to unflattened parameter IDs.

    Args:
        fsdp_param_info (FSDPParamInfo): The fsdp state and the target flatten
            parameter.
        flat_param_state (Dict[str, Any]): Entry for the flattened parameter
            in the "state" part of the optimizer state dict.
        to_save (bool): Whether to save the state on this rank.

    Returns:
        List[Dict[str, Any]]: A :class:`list` holding the entries in the
        "state" part of the optimizer state dict corresponding to the
        unflattened parameters comprising the flattened parameter if on the
        target rank or an empty :class:`list` otherwise. The final optimizer
        state dict will need to map these entries using the proper unflattened
        parameter IDs.
    """
    assert (
        not shard_state or to_save
    ), "If ``shard_state`` is True, ``to_save`` has to be True."
    consolidated_state = _communicate_optim_state(
        fsdp_param_info,
        flat_param_state,
    )
    if to_save:
        unflat_param_state = _unflatten_communicated_optim_state(
            fsdp_param_info,
            consolidated_state,
            shard_state,
        )
        for optim_state in unflat_param_state:
            for key in list(optim_state.keys()):
                state = optim_state[key]
                if isinstance(state, torch.Tensor):
                    optim_state[key] = state.cpu()
        return unflat_param_state
    else:
        return []


def _is_zero_dim_tensor(x: Any) -> bool:
    return torch.is_tensor(x) and x.dim() == 0


def _communicate_optim_state(
    fsdp_param_info: FSDPParamInfo,
    flat_param_state: Dict[str, Any],
) -> _ConsolidatedOptimState:
    """
    Communicates the optimizer state for a flattened parameter across ranks.
    All ranks will hold the entire non-sharded optimizer state on GPU.

    If ``N`` is the number of tensor optimizer states in the optimizer state
    dict, then the communication complexity is 0 if ``N = 0`` and ``N + 1``
    otherwise (where the plus 1 comes from all-gathering the padding per rank).

    Args:
        fsdp_param_info (FSDPParamInfo): The fsdp state and the target flatten
            parameter.
        flat_param_state (Dict[str, Any]): The entry in the "state" part of the
            optimizer state dict corresponding to the flattened parameter.

    Returns:
        ConsolidatedOptimState: Consolidated optimizer state for the target
            flattened parameter.
    """
    fsdp_state = fsdp_param_info.state
    flat_param = fsdp_param_info.flat_param
    state = _ConsolidatedOptimState()
    tensor_state, zero_dim_tensor_state, non_tensor_state = (
        state.tensor_state,
        state.zero_dim_tensor_state,
        state.non_tensor_state,
    )

    for state_name, value in sorted_items(flat_param_state):
        # Positive-dimension tensor state: communicate across ranks
        if torch.is_tensor(value) and value.dim() > 0:
            # If the parameter is not sharded, then neither is the
            # positive-dimension tensor state, so no need to communicate it --
            # we take the target rank's value
            if (
                fsdp_state.world_size == 1
                or fsdp_state.sharding_strategy == ShardingStrategy.NO_SHARD
            ):
                tensor_state[state_name] = value
                continue
            if not value.is_cuda:
                value = value.to(fsdp_state.compute_device)
            # Assume that positive-dimension tensor optimizer state
            # has the same shape as the sharded flattened parameter
            buffer_size = flat_param._full_param_padded.size()  # type: ignore[attr-defined]
            tensor_buffer = value.new_zeros(*buffer_size)
            dist.all_gather_into_tensor(
                tensor_buffer, value, group=fsdp_state.process_group
            )
            torch.cuda.synchronize()
            unpadded_numel = cast(
                nn.Parameter, flat_param._unpadded_unsharded_size
            ).numel()
            tensor_state[state_name] = tensor_buffer[:unpadded_numel]
        # Zero-dimension tensor state and non-tensor state: take this rank's
        # value directly
        else:
            if _is_zero_dim_tensor(value):
                zero_dim_tensor_state[state_name] = value
            else:
                non_tensor_state[state_name] = value
    return state


def _unflatten_communicated_optim_state(
    fsdp_param_info: FSDPParamInfo,
    state: _ConsolidatedOptimState,
    shard_state: bool,
) -> List[Dict[str, Any]]:
    """
    Unflattens the communicated optimizer state (given by ``tensor_state``,
    ``non_tensor_state``, and ``zero_dim_tensor_state``) for a single flattened
    parameter. This should only be called on the target rank.

    Args:
        fsdp_param_info (FSDPParamInfo): The fsdp state and the target flatten
            parameter.
        state (_ConsolidatedOptimState): Consolidated optimizer state.

    Returns:
        List[Dict[str, Any]]: A :class:`list` holding the entries in the
        "state" part of the optimizer state dict corresponding to the
        unflattened parameters comprising the flattened parameter. The final
        optimizer state dict will need to map these entries using the proper
        unflattened parameter IDs.
    """
    fsdp_state = fsdp_param_info.state
    flat_param = fsdp_param_info.flat_param
    unflat_param_state: List[Dict[str, Any]] = []
    flat_param_views: Dict[str, Iterator] = {}
    num_unflat_params = flat_param._num_params
    tensor_state, zero_dim_tensor_state, non_tensor_state = (
        state.tensor_state,
        state.zero_dim_tensor_state,
        state.non_tensor_state,
    )

    for _ in range(num_unflat_params):
        unflat_state_param = {}
        # Add positive-dimension tensor state: unflatten with views
        for state_name, flat_tensor in sorted_items(tensor_state):
            views_generated = state_name in flat_param_views
            if not views_generated:
                views = FlatParamHandle._get_unflat_views(flat_param, flat_tensor)
                flat_param_views[state_name] = views
            else:
                views = flat_param_views[state_name]
            optim_state: Union[torch.Tensor, ShardedTensor] = next(views)
            if shard_state:
                assert fsdp_state.process_group is not None
                optim_state = _ext_chunk_tensor(
                    optim_state,
                    fsdp_state.rank,
                    fsdp_state.world_size,
                    torch.cuda.device_count(),
                    fsdp_state.process_group,
                )
            unflat_state_param[state_name] = optim_state

        # Add zero-dimension tensor state: take the target rank's value
        for state_name, zero_dim_tensor in sorted_items(zero_dim_tensor_state):
            unflat_state_param[state_name] = zero_dim_tensor
        # Add non-tensor state: take the target rank's value
        for state_name, non_tensor in sorted_items(non_tensor_state):
            unflat_state_param[state_name] = non_tensor
        unflat_param_state.append(unflat_state_param)
    return unflat_param_state


def _flatten_optim_state_dict(
    optim_state_dict: Dict[str, Any],
    model: nn.Module,
    shard_state: bool,
    use_orig_params: bool = False,
    optim: Optional[torch.optim.Optimizer] = None,
) -> Dict[str, Any]:
    """
    Flattens the full optimizer state dict, still keying by unflattened
    parameter names. If ``shard_state=True``, then FSDP-managed
    ``FlatParameter`` 's optimizer states are sharded, and otherwise, they are
    kept unsharded.

    If ``use_orig_params`` is True, each rank will have all FSDP-managed
    parameters but some of these parameters may be empty due to the sharding.
    For a regular optim.Optimizer, states for those empty parameters will
    not be initialized. So, when aggregating the FQNs across ranks, no assert
    will be raised on a rank even if it does not have all the states -- it is
    valid and FSDP know how to aggregate them. However, FSDP has to ignore
    handling those parameters that are not managed by FSDP and do not exist on
    the local rank -- it is managed by other parallelism and FSDP does not
    know ho to handle/aggregate them.

    Note that ``_flatten_tensor_optim_state`` does not need ``optim`` to
    flatten/shard the state. However, NamedOptimizer and KeyedOptimizer require
    all the states even if the corresponding parameters are empty. To this end,
    ``optim`` will be used to to get the initial state of the empty parameters.
    ``optim`` should only be non-None if the ``optim` is KeyedOptimizer or
    NamedOptimizer.

