beam_search.py

from inspect import signature
from typing import Any, List, Callable, Tuple, Dict, cast, TypeVar, Optional
import copy
import warnings

from overrides import overrides
import torch

from allennlp.common import Registrable
from allennlp.common.checks import ConfigurationError
from allennlp.nn.util import min_value_of_dtype


StateType = Dict[str, torch.Tensor]
StepFunctionTypeWithTimestep = Callable[
    [torch.Tensor, StateType, int], Tuple[torch.Tensor, StateType]
]
StepFunctionTypeNoTimestep = Callable[[torch.Tensor, StateType], Tuple[torch.Tensor, StateType]]

StepFunctionType = TypeVar(
    "StepFunctionType", StepFunctionTypeWithTimestep, StepFunctionTypeNoTimestep
)
"""
The type of step function that can be passed to [`BeamSearch.search`](#search).

This can either be [`StepFunctionTypeWithTimestep`](#stepfunctiontypewithtimestep)
or [`StepFunctionTypeNoTimestep`](#stepfunctiontypenotimestep).
"""

ConstraintStateType = List[List[Dict[str, Any]]]


class Sampler(Registrable):
    """
    An abstract class that can be used to sample candidates (either nodes or beams)
    within `BeamSearch`.

    A `Sampler` just has three methods, `init_state()`, `sample_nodes()` and `sample_beams()`.

    `init_state()` takes three arguments:

    - a tensor of starting log probs with shape `(batch_size,, num_classes)`,
    - the batch size, an int,
    - and the number of classes, also an int.

    It returns a state dictionary with any state tensors needed for subsequent
    calls to `sample_nodes()` and `sample_beams()`.

    By default this method just returns an empty dictionary.

    Both `sample_nodes()` and `sample_beams()` should take three arguments:

    - tensor of normalized log probabilities with shape `(batch_size, num_examples)`,
    - an integer representing the number of samples to take for each example in the batch,
    - and a state dictionary which could contain any tensors needed for the `Sampler` to keep
      track of state.

    For `sample_nodes()`, `num_examples = num_classes`, but for `sample_beams`,
    `num_examples = beam_size * per_node_beam_size`.

    The return value should be a tuple containing:

    - a tensor of log probabilities of the sampled examples with shape `(batch_size, num_samples)`,
    - a tensor of indices of the sampled examples with shape `(batch_size, num_samples)`,
    - and the updated state dictionary.

    A default implementation of `sample_beams` is provided, which just deterministically
    picks the `k` examples with highest log probability.
    """

    default_implementation = "deterministic"

    def init_state(
        self, start_class_log_probabilities: torch.Tensor, batch_size: int, num_classes: int
    ) -> StateType:
        return {}

    def sample_nodes(
        self, log_probs: torch.Tensor, per_node_beam_size: int, state: StateType
    ) -> Tuple[torch.Tensor, torch.Tensor, StateType]:
        raise NotImplementedError

    def sample_beams(
        self, log_probs: torch.Tensor, beam_size: int, state: StateType
    ) -> Tuple[torch.Tensor, torch.Tensor, StateType]:
        selected_log_probs, selected_indices = torch.topk(log_probs, beam_size, dim=-1)
        return selected_log_probs, selected_indices, {}


@Sampler.register("deterministic")
class DeterministicSampler(Sampler):
    """
    A `Sampler` that just deterministically returns the `k` nodes or beams with highest
    log probability.
    """

    @overrides
    def sample_nodes(
        self, log_probs: torch.Tensor, per_node_beam_size: int, state: StateType
    ) -> Tuple[torch.Tensor, torch.Tensor, StateType]:
        selected_log_probs, selected_indices = torch.topk(log_probs, per_node_beam_size, dim=-1)
        return selected_log_probs, selected_indices, {}


@Sampler.register("multinomial")
class MultinomialSampler(Sampler):
    """
    A `Sampler` which samples nodes from the given multinomial distribution. Beams are sampled
    in the default, non-deterministic way.

    # Parameters

    temperature : `float`, optional (default = `1.0`)
        A `temperature` below 1.0 produces a sharper probability distribution and a `temperature` above 1.0
        produces a flatter probability distribution.
    with_replacement : `bool`, optional (default = `False`)
        Whether to sample with replacement.
    """

    def __init__(
        self,
        temperature: float = 1.0,
        with_replacement: bool = False,
    ) -> None:
        self.temperature = temperature
        self.with_replacement = with_replacement

    @overrides
    def sample_nodes(
        self, log_probs: torch.Tensor, per_node_beam_size: int, state: StateType
    ) -> Tuple[torch.Tensor, torch.Tensor, StateType]:
        if self.temperature != 1.0:
            _probabilities = torch.nn.functional.softmax(log_probs / self.temperature, dim=-1)
        else:
            _probabilities = log_probs.exp()

        selected_indices = torch.multinomial(
            _probabilities, per_node_beam_size, replacement=self.with_replacement
        )

        return torch.gather(log_probs, 1, selected_indices), selected_indices, state


@Sampler.register("top-k")
class TopKSampler(Sampler):
    """
    A `Sampler` which redistributes the probability mass function for nodes among the
    top `k` choices, then samples from that subset after re-normalizing the probabilities.

    Beams are sampled in the default, deterministic way.

    # Parameters

    k : `int`, optional (default = `1`)
        The number of top choices to be selected from.
    temperature : `float`, optional (default = `1.0`)
        A `temperature` below 1.0 produces a sharper probability distribution and a `temperature`
        above 1.0 produces a flatter probability distribution.
    with_replacement: `bool`, optional, (default = `False`)
        If set to `True`, samples will be selected with replacement from the top k choices.
    """

    def __init__(
        self,
        k: int = 1,
        temperature: float = 1.0,
        with_replacement: bool = False,
    ):
        self.k = k
        self.temperature = temperature or 1.0
        self.with_replacement = with_replacement

    @overrides
    def sample_nodes(
        self, log_probs: torch.Tensor, per_node_beam_size: int, state: StateType
    ) -> Tuple[torch.Tensor, torch.Tensor, StateType]:
        if not per_node_beam_size <= self.k <= log_probs.size()[1]:
            raise ValueError(
                "k must be a postive integer no less than per_node_beam_size and no greater than vocabulary size"
            )

        # shape (both): (batch_size, k)
        top_k_log_probs, top_k_indices = log_probs.topk(self.k, dim=-1)

        # Apply temperature if necessary.
        # shape: (batch_size, k)
        if self.temperature != 1.0:
            top_k_log_probs = top_k_log_probs / self.temperature

        # Re-normalize the subset.
        # shape: (batch_size, k)
        normalized_top_k_probs = torch.nn.functional.softmax(top_k_log_probs, dim=-1)

        # Sample from the re-normalized subset.
        # NOTE: These indices are not indices into `log_probs`, they are indices into `top_k_log_probs`.
        # shape: (batch_size, per_node_beam_size)
        sampled_indices = torch.multinomial(
            normalized_top_k_probs, per_node_beam_size, replacement=self.with_replacement
        )

        # Convert `sampled_indices` back to indices in the original `log_probs` tensor.
        # shape: (batch_size, per_node_beam_size)
        indices = top_k_indices.gather(-1, sampled_indices)

        return log_probs.gather(1, indices), indices, state


@Sampler.register("top-p")
class TopPSampler(Sampler):
    """
    A `Sampler` which redistributes the probability mass function for nodes among
    the top choices with a cumulative probability of at least `p`, then samples from that subset
    after re-normalizing the probabilities.

    Beams are sampled in the default, deterministic way.