    Returns:
        Dict[str, Any]: The flattened optimizer state dict.
    """
    unflat_osd = optim_state_dict
    if "state" not in unflat_osd or "param_groups" not in unflat_osd:
        raise ValueError(
            '`optim_state_dict` must have the keys "state" and '
            '"param_groups" to be a valid optimizer state dict'
        )
    param_to_fqns = _get_param_to_fqns(model)
    fqn_to_fsdp_param_info = _get_fqn_to_fsdp_param_info(model)

    # Construct the "state" part
    flat_osd_state: Dict[Union[_OptimStateKey, str], Any] = {}
    unflat_osd_state = unflat_osd["state"]
    all_state_keys = set(unflat_osd_state.keys())

    # local_state_dict is used to construct states of empty parameters.
    # This should only be used if is_named_optimizer=True.
    local_state_dict: Dict[str, Any] = {}
    local_state_clean_fqns: Dict[str, str] = {}
    if optim is not None:
        local_state_dict = optim.state_dict()["state"]
        for fqn in local_state_dict.keys():
            clean_fqn = clean_tensor_name(fqn)
            local_state_clean_fqns[clean_fqn] = fqn

    for param, unflat_param_names in param_to_fqns.items():
        fqn = unflat_param_names[0]
        if fqn not in unflat_osd_state:
            continue
        all_state_keys.difference_update(unflat_param_names)
        if fqn in fqn_to_fsdp_param_info:
            fsdp_param_info = fqn_to_fsdp_param_info[fqn]
            if use_orig_params:
                assert (
                    shard_state
                ), "If use_orig_params is True, shard_state must be True."
                flat_state = _shard_orig_param_state(
                    fsdp_param_info,
                    fqn,
                    unflat_osd_state[fqn],
                )
            else:
                flat_state = _flatten_optim_state(
                    fsdp_param_info,
                    unflat_osd_state,
                    unflat_param_names,
                    shard_state,
                )
            key = _OptimStateKey(tuple(unflat_param_names), True)
            # Only include non-empty states since as expected by
            # `torch.optim.Optimizer` s unless the optimizer is KeyedOptimizer
            # or NamedOptimizer.
            if flat_state:
                flat_osd_state[key] = flat_state
            elif optim is not None:  # NamedOptimizer or KeyedOptimizer case.
                assert len(unflat_param_names) == 1
                local_wrapped_fqn = local_state_clean_fqns.get(fqn, "")
                if local_wrapped_fqn:
                    flat_osd_state[key] = copy.deepcopy(
                        local_state_dict[local_wrapped_fqn]
                    )
        else:  # do not flatten non-FSDP parameters' states
            assert len(unflat_param_names) == 1
            key = _OptimStateKey(tuple(unflat_param_names), False)
            flat_osd_state[key] = copy.copy(unflat_osd_state[fqn])

    # Handle user-defined state, states that are not accosiated with parameters.
    for key in all_state_keys:
        flat_osd_state[key] = copy.copy(unflat_osd_state[key])

    # Construct the "param_groups" part -- copy as is since it will be
    # rekeyed later according to the target rank's optimizer
    flat_osd_param_groups = copy.deepcopy(unflat_osd["param_groups"])
    return {"state": flat_osd_state, "param_groups": flat_osd_param_groups}


def _flatten_optim_state(
    fsdp_param_info: FSDPParamInfo,
    unflat_osd_state: Dict[str, Dict[str, Any]],
    unflat_param_names: List[str],
    shard_state: bool,
) -> Dict[str, Any]:
    """
    Flattens the optimizer state in ``full_optim_state_dict`` for a single
    flattened parameter in ``fsdp_param_info`` corresponding to the unflattened
    parameter names in ``unflat_param_names``.

    Args:
        unflat_osd_state (Dict[str, Dict[str, Any]]): The "state" part of the
            optimizer state dict corresponding to the unflattened parameters.
        unflat_param_names (List[str]): A :class:`list` of unflattened
            parameter names corresponding to the flattened parameter
            ``flat_param``.
        fsdp_param_info (FSDPParamInfo): The fsdp state and the target flatten
            parameter.
        shard_state (bool): Whether to shard flattened positive-dimension
            tensor state; if ``False``, then the full flattened tensor is
            kept in the returned :class:`dict.

    Returns:
        Dict[str, Any]: A :class:`dict` mapping state names to their values for
        a particular flattened parameter. The sharded optimizer state dict's
        "state" part will map a key to this returned value.
    """
    fsdp_state = fsdp_param_info.state
    flat_param = fsdp_param_info.flat_param
    num_unflat_params = len(unflat_param_names)
    assert num_unflat_params > 0, (
        "Expects at least one unflattened parameter corresponding to the "
        "flattened parameter"
    )
    unflat_param_shapes = flat_param._shapes
    num_unflat_param_shapes = len(unflat_param_shapes)
    assert (
        num_unflat_params == num_unflat_param_shapes
    ), f"Expects {num_unflat_params} shapes but got {num_unflat_param_shapes}"

    # Check if these unflattened parameters have any optimizer state
    has_state = [
        bool(unflat_param_name in unflat_osd_state)
        for unflat_param_name in unflat_param_names
    ]
    # If none of the unflattened parameters comprising this flattened parameter
    # have any state, then we do not want an entry in the optimizer state dict
    if not any(has_state):
        return {}  # no need to flatten any state
    # There may still be some unflattened parameters with state and some
    # without
    unflat_param_states = [
        _gather_state_dict(
            unflat_osd_state[unflat_param_name], pg=fsdp_state.process_group
        )
        if unflat_param_name in unflat_osd_state
        else None
        for unflat_param_name in unflat_param_names
    ]
    # Check that the unflattened parameters have the same state names
    state_names = None
    for unflat_param_state in unflat_param_states:
        if unflat_param_state is None:
            continue
        if state_names is None:
            state_names = set(unflat_param_state.keys())
        else:
            if state_names != set(unflat_param_state.keys()):
                raise ValueError(
                    "Differing optimizer state names for the unflattened "
                    f"parameters: {unflat_param_names}"
                )
    assert state_names is not None

    # Flatten the state
    flat_state: Dict[str, Any] = {}
    for state_name in state_names:
        state_values = [
            unflat_param_state[state_name] if unflat_param_state is not None else None
            for unflat_param_state in unflat_param_states
        ]
        non_none_state_values = [v for v in state_values if v is not None]
        are_pos_dim_tensors = are_zero_dim_tensors = are_non_tensors = True
        for v in non_none_state_values:
            are_pos_dim_tensors &= torch.is_tensor(v) and v.dim() > 0
            are_zero_dim_tensors &= _is_zero_dim_tensor(v)
            are_non_tensors &= not torch.is_tensor(v)
        types = {type(v) for v in non_none_state_values}
        if len(types) != 1 or not (
            are_pos_dim_tensors or are_zero_dim_tensors or are_non_tensors
        ):
            raise ValueError(
                f"Differing optimizer state types for state {state_name}, "
                f"values {non_none_state_values}, and unflattened parameter "
                f"names {unflat_param_names}"
            )
        if are_pos_dim_tensors:
            flat_tensor = _flatten_tensor_optim_state(
                state_name,
                state_values,
                unflat_param_names,
                unflat_param_shapes,
                flat_param,
            )
            if shard_state:
                # Shard the flattened tensor immediately to minimize max memory
                # usage
                sharded_flat_tensor, _ = FlatParamHandle._get_shard(
                    flat_tensor,
                    fsdp_state.rank,
                    fsdp_state.world_size,
                )
                flat_state[state_name] = sharded_flat_tensor
            else:
                flat_state[state_name] = flat_tensor
        elif are_zero_dim_tensors:
            flat_state[state_name] = _flatten_zero_dim_tensor_optim_state(
                state_name,
                state_values,
                unflat_param_names,
            )
        else:
            assert are_non_tensors
            flat_state[state_name] = _flatten_non_tensor_optim_state(
                state_name,
                state_values,
                unflat_param_names,
            )

    return flat_state


def _flatten_tensor_optim_state(
    state_name: str,
    pos_dim_tensors: List[torch.Tensor],
    unflat_param_names: List[str],
    unflat_param_shapes: Sequence[torch.Size],
    flat_param: FlatParameter,
) -> torch.Tensor:
    """
    Flattens the positive-dimension tensor optimizer state given by the values
    ``tensors`` for the state ``state_name`` for a single flattened parameter
    ``flat_param`` corresponding to the unflattened parameter names
    ``unflat_param_names`` and unflatted parameter shapes
    ``unflat_param_shapes``. This flattens each unflattened parameter's tensor
    state into one tensor.

    NOTE: We use zero tensors for any unflattened parameters without state
    since some value is required to fill those entries. This assumes that the
    zero tensor is mathematically equivalent to having no state, which is true
    for Adam's "exp_avg" and "exp_avg_sq" but may not be true for all
    optimizers.

    Args:
        state_name (str): Optimizer state name.
        pos_dim_tensors (List[torch.Tensor]): Positive-dimension tensor
            optimizer state values for the unflattened parameters corresponding
            to the single flattened parameter.
        unflat_param_names (List[str]): A :class:`list` of unflattened
            parameter names corresponding to the single flattened parameter.
        unflat_param_shapes (List[torch.Size]): Unflattened parameter shapes
            corresponding to the single flattened parameter.
        flat_param (FlatParameter): The flattened parameter.

    Returns:
        torch.Tensor: A flattened tensor containing the optimizer state
        corresponding to ``state_name`` constructed by concatenating the
        unflattened parameter tensor states in ``pos_dim_tensors`` (using zero
        tensors for any unflattened parameters without the state).
    """
    non_none_tensors = [t for t in pos_dim_tensors if t is not None]
    # Check that all are tensors with the same dtype
    dtypes = {t.dtype for t in non_none_tensors}
    if len(dtypes) != 1:
        raise ValueError(
            "All unflattened parameters comprising a single flattened "
            "parameter must have positive-dimension tensor state with the "
            f"same dtype but got dtypes {dtypes} for state {state_name} and "
            f"unflattened parameter names {unflat_param_names}"
        )
    dtype = next(iter(dtypes))
    # Check that each tensor state matches its parameter's shape
    for tensor, shape in zip(pos_dim_tensors, unflat_param_shapes):
        if tensor is None and len(shape) == 0:
            raise ValueError("Flattening a zero-dimension parameter is not supported")
        elif tensor is not None and tensor.shape != shape:
            raise ValueError(
                "Tensor optimizer state does not have same shape as its "
                f"parameter: {tensor.shape} {shape}"
            )
    # Flatten the tensor states: we do not need to add any padding since the
    # flattened optimizer state tensor sharded via `_get_shard()`, which pads
    # the shard as needed (just like for the flattened parameter)
    cpu_device = torch.device("cpu")
    tensors = [
        torch.flatten(state_value.to(cpu_device))
        if state_value is not None
        else torch.flatten(
            torch.zeros(
                size=shape,
                dtype=dtype,
                device=cpu_device,
            )
        )
        for state_value, shape in zip(pos_dim_tensors, unflat_param_shapes)
    ]
    flat_tensor = torch.cat(tensors)
    flat_param_shape = flat_param._unpadded_unsharded_size  # type: ignore[attr-defined]
    assert flat_tensor.shape == flat_param_shape, (
        f"tensor optim state: {flat_tensor.shape} "
        f"flattened parameter: {flat_param_shape}"
    )
    return flat_tensor


def _flatten_zero_dim_tensor_optim_state(
    state_name: str,
    zero_dim_tensors: List[torch.Tensor],
    unflat_param_names: List[str],
) -> torch.Tensor:
    """
    Flattens the zero-dimension tensor optimizer state given by the values
    ``zero_dim_tensors`` for the state ``state_name`` for a single flattened
    parameter corresponding to the unflattened parameter names
    ``unflat_param_names`` by enforcing that all tensors are the same and using
    that common value.

    NOTE: The requirement that the tensors are the same across all unflattened
    parameters comprising the flattened parameter is needed to maintain the
    invariant that FSDP performs the same computation as its non-sharded
    equivalent. This means that none of the unflattened parameters can be
    missing this state since imposing a value may differ from having no value.
    For example, for Adam's "step", no value means maximum bias correction,
    while having some positive value means less bias correction.

    Args:
        state_name (str): Optimizer state name.
        zero_dim_tensors (List[torch.Tensor]): Zero-dimension optimizer state
            for the unflattened parameters corresponding to the single
            flattened parameter.
        unflat_param_names (List[str]): A :class:`list` of unflattened
            parameter names corresponding to the single flattened parameter.

    Returns:
        torch.Tensor: A zero-dimensional tensor giving the value of the state
        ``state_name`` for all unflattened parameters corresponding to the
        names ``unflat_param_names``.
    """
    non_none_tensors = [t for t in zero_dim_tensors if t is not None]
    # Enforce that all have the same value and dtype
    values_set = {t.item() if t is not None else None for t in zero_dim_tensors}
    dtypes = {t.dtype if t is not None else None for t in zero_dim_tensors}
    if (
        len(non_none_tensors) != len(zero_dim_tensors)
        or len(values_set) != 1
        or len(dtypes) != 1
    ):
        raise ValueError(
            "All unflattened parameters comprising a single flattened "
            "parameter must have scalar state with the same value and dtype "
            f"but got values {values_set} and dtypes {dtypes} for state "
            f"{state_name} and unflattened parameter names "
            f"{unflat_param_names}"
        )
    value = next(iter(values_set))
    dtype = next(iter(dtypes))
    return torch.tensor(value, dtype=dtype, device=torch.device("cpu"))


def _flatten_non_tensor_optim_state(
    state_name: str,
    non_tensors: List[Any],
    unflat_param_names: List[str],
) -> Any:
    """
    Flattens the non-tensor optimizer state given by the values ``non_tensors``
    for the state ``state_name`` for a single flattened parameter corresponding
    to the unflattened parameter names ``unflat_param_names`` by enforcing that
    all values are the same and using that common value.

    See the note in :func:`_flatten_zero_dim_tensor_optim_state`.

    Args:
        state_name (str): Optimizer state name.
        non_tensors (List[Any]): Non-tensor optimizer state for the unflattened
            parameters corresponding to the single flattened parameter.
        unflat_param_names (List[str]): A :class:`list` of unflattened
            parameter names corresponding to the single flattened parameter.

    Returns:
        Any: A non-tensor giving the value of the state ``state_name`` for all
        unflattened parameters corresponding to the names
        ``unflat_param_names``.
    """
    non_none_non_tensors = [nt for nt in non_tensors if nt is not None]
    # Enforce that all have the same value (same type already checked)
    non_tensor_set = set(non_tensors)
    if len(non_none_non_tensors) != len(non_tensors) or len(non_tensor_set) != 1:
        raise ValueError(
            "All unflattened parameters comprising a single flattened "
            "parameter must have scalar state with the same value and dtype "
            f"but got values {non_tensor_set} for state {state_name} and  "
            f"unflattened parameter names {unflat_param_names}"
        )
    non_tensor = next(iter(non_tensor_set))
    return non_tensor


def _process_pos_dim_tensor_state(
    flat_optim_state_dict: Dict[str, Any],
    world_size: int,
) -> Dict[str, Any]:
    """
    Processes positive-dimension tensor states in ``flat_optim_state_dict`` by
    replacing them with metadata. This is done so the processed optimizer state
    dict can be broadcast from rank 0 to all ranks without copying those tensor
    states, and thus, this is meant to only be called on rank 0.