    # Parameters

    p : `float`, optional (default = `0.9`)
        The cumulative probability cutoff threshold. A higher value of `p` will result in more possible
        examples to sample from. If `with_replacement` is `False` and the number of possible samples is
        insufficient to sample without replacement from when calling `sample_nodes`, then the top
        `per_node_beam_size` examples will be chosen.
    temperature : `float`, optional (default = `1.0`)
        A `temperature` below 1.0 produces a sharper probability distribution and a `temperature`
        above 1.0 produces a flatter probability distribution.
    with_replacement : `bool`, optional, (default = `False`)
        If set to `True`, samples will be selected with replacement from the top choices.
    """

    def __init__(
        self,
        p: float = 0.9,
        temperature: float = 1.0,
        with_replacement: bool = False,
    ):
        if p < 0.0 or p > 1.0:
            raise ValueError("p must be a positive float no greater than 1.0")
        self.p = p
        self.temperature = temperature or 1.0
        self.with_replacement = with_replacement

    @overrides
    def sample_nodes(
        self, log_probs: torch.Tensor, per_node_beam_size: int, state: StateType
    ) -> Tuple[torch.Tensor, torch.Tensor, StateType]:
        if not per_node_beam_size <= log_probs.size()[1]:
            raise ValueError("per_node_beam_size cannot be greater than vocabulary size")

        # First apply temperature coefficient:
        if self.temperature != 1.0:
            _log_probs = torch.nn.functional.log_softmax(log_probs / self.temperature, dim=-1)
        else:
            _log_probs = log_probs

        # Sort the probabilities in descending order to then find cumulative sum
        log_probs_descending, sorting_indices = torch.sort(_log_probs, descending=True)

        # shape: (batch_size, num_classes)
        probabilities_descending = log_probs_descending.exp()
        probabilities_summed = torch.cumsum(probabilities_descending, dim=-1)

        # Create a mask for filtering out probabilities that don't make the top `p`.
        # shape: (batch_size, num_classes)
        exclusion_mask = probabilities_summed >= self.p

        # We want to include the first index where probabilities_summed >= p, so we shift over one.
        exclusion_mask[..., 1:] = exclusion_mask[..., :-1].clone()
        exclusion_mask[..., 0] = False

        # Make sure there's at least `per_node_beam_size` options to be selected.
        if not self.with_replacement:
            exclusion_mask[..., :per_node_beam_size] = False

        log_probs_descending[exclusion_mask] = min_value_of_dtype(log_probs.dtype)

        # Now re-normalized the included log probs.
        # shape: (batch_size, num_classes)
        filtered_probabilities = torch.nn.functional.softmax(log_probs_descending, dim=-1)

        # Sample from the re-normalized subset.
        # NOTE: These indices are not indices into `log_probs`, they are indices into `log_probs_descending`.
        # shape: (batch_size, per_node_beam_size)
        sampled_indices = torch.multinomial(
            filtered_probabilities, per_node_beam_size, replacement=self.with_replacement
        )

        # Convert `sampled_indices` back to indices in the original `log_probs` tensor.
        # shape: (batch_size, per_node_beam_size)
        selected_indices = sorting_indices.gather(-1, sampled_indices)

        # Return (selected log probabilities, selected classes)
        # shape: (len(log_probs),1) , (len(log_probs), 1)
        return torch.gather(log_probs, 1, selected_indices), selected_indices, state


@Sampler.register("gumbel")
class GumbelSampler(Sampler):
    """
    A `Sampler` which uses the Gumbel-Top-K trick to sample without replacement. See
    [*Stochastic Beams and Where to Find Them: The Gumbel-Top-k Trick for Sampling
    Sequences Without Replacement*, W Kool, H Van Hoof and M Welling, 2010]
    (https://api.semanticscholar.org/CorpusID:76662039).