    Args:
        flat_optim_state_dict (Dict[str, Any]): Flattened optimizer state dict
            with the positive-dimension tensor states unsharded.

    Returns:
        Dict[str, Any]: The flattened optimizer state dict with positive-
        dimension tensor states replaced by metadata.
    """
    flat_osd = flat_optim_state_dict  # alias
    no_tensor_osd: Dict[str, Any] = {"state": {}}
    for key, param_state in flat_osd["state"].items():
        no_tensor_osd["state"][key] = {}
        for state_name, value in sorted_items(param_state):
            is_pos_dim_tensor_state = torch.is_tensor(value) and value.dim() > 0
            if not is_pos_dim_tensor_state:
                no_tensor_osd["state"][key][state_name] = value
                continue
            if key.is_fsdp_managed:  # FSDP parameter
                sharded_size = FlatParamHandle._get_sharded_size(
                    value, rank=0, world_size=world_size
                )
                assert len(sharded_size) == 1, f"{sharded_size}"
                info = _PosDimTensorInfo(sharded_size, value.dtype)
            else:  # non-FSDP parameter
                info = _PosDimTensorInfo(value.shape, value.dtype)
            no_tensor_osd["state"][key][state_name] = info
    no_tensor_osd["param_groups"] = flat_osd["param_groups"]
    return no_tensor_osd


def _broadcast_processed_optim_state_dict(
    processed_optim_state_dict: Optional[Dict[str, Any]],
    rank: int,
    group,
) -> Dict[str, Any]:
    """
    Broadcasts the processed optimizer state dict from rank 0 to all ranks.

    Args:
        processed_optim_state_dict (Optional[Dict[str, Any]]): The flattened
            optimizer state dict with positive-dimension tensor states replaced
            with metadata if on rank 0; ignored otherwise.

    Returns:
        Dict[str, Any]: The processed optimizer state dict.
    """
    # Broadcast the two data structures rank 0 to all ranks
    obj_list = [processed_optim_state_dict] if rank == 0 else [None]
    dist.broadcast_object_list(obj_list, src=0, group=group)
    processed_optim_state_dict = obj_list[0]  # type: ignore[assignment]
    assert processed_optim_state_dict is not None
    # Keep zero-dimension tensors on CPU
    return processed_optim_state_dict


def _broadcast_pos_dim_tensor_states(
    processed_optim_state_dict: Dict[str, Any],
    flat_optim_state_dict: Optional[Dict[str, Any]],
    rank: int,
    world_size: int,
    group,
    broadcast_device: torch.device,
) -> Dict[str, Any]:
    """
    Takes ``processed_optim_state_dict``, which has metadata in place of
    positive-dimension tensor states, and broadcasts those tensor states from
    rank 0 to all ranks. For tensor states corresponding to FSDP parameters,
    rank 0 shards the tensor and broadcasts shard-by-shard, and for tensor
    states corresponding to non-FSDP parameters, rank 0 broadcasts the full
    tensor.

    Args:
        processed_optim_state_dict (Dict[str, Any]): The flattened optimizer
            state dict with positive-dimension tensor states replaced with
            metadata; this should be returned by
            :meth:`_process_pos_dim_tensor_state` and non-empty on all ranks.
        flat_optim_state_dict (Optional[Dict[str, Any]]): The flattened
            unsharded optimizer state dict with the actual positive-dimension
            tensor states if on rank 0; ignored on nonzero ranks.

    Returns:
        Dict[str, Any]: The optimizer state dict with the positive-dimension
        tensor state correctly populated via ``broadcast()`` s from rank 0.
    """
    assert (
        rank != 0 or flat_optim_state_dict is not None
    ), "Expects rank 0 to pass in the flattened optimizer state dict"
    no_tensor_osd = processed_optim_state_dict  # alias
    flat_osd = flat_optim_state_dict  # alias
    for key, param_state in no_tensor_osd["state"].items():
        for state_name, value in sorted_items(param_state):
            is_pos_dim_tensor_state = isinstance(value, _PosDimTensorInfo)
            if not is_pos_dim_tensor_state:
                continue
            if rank == 0:
                assert flat_osd is not None
                unsharded_tensor = flat_osd["state"][key][state_name]
            else:
                unsharded_tensor = None
            shape, dtype = value.shape, value.dtype
            if key.is_fsdp_managed:  # FSDP parameter
                _broadcast_sharded_pos_dim_tensor_state(
                    unsharded_tensor,
                    param_state,
                    state_name,
                    shape,
                    dtype,
                    broadcast_device,
                    rank,
                    world_size,
                    group,
                )  # modify `param_state` destructively
            else:  # non-FSDP parameter
                _broadcast_unsharded_pos_dim_tensor_state(
                    unsharded_tensor,
                    param_state,
                    state_name,
                    shape,
                    dtype,
                    broadcast_device,
                    rank,
                    group,
                )  # modify `param_state` destructively
    return no_tensor_osd


def _broadcast_sharded_pos_dim_tensor_state(
    unsharded_tensor: Optional[torch.Tensor],
    param_state: Dict[str, Any],
    state_name: str,
    shape: torch.Size,
    dtype: torch.dtype,
    broadcast_device: torch.device,
    rank: int,
    world_size: int,
    group,
) -> None:
    """
    Broadcasts positive-dimension tensor state for the state ``state_name``
    corresponding to an FSDP parameter shard-by-shard, only to be saved on the
    relevant rank. This modifies ``param_state`` destructively.

    Args:
        unsharded_tensor (Optional[torch.Tensor]): Unsharded tensor from which
            to broadcast shards if on rank 0; ignored otherwise.
        shape (torch.Size): Shape of the sharded tensor; same on all ranks.
    """
    get_shard: Optional[functools.partial[Tuple[torch.Tensor, int]]] = None
    if rank == 0:
        assert (
            unsharded_tensor is not None
        ), "Expects rank 0 to pass in the unsharded tensor"
        get_shard = functools.partial(
            FlatParamHandle._get_shard,
            unsharded_tensor,
        )
    for target_rank in range(1, world_size):
        if rank == 0:
            assert get_shard is not None
            sharded_tensor = get_shard(target_rank, world_size)[0].to(broadcast_device)
        else:
            sharded_tensor = torch.zeros(
                shape,
                requires_grad=False,
                dtype=dtype,
                device=broadcast_device,
            )
        dist.broadcast(sharded_tensor, src=0, group=group)
        # Only keep the shard on the target rank and keep it on the broadcast
        # device, which is typically GPU
        if rank == target_rank:
            param_state[state_name] = sharded_tensor
        else:
            del sharded_tensor
    # Lastly, shard on rank 0
    if rank != 0:
        return
    param_state[state_name] = get_shard(0, world_size)[0].to(broadcast_device)  # type: ignore[misc]


def _broadcast_unsharded_pos_dim_tensor_state(
    unsharded_tensor: Optional[torch.Tensor],
    param_state: Dict[str, Any],
    state_name: str,
    shape: torch.Size,
    dtype: torch.dtype,
    broadcast_device: torch.device,
    rank: int,
    group,
) -> None:
    """
    Broadcasts positive-dimension tensor state for the state ``state_name``
    corresponding to an unsharded non-FSDP parameter from rank 0 to all ranks.
    This modifies ``param_state`` destructively.