    # Parameters

    temperature : `float`, optional (default = `1.0`)
        A `temperature` below 1.0 produces a sharper probability distribution and a `temperature`
        above 1.0 produces a flatter probability distribution.
    """

    def __init__(self, temperature: float = 1.0):
        self.temperature = temperature

    @overrides
    def init_state(
        self, start_class_log_probabilities: torch.Tensor, batch_size: int, num_classes: int
    ) -> StateType:
        # shape: (batch_size, num_classes)
        zeros = start_class_log_probabilities.new_zeros((batch_size, num_classes))

        # shape: (batch_size, num_classes)
        G_phi_S = self.gumbel_with_max(start_class_log_probabilities, zeros)

        return {"G_phi_S": G_phi_S}

    @overrides
    def sample_nodes(
        self,
        log_probs: torch.Tensor,
        per_node_beam_size: int,
        state: StateType,
    ) -> Tuple[torch.Tensor, torch.Tensor, StateType]:
        # First apply temperature coefficient:
        # shape: (batch_size * beam_size, num_classes)
        if self.temperature != 1.0:
            _log_probs = torch.nn.functional.log_softmax(log_probs / self.temperature, dim=-1)
        else:
            _log_probs = log_probs

        # shape: (group_size,)
        phi_S = state["phi_S"]

        # shape: (group_size, num_classes)
        phi_S = phi_S.unsqueeze(-1).expand_as(_log_probs)

        # shape: (group_size, num_classes)
        phi_S_new = phi_S + _log_probs

        # shape: (group_size, 1)
        G_phi_S = state["G_phi_S"].unsqueeze(-1)

        # shape: (group_size, num_classes)
        G_phi_S_new = self.gumbel_with_max(phi_S_new, G_phi_S)

        # Replace NaNs with very negative number.
        # shape: (group_size, num_classes)
        #  G_phi_S_new[G_phi_S_new.isnan()] = min_value_of_dtype(G_phi_S_new.dtype)

        # shape (both): (group_size, per_node_beam_size)
        top_G_phi_S_new, top_indices = torch.topk(G_phi_S_new, per_node_beam_size, dim=-1)

        # shape: (group_size, per_node_beam_size)
        top_log_probs = log_probs.gather(1, top_indices)

        return top_log_probs, top_indices, {"G_phi_S": top_G_phi_S_new}

    @overrides
    def sample_beams(
        self,
        log_probs: torch.Tensor,
        beam_size: int,
        state: StateType,
    ) -> Tuple[torch.Tensor, torch.Tensor, StateType]:
        """
        Returns the beams with the highest perturbed log probabilities.
        """
        # shape (log_probs): (batch_size, beam_size * per_node_beam_size)

        batch_size = log_probs.size()[0]

        # shape: (batch_size * beam_size, per_node_beam_size)
        G_phi_S = state["G_phi_S"]

        # shape: (batch_size, beam_size * per_node_beam_size)
        G_phi_S = G_phi_S.reshape_as(log_probs)

        # shape (both): (batch_size, beam_size)
        G_phi_S_new, selected_indices = torch.topk(G_phi_S, beam_size, dim=-1)

        # shape: (batch_size, beam_size)
        selected_log_probs = log_probs.gather(1, selected_indices)

        # Now sort the selected beams by their true log prob.
        # shape (all): (batch_size, beam_size)
        selected_log_probs, sort_indices = selected_log_probs.sort(dim=-1, descending=True)
        selected_indices = selected_indices.gather(1, sort_indices)
        G_phi_S_new = G_phi_S_new.gather(1, sort_indices)

        # shape: (batch_size * beam_size,)
        G_phi_S_new = G_phi_S_new.reshape(batch_size * beam_size)

        # shape: (batch_size * beam_size,)
        phi_S = selected_log_probs.reshape(batch_size * beam_size)

        return selected_log_probs, selected_indices, {"G_phi_S": G_phi_S_new, "phi_S": phi_S}

    def gumbel(self, phi) -> torch.Tensor:
        """
        Sample `Gumbel(phi)`.