    Args:
        unsharded_tensor (Optional[torch.Tensor]): Unsharded tensor to
            broadcast if on rank 0; ignored otherwise.
    """
    if rank == 0:
        assert (
            unsharded_tensor is not None
        ), "Expects rank 0 to pass in the unsharded tensor"
        assert (
            shape == unsharded_tensor.shape
        ), f"Shape mismatch: {shape} {unsharded_tensor.shape}"
        assert (
            dtype == unsharded_tensor.dtype
        ), f"dtype mismatch: {dtype} {unsharded_tensor.dtype}"
        unsharded_tensor = unsharded_tensor.to(broadcast_device)
    else:
        unsharded_tensor = torch.zeros(
            shape,
            requires_grad=False,
            dtype=dtype,
            device=broadcast_device,
        )
    dist.broadcast(unsharded_tensor, src=0, group=group)
    # Keep the tensor on the broadcast device, which is typically GPU
    param_state[state_name] = unsharded_tensor


def _rekey_sharded_optim_state_dict(
    sharded_osd: Dict[str, Any],
    model: nn.Module,
    optim: torch.optim.Optimizer,
    optim_input: Optional[
        Union[
            List[Dict[str, Any]],
            Iterable[nn.Parameter],
        ]
    ],
    using_optim_input: bool,
    is_named_optimizer: bool = False,
) -> Dict[str, Any]:
    """
    Rekeys the optimizer state dict from unflattened parameter names to
    flattened parameter IDs according to the calling rank's ``optim``, which
    may be different across ranks. In particular, the unflattened parameter
    names are represented as :class:`_OptimStateKey` s.
    """
    param_to_fqns = _get_param_to_fqns(model)
    flat_param_to_fqn = _get_flat_param_to_fqn(model)
    param_to_param_key: Dict[nn.Parameter, Union[int, str]] = cast(
        Dict[nn.Parameter, Union[int, str]],
        (
            _get_param_to_param_id_from_optim_input(model, optim_input)
            if using_optim_input
            else _get_param_to_param_key(
                optim, model, is_named_optimizer, param_to_fqns, flat_param_to_fqn
            )
        ),
    )
    # All parameter keys in `param_to_param_key` should be in
    # `param_to_fqns` -- strict inequality follows when not all parameters are
    # passed to the optimizer
    assert len(param_to_param_key) <= len(param_to_fqns)

    unflat_param_names_to_flat_param_key: Dict[
        Tuple[str, ...], Union[int, str]
    ] = {}  # for "state"
    unflat_param_name_to_flat_param_key: Dict[
        str, Union[int, str]
    ] = {}  # for "param_groups"
    for param, unflat_param_names in param_to_fqns.items():
        if param not in param_to_param_key:
            # This parameter was not passed to the optimizer
            continue
        flat_param_key = param_to_param_key[param]
        unflat_param_names_to_flat_param_key[tuple(unflat_param_names)] = flat_param_key
        for unflat_param_name in unflat_param_names:
            unflat_param_name_to_flat_param_key[unflat_param_name] = flat_param_key

    sharded_osd_state = sharded_osd["state"]
    rekeyed_osd_state: Dict[Union[str, int], Any] = {}
    for key, param_state in sharded_osd_state.items():
        if isinstance(key, str):
            rekeyed_osd_state[key] = param_state
            continue
        flat_param_key = unflat_param_names_to_flat_param_key.get(
            key.unflat_param_names, key.unflat_param_names
        )
        rekeyed_osd_state[flat_param_key] = param_state

    rekeyed_osd_param_groups: List[Dict[str, Any]] = []
    for unflat_param_group in sharded_osd["param_groups"]:
        flat_param_group = copy.deepcopy(unflat_param_group)
        flat_param_keys = sorted(
            {
                unflat_param_name_to_flat_param_key[unflat_param_name]
                for unflat_param_name in unflat_param_group["params"]
            }
        )
        flat_param_group["params"] = flat_param_keys
        rekeyed_osd_param_groups.append(flat_param_group)

    return {"state": rekeyed_osd_state, "param_groups": rekeyed_osd_param_groups}


def _get_param_id_to_param_from_optim_input(
    model: nn.Module,
    optim_input: Optional[
        Union[
            List[Dict[str, Any]],
            Iterable[nn.Parameter],
        ]
    ] = None,
) -> Dict[int, nn.Parameter]:
    """
    Constructs a mapping from parameter IDs to parameters. This may be used
    both for models with ``FlatParameter`` s and without.

    NOTE: This method is only preserved for backward compatibility. The method
    :meth:`_get_param_key_to_param` is the preferred code path that does not
    rely on ``optim_input``.

    NOTE: We critically assume that, whether the optimizer input is a list of
    parameters or a list of parameter groups, :class:`torch.optim.Optimizer`
    enumerates the parameter IDs in order. In other words, for a parameter list
    input, the parameter IDs should be in that list order, and for a parameter
    groups input, the parameter IDs should be in order within each parameter
    group and in order across parameter groups.

    Args:
        model (nn.Module): Model whose parameters are passed into the
            optimizer.
        optim_input (Optional[Union[List[Dict[str, Any]],
        Iterable[nn.Parameter]]]): Input passed into the optimizer
            representing either a :class:`list` of parameter groups or an
            iterable of parameters; if ``None``, then this method assumes the
            input was ``model.parameters()``. (Default: ``None``)

    Returns:
        List[nn.Parameter]: Mapping from parameter IDs to parameters,
        where the parameter ID is implicitly the index in the :class:`list`.
    """
    # Assume the standard case of passing `model.parameters()` to the optimizer
    # if `optim_input` is not specified
    if optim_input is None:
        return {pid: param for pid, param in enumerate(model.parameters())}
    try:
        params = cast(List[nn.Parameter], list(optim_input))
    except TypeError as e:
        raise TypeError(
            "Optimizer input should be an iterable of Tensors or dicts, "
            f"but got {optim_input}"
        ) from e
    if len(params) == 0:
        raise ValueError("Optimizer input should not be empty")

    # Check if the optimizer input represents tensors or parameter groups
    all_tensors = True
    all_dicts = True
    for param in params:
        all_tensors &= isinstance(param, torch.Tensor)
        all_dicts &= isinstance(param, dict)
    if not all_tensors and not all_dicts:
        raise TypeError("Optimizer input should be an iterable of Tensors or dicts")
    if all_tensors:
        return {pid: param for pid, param in enumerate(params)}
    assert all_dicts
    param_id_to_param: List[nn.Parameter] = []
    for param_group in params:
        has_params_key = "params" in param_group  # type: ignore[operator]
        assert has_params_key, (
            'A parameter group should map "params" to a list of the '
            "parameters in the group"
        )
        for param in param_group["params"]:  # type: ignore[index]
            # Implicitly map `flat_param_id` (current length of the list) to
            # `param`
            param_id_to_param.append(param)
    return {pid: param for pid, param in enumerate(param_id_to_param)}


def _get_flat_param_to_fqn(model: torch.nn.Module) -> Dict[nn.Parameter, str]:
    def module_fn(module, prefix, flat_param_to_fqn):
        for param_name, param in module.named_parameters(recurse=False):
            if type(param) is not FlatParameter:
                continue
            fqn = clean_tensor_name(prefix + param_name)
            flat_param_to_fqn[param] = fqn

    def return_fn(flat_param_to_fqn):
        return flat_param_to_fqn

    flat_param_to_fqn_ret: Dict[torch.nn.Parameter, str] = {}
    return _apply_to_modules(
        model,
        module_fn,
        return_fn,
        [fqn for fqn, _ in model.named_parameters()],
        flat_param_to_fqn_ret,
    )


def _get_param_key_to_param(
    optim: torch.optim.Optimizer,
    model: Optional[nn.Module] = None,
    is_named_optimizer: bool = False,
    param_to_fqns: Optional[Dict[nn.Parameter, List[str]]] = None,
    flat_param_to_fqn: Optional[Dict[nn.Parameter, str]] = None,
) -> Dict[Union[int, str], nn.Parameter]:
    """
    Constructs a mapping from parameter keys to parameters. For the regular
    optimizers, the keys are parameter IDs. For NamedOptimizer, the keys
    are FQNs. This API may be used both for models with ``FlatParameter`` s and
    without.
    """
    clean_fqn_to_curr_fqn: Dict[str, str] = {}
    if is_named_optimizer:
        assert (
            param_to_fqns is not None and flat_param_to_fqn is not None
        ), "The optimizer is a NamedOptimizer, `param_to_fqns` must not be None."
        assert model is not None
        for key, _ in model.named_parameters():
            clean_fqn_to_curr_fqn[clean_tensor_name(key)] = key