        `phi` should have shape `(batch_size, num_classes)`.
        """
        return -torch.log(-torch.log(torch.rand_like(phi))) + phi

    def gumbel_with_max(self, phi, T) -> torch.Tensor:
        """
        Sample `Gumbel(phi)` conditioned on the maximum value being equal to `T`.

        `phi` should have shape `(batch_size, num_classes)` and `T` should have
        shape `(batch_size, 1)`.
        """
        # Shape: (batch_size, num_classes)
        G_phi = self.gumbel(phi)

        # Now we find the maximum from these samples.
        # Shape: (batch_size, )
        Z, _ = G_phi.max(dim=-1)

        # Shape: (batch_size, num_classes)
        v = T - G_phi + torch.log1p(-torch.exp(G_phi - Z.unsqueeze(-1)))

        # Shape: (batch_size, num_classes)
        return T - torch.nn.functional.relu(v) - torch.log1p(torch.exp(-v.abs()))


class FinalSequenceScorer(Registrable):
    """
    An abstract class that can be used to score the final generated sequences found
    by beam search. Given the predicted sequences and the corresponding log probabilities of
    those sequences, the class calculates and returns the final score of the sequences.

    The default implementation scores the sequences using the sum of the log probabilities of
    the sequence, which is passed as input.
    """

    default_implementation = "sequence-log-prob"

    def score(
        self, predictions: torch.Tensor, log_probabilities: torch.Tensor, end_index: int
    ) -> torch.Tensor:
        """
        Score the final predictions found by beam search.

        # Parameters

        predictions : `torch.Tensor`
            A tensor containing the initial predictions with shape `(batch_size, beam_size, max_steps)`.

        log_probabilities : `torch.Tensor`
            A tensor containing the log probabilities of the sequence, defined as the sum
            of the log probabilities per token, with shape `(batch_size, beam_size)`.

        end_index : `int`
            The index of the end symbol.

        # Returns

        `torch.Tensor`
            A tensor of the final sequence scores of shape `(batch_size, beam_size)`.
        """
        raise NotImplementedError


@FinalSequenceScorer.register("sequence-log-prob")
class SequenceLogProbabilityScorer(FinalSequenceScorer):
    """
    A `FinalSequenceScorer` which scores the sequences by the sum of the log probabilities
    across the sequence's tokens.
    """

    @overrides
    def score(
        self, predictions: torch.Tensor, log_probabilities: torch.Tensor, end_index: int
    ) -> torch.Tensor:
        # The sum of the sequence log probabilities is the input parameter, so just
        # return it.
        return log_probabilities


@FinalSequenceScorer.register("length-normalized-sequence-log-prob")
class LengthNormalizedSequenceLogProbabilityScorer(FinalSequenceScorer):
    """
    A `FinalSequenceScorer` which scores the sequences by the average log probability of the
    tokens in the sequence. It optionally includes a length penalty which promotes
    or demotes sequences based on their lengths. The final score for a sequence will
    be `(sequence_log_probability) / (sequence_length ** length_penalty)`. The sequence length
    here includes the end token.

    # Parameters

    length_penalty : `float`, optional (default = `1.0`)
        The length penalty to use. A value of 1.0 means no length penalty is used.
        A value > 1.0 favors longer sequences, and < 1.0 favors shorter sequences.
    """

    def __init__(self, length_penalty: float = 1.0):
        super().__init__()
        self.length_penalty = length_penalty

    @overrides
    def score(
        self, predictions: torch.Tensor, log_probabilities: torch.Tensor, end_index: int
    ) -> torch.Tensor:
        # shape: (batch_size, beam_size)
        lengths = (predictions != end_index).long().sum(dim=2)

        # If the sequence ended during beam search, the `log_probabilities` will include
        # the transition to the end token. Therefore, in such situations, `lengths` is
        # actually off by 1. This corrects for that.
        # shape: (batch_size, beam_size)
        is_end_token = predictions[:, :, -1] == end_index
        lengths += is_end_token.long()

        # shape: (batch_size, beam_size)
        average_log_probs = log_probabilities / (lengths ** self.length_penalty)
        return average_log_probs


class Constraint(Registrable):
    """
    An abstract class that can be used to enforce constraints on the output predictions
    by manipulating the class log probabilities during beam search.