    param_key_to_param: Dict[Union[str, int], nn.Parameter] = {}
    pid = 0
    for param_group in optim.param_groups:
        if is_named_optimizer:
            for param in param_group["params"]:
                assert flat_param_to_fqn is not None
                if param in flat_param_to_fqn:
                    # FlatParameter case
                    key = flat_param_to_fqn[param]
                else:
                    assert param_to_fqns is not None
                    # use_orig_params case
                    assert len(param_to_fqns[param]) == 1
                    key = param_to_fqns[param][0]
                key = clean_fqn_to_curr_fqn[key]
                param_key_to_param[key] = param
        else:
            for param in param_group["params"]:
                param_key_to_param[pid] = param
                pid += 1

    return param_key_to_param


def _get_param_to_param_key(
    optim: torch.optim.Optimizer,
    model: Optional[nn.Module] = None,
    is_named_optimizer: bool = False,
    param_to_fqns: Optional[Dict[nn.Parameter, List[str]]] = None,
    flat_param_to_fqn: Optional[Dict[nn.Parameter, str]] = None,
) -> Dict[nn.Parameter, Union[int, str]]:
    """
    Constructs the inverse mapping of :func:`_get_param_key_to_param`. This API
    only supports the case where `optim` is a regular optimizer, not NamedOptimizer.
    So the parameter keys will be parameter id.
    """
    param_id_to_param = _get_param_key_to_param(
        optim, model, is_named_optimizer, param_to_fqns, flat_param_to_fqn
    )
    return {param: param_id for param_id, param in param_id_to_param.items()}


def _get_param_to_param_id_from_optim_input(
    model: nn.Module,
    optim_input: Optional[
        Union[
            List[Dict[str, Any]],
            Iterable[nn.Parameter],
        ]
    ] = None,
) -> Dict[nn.Parameter, int]:
    """Constructs the inverse mapping of :func:`_get_param_id_to_param_from_optim_input`."""
    param_id_to_param = _get_param_id_to_param_from_optim_input(model, optim_input)
    return {param: param_id for param_id, param in param_id_to_param.items()}


def _check_missing_keys_on_rank(
    r0_optim_state_keys: List[_OptimStateKey],
    optim_state_key_to_param_key: Dict[_OptimStateKey, Union[str, int]],
    param_key_to_param: Dict[Union[str, int], nn.Parameter],
    group: Optional[dist.ProcessGroup],
) -> None:
    # Ensure that all ranks have at least the optimizer states needed by
    # rank 0's optimizer
    missing_keys: List[_OptimStateKey] = []
    for r0_optim_state_key in r0_optim_state_keys:
        if r0_optim_state_key not in optim_state_key_to_param_key:
            # A parameter from rank 0's optimizer does not exist for this
            # rank's optimizer
            missing_keys.append(r0_optim_state_key)
            continue
        param_key = optim_state_key_to_param_key[r0_optim_state_key]
        if isinstance(param_key, int):
            assert param_key >= 0 and param_key < len(
                param_key_to_param
            ), "Check the `param_key_to_param` construction"
    device = torch.device("cuda", torch.cuda.current_device())
    num_missing = torch.tensor([len(missing_keys)], dtype=torch.int32, device=device)
    dist.all_reduce(num_missing, group=group)
    if num_missing.item() > 0:
        obj_list = [None for _ in range(dist.get_world_size(group))]
        dist.all_gather_object(obj_list, missing_keys, group=group)
        error_msg = (
            "FSDP currently requires each rank to have at least the "
            "optimizer states needed by rank 0's optimizer but some ranks "
            "are missing some of those states"
        )
        for rank, keys in enumerate(obj_list):
            keys = cast(List[_OptimStateKey], keys)
            if len(keys) > 0:
                error_msg += (
                    f"\nRank {rank} is missing states for the parameters: "
                    f"{[key.unflat_param_names for key in keys]}"
                )
        raise RuntimeError(error_msg)


def _map_param_key_to_optim_keys(
    optim_state_dict: Dict[str, Any],
    group: Optional[dist.ProcessGroup],
    param_key_to_param: Dict[Union[int, str], nn.Parameter],
    param_to_fqns: Dict[nn.Parameter, List[str]],
    fqn_to_fsdp_param_info: Dict[str, FSDPParamInfo],
    merge_keys: bool = False,
) -> Tuple[List[_OptimStateKey], Dict[_OptimStateKey, Union[int, str]]]:
    """
    Construct the local mapping between the ``_OptimStateKey`` and parameter keys
    and all the ``_OptimStateKey`` across ranks. If ``merge_keys`` is False, rank0
    must contain all the ``_OptimStateKey``, an exception will be raised otherwise.
    Note that ``merge_keys`` should equal to ``use_orig_params``.
    """
    rank = dist.get_rank(group)
    optim_state_key_to_param_key: Dict[_OptimStateKey, Union[int, str]] = {}  # local
    all_optim_state_keys: List[_OptimStateKey] = []

    for param_key, param in param_key_to_param.items():
        # Do not include parameters without state to avoid empty mappings
        # just like in normal `torch.optim.Optimizer.state_dict()`
        if param_key not in optim_state_dict["state"]:
            continue
        fqns = param_to_fqns[param]
        is_fsdp_managed = isinstance(param, FlatParameter)
        if is_fsdp_managed:
            assert fqns[0] in fqn_to_fsdp_param_info, (
                fqns[0],
                list(fqn_to_fsdp_param_info.keys()),
            )
        is_fsdp_managed = fqns[0] in fqn_to_fsdp_param_info
        optim_state_key = _OptimStateKey(
            unflat_param_names=tuple(fqns),
            is_fsdp_managed=is_fsdp_managed,
        )
        if rank == 0 or merge_keys:
            all_optim_state_keys.append(optim_state_key)
        optim_state_key_to_param_key[optim_state_key] = param_key

    if merge_keys:
        all_keys: List[List[_OptimStateKey]] = [
            [] for _ in range(dist.get_world_size(group))
        ]
        dist.all_gather_object(all_keys, all_optim_state_keys, group=group)
        merge_all_optim_state_keys = [
            key for local_keys in all_keys for key in local_keys
        ]
        all_optim_state_keys = sorted(set(merge_all_optim_state_keys))
    else:
        key_obj_list: List[Optional[List[_OptimStateKey]]] = (
            [all_optim_state_keys] if rank == 0 else [None]
        )
        dist.broadcast_object_list(key_obj_list, src=0, group=group)
        assert key_obj_list[0] is not None
        all_optim_state_keys = key_obj_list[0]
        _check_missing_keys_on_rank(
            all_optim_state_keys,
            optim_state_key_to_param_key,
            param_key_to_param,
            group,
        )

    return all_optim_state_keys, optim_state_key_to_param_key


def _unflatten_param_groups(
    state_dict: Dict[str, Any],
    param_key_to_param: Dict[Union[int, str], nn.Parameter],
    param_to_fqns: Dict[nn.Parameter, List[str]],
) -> List[Dict[str, Any]]:
    param_groups: List[Dict[str, Any]] = []
    for flat_param_group in state_dict["param_groups"]:
        unflat_param_group = copy.deepcopy(flat_param_group)
        param_group_params = [
            param_key_to_param[flat_param_key]
            for flat_param_key in flat_param_group["params"]
        ]
        nested_unflat_param_names = [
            param_to_fqns[param] for param in param_group_params
        ]
        unflat_param_group["params"] = [
            unflat_param_name
            for unflat_param_names in nested_unflat_param_names
            for unflat_param_name in unflat_param_names
        ]  # flatten the list of lists
        param_groups.append(unflat_param_group)
    return param_groups