    A `Constraint` just has three methods that need to be implemented by subclasses:
    `init_state()`, `apply()` and `_update_state()`.

    `init_state()` takes one argument:

    - the batch size, an int

    It returns a constraint state, which is a nested list of dictionaries, with any state needed for subsequent
    calls to `apply()` and `update_state()`. The length of the outer list should be equal to `batch_size`.
    Each inner list should be of length 1.

    `apply()` takes two arguments:

    - the constraint state, which is a nested list of dictionaries. The length of the outer list is `batch_size`
    and the length of each inner list is `beam_size` except on the first time `apply()` is called when it is 1.
    - `class_log_probabilities`, a tensor of shape `(batch_size, beam_size, num_classes)` that contains the
    log probabilities for the classes during search. The first time `apply()` is called, `beam_size = 1`.

    The `apply()` method should return new `class_log_probabilities` that enforce the constraint
    for this step of beam search. For instance, it may prevent a specific class from being selected by setting
    the corresponding log probability to a negligible value such as `float("-inf")` or
    `min_value_of_dtype(class_log_probabilities.dtype)`.

    `_update_state()` takes two arguments:

    - the copied parent constraint state, which is a nested list of dictionaries. `state[i][j]` contains the
    copied state for the parent of `last_prediction[i, j]`. It is unique to that batch and beam, so it can be
    directly edited in-place without affecting the others.
    - last_prediction, a tensor of shape `(batch_size, beam_size)` containing the predictions from the last
    step of beam search.

    The `_update_state()` function should return a new constraint state, a nested list of dictionaries of
    length `batch_size` and inner list of length `beam_size`, one for each of the predictions in `last_prediction`.

    """

    def init_state(
        self,
        batch_size: int,
    ) -> ConstraintStateType:
        raise NotImplementedError

    def apply(
        self,
        state: ConstraintStateType,
        class_log_probabilities: torch.Tensor,
    ) -> torch.Tensor:
        raise NotImplementedError

    @staticmethod
    def _copy_state(
        state: ConstraintStateType,
        batch_size: int,
        beam_size: int,
        last_backpointer: Optional[torch.Tensor] = None,
    ) -> ConstraintStateType:
        """
        Copies the `state` . This method copies the data in `state` using `copy.deepcopy()`. If this
        is not appropriate for your constraint, you will need to implement the copying yourself.
        """
        new_state = []
        for i in range(batch_size):
            batch_state = []
            for j in range(beam_size):
                if last_backpointer is None:
                    # This is the first prediction, so the backpointer is 0
                    backpointer = 0
                else:
                    backpointer = last_backpointer[i, j].item()
                batch_state.append(copy.deepcopy(state[i][backpointer]))
            new_state.append(batch_state)
        return new_state

    def update_state(
        self,
        state: ConstraintStateType,
        last_prediction: torch.Tensor,
        last_backpointer: Optional[torch.Tensor] = None,
    ) -> ConstraintStateType:
        batch_size, beam_size = last_prediction.size()
        new_state = self._copy_state(state, batch_size, beam_size, last_backpointer)
        return self._update_state(new_state, last_prediction)

    def _update_state(
        self,
        state: ConstraintStateType,
        last_prediction: torch.Tensor,
    ) -> ConstraintStateType:
        raise NotImplementedError


@Constraint.register("repeated-ngram-blocking")
class RepeatedNGramBlockingConstraint(Constraint):
    def __init__(self, ngram_size: int) -> None:
        super().__init__()
        self.ngram_size = ngram_size

    @overrides
    def init_state(
        self,
        batch_size: int,
    ) -> ConstraintStateType:
        return [[{"seen_ngrams": {}, "current_prefix": []}] for _ in range(batch_size)]