def _is_named_optimizer(optim_state_dict: Dict[str, Any]) -> bool:
    state = optim_state_dict.get("state", None)
    if not state:
        # If we cannot find a state, assume it is not NamedOptimizer as
        # NamedOptimizer has eagerly initialization.
        return False
    try:
        key = next(iter(state.keys()))
    except Exception as e:
        raise Exception(optim_state_dict) from e
    return isinstance(key, str)


def _optim_state_dict(
    model: nn.Module,
    optim: torch.optim.Optimizer,
    optim_state_dict: Dict[str, Any],
    optim_input: Optional[
        Union[
            List[Dict[str, Any]],
            Iterable[nn.Parameter],
        ]
    ],
    rank0_only: bool,
    shard_state: bool,
    group: Optional[dist.ProcessGroup],
    using_optim_input: bool,
    use_orig_params: bool = False,
) -> Dict[str, Any]:
    """
    Consolidates the optimizer state and returns it as a :class:`dict`
    following the convention of :meth:`torch.optim.Optimizer.state_dict`,
    i.e. with keys ``"state"`` and ``"param_groups"``.
    The flattened parameters in ``FSDP`` modules contained in ``model``
    are mapped back to their unflattened parameters.

    Parameter keys are not well-defined. For a regular optimizer, the optimizer
    state_dict contains a mapping from parameter IDs to parameter states.
    Parameter IDs are the order of parameters in ``optim.param_groups()`` across
    all the groups. This API also allows user to pass ``optim_input`` for the
    mapping between parameters and parameter IDs. Using ``optim_input`` is being
    deprecated.

    If the optimizer is a ``NamedOptimizer``, the optimizer state_dict does not
    contain parameter IDs mapping but a mapping from parameter FQNs to parameter
    states. This API finds the mapping from FQNs to parameters if the optimizer
    is a ``NamedOptimizer``.

    If ``use_orig_params`` is True, each rank will have all FSDP-managed
    parameters but some of these parameters may be empty due to the sharding.
    For a regular optim.Optimizer, states for those empty parameters will
    not be initialized. So, when aggregating the FQNs across ranks, no assert
    will be raised on a rank even if it does not have all the states -- it is
    valid and FSDP know how to aggregate them. However, FSDP has to ignore
    handling those parameters that are not managed by FSDP and do not exist on
    the local rank -- it is managed by other parallelism and FSDP does not
    know ho to handle/aggregate them.

    Args:
        model (nn.Module): Root module (which may or may not be a
            :class:`FullyShardedDataParallel` instance) whose parameters
            were passed into the optimizer ``optim``.
        optim (torch.optim.Optimizer): Optimizer for ``model`` 's
            parameters.
        rank0_only (bool): If ``True``, saves the populated :class:`dict`
            only on rank 0; if ``False``, saves it on all ranks. (Default:
            ``True``)
        shard_state (bool): If ``True``, shard and distribute all
            non-zero-dimension states.

    Returns:
        Dict[str, Any]: A :class:`dict` containing the optimizer state for
        ``model`` 's original unflattened parameters and including keys
        "state" and "param_groups" following the convention of
        :meth:`torch.optim.Optimizer.state_dict`. If ``rank0_only=False``,
        then nonzero ranks return an empty :class:`dict`.
    """
    _clear_grads_if_needed(traversal_utils._get_fsdp_handles(model))
    to_save = not rank0_only or (dist.get_rank(group) == 0 or shard_state)
    fsdp_osd: Dict[str, Any] = {"state": {}, "param_groups": []} if to_save else {}
    fsdp_osd_state: Dict[str, Any] = fsdp_osd["state"] if to_save else {}
    param_to_fqns = _get_param_to_fqns(model)
    flat_param_to_fqn = _get_flat_param_to_fqn(model)
    is_named_optimizer = _is_named_optimizer(optim_state_dict)

    param_key_to_param = cast(
        Dict[Union[int, str], nn.Parameter],
        (
            _get_param_id_to_param_from_optim_input(model, optim_input)
            if using_optim_input
            else _get_param_key_to_param(
                optim, model, is_named_optimizer, param_to_fqns, flat_param_to_fqn
            )
        ),
    )
    fqn_to_fsdp_param_info = _get_fqn_to_fsdp_param_info(model)

    all_optim_state_keys, optim_state_key_to_param_key = _map_param_key_to_optim_keys(
        optim_state_dict,
        group,
        param_key_to_param,
        param_to_fqns,
        fqn_to_fsdp_param_info,
        merge_keys=use_orig_params,
    )

    # Iterate in rank 0's flattened parameter ID order to ensure aligned
    # all-gathers across ranks
    for optim_state_key in all_optim_state_keys:
        param_key: Union[str, int, None] = optim_state_key_to_param_key.get(
            optim_state_key, None
        )

        if param_key is None:
            assert use_orig_params, (
                "If use_orig_params is False, we must be able to find the "
                f"corresponding param id. {optim_state_key} {param_key}"
            )
            if not optim_state_key.is_fsdp_managed:
                continue

        if optim_state_key.is_fsdp_managed:
            # If there are multiple unflat_param_names (not use_orig_params),
            # they share the same FSDPParamInfo. So the first unflat_param_name
            # is sufficient to fetch the FSDPParamInfo.
            fqn = optim_state_key.unflat_param_names[0]
            fsdp_param_info = fqn_to_fsdp_param_info[fqn]
            if use_orig_params:
                state = (
                    {} if param_key is None else optim_state_dict["state"][param_key]
                )
                unflat_state = [
                    _gather_orig_param_state(
                        fsdp_param_info, fqn, state, shard_state, group
                    )
                ]
            else:
                unflat_state = _unflatten_optim_state(
                    fsdp_param_info,
                    optim_state_dict["state"][param_key],
                    to_save,
                    shard_state,
                )
            if to_save:
                assert len(unflat_state) == len(optim_state_key.unflat_param_names)
                for unflat_param_name, unflat_param_state in zip(
                    optim_state_key.unflat_param_names,
                    unflat_state,
                ):
                    fsdp_osd_state[unflat_param_name] = unflat_param_state
        elif to_save:
            assert len(optim_state_key.unflat_param_names) == 1
            unflat_param_name = optim_state_key.unflat_param_names[0]
            fsdp_osd_state[unflat_param_name] = copy.copy(
                optim_state_dict["state"][param_key]
            )
            for state_name, value in sorted_items(fsdp_osd_state[unflat_param_name]):
                if torch.is_tensor(value):
                    fsdp_osd_state[unflat_param_name][state_name] = value.cpu()

    if to_save:
        flat_param_fqns = set(flat_param_to_fqn.values())
        for key, value in optim_state_dict["state"].items():
            if key in fsdp_osd_state:
                continue
            if key in flat_param_fqns:
                continue
            if key in param_key_to_param:
                continue
            # This key is not recognized by FSDP. It may be a user-defined state
            # or some parameters state that FSDP is unable to map from
            # ``optim.param_groups``.
            warnings.warn(
                f"Found a optim state, {key}, that FSDP cannot process. FSDP "
                "will directly copy everything to the returned state_dict. In "
                "most cases, this is a user-defined state that is not "
                "associated with any particular parameter. Another possible "
                "case is this state is managed by DMP. Otherwise, there may "
                " be a mismatched assumption of optim_state_dict of this mode."
            )
            fsdp_osd_state[key] = value

        fsdp_osd["param_groups"] = _unflatten_param_groups(
            optim_state_dict, param_key_to_param, param_to_fqns
        )

    return fsdp_osd


def _get_fqn_to_fsdp_param_info(model: nn.Module) -> Dict[str, FSDPParamInfo]:
    """
    Construct the mapping from a param's fqn to its corresponding ``FSDPParamInfo``
    if the param is managed by FSDP. ``FlatParameter._fqns`` only stores the first
    FQN of a shared parameter. So the keys in the mapping are guaranteed to map
    to unique parameters.
    """

    def module_fn(module, prefix, fqn_to_param_info):
        fsdp_state = _get_module_fsdp_state_if_fully_sharded_module(module)
        if fsdp_state is None:
            return
        _lazy_init(fsdp_state, module)
        handles = _module_handles(fsdp_state, module)
        if not handles:
            return
        flat_param = handles[0].flat_param
        fsdp_param_info = FSDPParamInfo(fsdp_state, flat_param, {})
        for idx, local_fqn in enumerate(flat_param._fqns):
            fqn = clean_tensor_name(prefix + local_fqn)
            if fqn in fqn_to_param_info:
                assert fqn_to_param_info[fqn].flat_param == flat_param
            fqn_to_param_info[fqn] = fsdp_param_info
            fsdp_param_info.param_indices[fqn] = idx

    def return_fn(fqn_to_param_info):
        return fqn_to_param_info

    fqn_to_param_info: Dict[str, FSDPParamInfo] = {}
    # FlatParameter._fqns stores the local fqn, starting from the root of the
    # FSDP. Using _apply_to_modules() with model (may not be the FSDP root
    # module) allows us to construct the global fqn.
    return _apply_to_modules(
        model,
        module_fn,
        return_fn,
        [fqn for fqn, _ in model.named_parameters()],
        fqn_to_param_info,
    )


@dataclass
class StateInfo:
    tensors: Dict[str, _PosDimTensorInfo]
    scalar_tensors: Dict[str, torch.Tensor]
    non_tensors: Dict[str, Any]


@dataclass
class AllGatherInfo:
    tensors: List[torch.Tensor]
    numels: List[int]
    work: Optional[dist.Work]


def _all_gather_optim_state(
    fsdp_state: _FSDPState,
    optim_state: Dict[str, Any],
    group=None,
) -> Dict[str, Any]:
    """
    All-gathering state from all the ranks. This API is slow as it uses
    ``all_gather_object``. However, optim state_dict is not in the critical path.
    We can fuse the communication across differnt state if the performance
    becomes a problem.
    """
    # Allgather the scalar tensor state, non-tensor states and tensors metadata.
    processed_state = StateInfo({}, {}, {})
    for state_name, value in sorted_items(optim_state):
        if torch.is_tensor(value):
            if value.dim() == 0:
                # Ensure that `step` is on CPU.
                processed_state.scalar_tensors[state_name] = value.cpu()
            else:
                processed_state.tensors[state_name] = _PosDimTensorInfo(
                    value.shape, value.dtype
                )
        else:
            processed_state.non_tensors = value
    object_list: List[StateInfo] = [
        processed_state for _ in range(fsdp_state.world_size)
    ]
    dist.all_gather_object(object_list, processed_state, group=group)

    # Convert the gathered, pre-proccessed state of each rank to the original one.
    gathered_state: Dict[str, Any] = {}

    all_tensor_states = sorted(
        {n for state in object_list for n in state.tensors.keys()}
    )
    empty_ranks: Set[int] = set()
    for name in all_tensor_states:
        numels = []
        dtype = torch.float
        _empty_ranks: Set[int] = set()
        for rank, object_state in enumerate(object_list):
            numels.append(0)
            info = object_state.tensors.get(name, None)
            if info is not None:
                numels[-1] = info.shape.numel()
                dtype = info.dtype
            if numels[-1] == 0:
                _empty_ranks.add(rank)

        empty_func = functools.partial(
            torch.empty, dtype=dtype, device=fsdp_state.compute_device
        )
        if empty_ranks:
            assert empty_ranks == _empty_ranks
        empty_ranks = _empty_ranks
        local_state = optim_state.get(name, empty_func(0))
        local_state = local_state.to(fsdp_state.compute_device)
        tensors = [
            empty_func(numel) if rank != fsdp_state.rank else local_state
            for rank, numel in enumerate(numels)
        ]
        work = dist.all_gather(
            tensors, local_state, group=fsdp_state.process_group, async_op=True
        )
        gathered_state[name] = AllGatherInfo(tensors, numels, work)

    for rank, object_state in enumerate(object_list):
        if rank in empty_ranks:
            continue
        for name, non_tensor_value in object_state.non_tensors.items():
            curr_non_tensor_value = gathered_state.get(name, None)
            assert (
                curr_non_tensor_value is None
                or curr_non_tensor_value == non_tensor_value
            ), f"Different ranks have different values for {name}."
            gathered_state[name] = non_tensor_value

        for name, scalar_tensor_value in object_state.scalar_tensors.items():
            curr_scalar_tensor_value = gathered_state.get(name, None)
            assert curr_scalar_tensor_value is None or torch.equal(
                scalar_tensor_value, curr_scalar_tensor_value
            ), f"Different ranks have different values for {name}."
            gathered_state[name] = scalar_tensor_value

    for name, value in list(gathered_state.items()):
        if not isinstance(value, AllGatherInfo):
            continue
        assert value.work is not None
        value.work.wait()
        gathered_state[name] = torch.cat(
            [
                rank_tensor[:rank_numel]
                for rank_tensor, rank_numel in zip(value.tensors, value.numels)
                if rank_numel > 0
            ]
        )

    return gathered_state


def _gather_orig_param_state(
    fsdp_param_info: FSDPParamInfo,
    fqn: str,
    optim_state: Dict[str, Any],
    shard_state: bool,
    group=None,
) -> Dict[str, Any]:
    """
    Gather the optimizer state for the original parameter with the name ``fqn``.
    This API should only be used when ``use_orig_params`` is True.
    """
    fsdp_state = fsdp_param_info.state
    assert (
        fsdp_state._use_orig_params
    ), "_gather_orig_param_state only support use_orig_params=True case"
    flat_param = fsdp_param_info.flat_param
    param_idx = fsdp_param_info.param_indices[fqn]
    if (
        fsdp_state.world_size == 1
        or fsdp_state.sharding_strategy == ShardingStrategy.NO_SHARD
    ):
        return optim_state

    gathered_state = _all_gather_optim_state(fsdp_state, optim_state, group=group)

    # Unflatten state values.
    for state_name, value in list(gathered_state.items()):
        if not torch.is_tensor(value) or value.dim() == 0:
            continue

        value = value[: flat_param._numels[param_idx]].reshape(
            flat_param._shapes[param_idx]
        )
        if shard_state:
            assert fsdp_state.process_group is not None
            value = _ext_chunk_tensor(
                value,
                fsdp_state.rank,
                fsdp_state.world_size,
                torch.cuda.device_count(),
                fsdp_state.process_group,
            )
        value = value.cpu()
        gathered_state[state_name] = value
    return gathered_state


def _shard_orig_param_state(
    fsdp_param_info: FSDPParamInfo,
    fqn: str,
    optim_state: Dict[str, Any],
) -> Dict[str, Any]:
    """
    Shard the optimizer state for the original parameter with the name ``fqn``.
    This API should only be used when ``use_orig_params`` is True.
    """
    if not optim_state:
        return {}
    fsdp_state = fsdp_param_info.state
    flat_param = fsdp_param_info.flat_param
    param_idx = fsdp_param_info.param_indices[fqn]

    optim_state = _gather_state_dict(optim_state, fsdp_state.process_group)
    start, end = flat_param._shard_indices  # type: ignore[attr-defined]
    if not (start <= param_idx <= end and flat_param._shard_param_offsets):  # type: ignore[attr-defined]
        return {}
    param_start, param_end = flat_param._shard_param_offsets[param_idx - start]  # type: ignore[attr-defined]

    # Flatten and shard the state.
    new_optim_state: Dict[str, Any] = {}
    for state_name, value in optim_state.items():
        if torch.is_tensor(value) and value.dim() > 0:
            value = value.flatten()[param_start : param_end + 1]
        new_optim_state[state_name] = value
    return new_optim_state