    @overrides
    def apply(
        self,
        state: ConstraintStateType,
        class_log_probabilities: torch.Tensor,
    ) -> torch.Tensor:
        for i, batch in enumerate(state):
            for j, beam in enumerate(batch):
                current_prefix = tuple(beam["current_prefix"])
                seen_ngrams = beam["seen_ngrams"]
                try:
                    disallowed_indices = seen_ngrams[current_prefix]
                    class_log_probabilities[i, j, disallowed_indices] = min_value_of_dtype(
                        class_log_probabilities.dtype
                    )
                except KeyError:
                    # We have not seen this prefix before, so there is no index
                    # that needs to be blocked
                    pass
        return class_log_probabilities

    @overrides
    def _update_state(
        self,
        state: ConstraintStateType,
        last_prediction: torch.Tensor,
    ) -> ConstraintStateType:
        for i, batch in enumerate(state):
            for j, beam in enumerate(batch):
                prediction = last_prediction[i, j].item()
                prefix = beam["current_prefix"]
                seen_ngrams = beam["seen_ngrams"]

                if len(prefix) == self.ngram_size - 1:
                    # This is a new ngram that we have to remember
                    if tuple(prefix) not in seen_ngrams:
                        seen_ngrams[tuple(prefix)] = []
                    seen_ngrams[tuple(prefix)].append(prediction)

                # Create the new prefix, removing the oldest index if the prefix
                # is too long
                prefix.append(prediction)
                if len(prefix) == self.ngram_size:
                    prefix.pop(0)
        return state


class BeamSearch(Registrable):
    """
    Implements the beam search algorithm for decoding the most likely sequences.

    # Parameters

    end_index : `int`
        The index of the "stop" or "end" token in the target vocabulary.

    max_steps : `int`, optional (default = `50`)
        The maximum number of decoding steps to take, i.e. the maximum length
        of the predicted sequences.

    beam_size : `int`, optional (default = `10`)
        The width of the beam used.

    per_node_beam_size : `int`, optional (default = `beam_size`)
        The maximum number of candidates to consider per node, at each step in the search.
        If not given, this just defaults to `beam_size`. Setting this parameter
        to a number smaller than `beam_size` may give better results, as it can introduce
        more diversity into the search. See
        [*Beam Search Strategies for Neural Machine Translation*, Freitag and Al-Onaizan, 2017]
        (https://api.semanticscholar.org/CorpusID:2229477).

    sampler : `Sampler`, optional (default = `None`)
        An optional `Sampler` which is used to pick next candidate nodes and beams.
        If not specified, `DeterministicSampler` will be used, which just takes the
        `per_node_beam_size` most likely nodes and the `beam_size` most likely beams.

        Using the [`GumbelSampler`](#gumbelsampler), on the other hand, will give you
        [Stochastic Beam Search](https://api.semanticscholar.org/CorpusID:76662039).

    min_steps : `int`, optional (default = `None`)
        The minimum number of decoding steps to take, i.e. the minimum length of
        the predicted sequences. This does not include the start or end tokens. If `None`,
        no minimum is enforced.

    final_sequence_scorer : `FinalSequenceScorer`, optional (default = `None`)
        An optional `FinalSequenceScorer` which is used to score the final generated sequences.
        The output from this module is what is returned by the `search` method. If not
        specified, `SequenceLogProbabilityScorer` will be used, which scores the sequences
        by the sum of the token log probabilities.

    constraints: `List[Constraint]`, optional (default = `None`)
        An optional list of `Constraint`s which should be applied during beam search. If not
        provided, no constraints will be enforced.
    """

    default_implementation = "beam_search"

    def __init__(
        self,
        end_index: int,
        max_steps: int = 50,
        beam_size: int = 10,
        per_node_beam_size: int = None,
        sampler: Sampler = None,
        min_steps: Optional[int] = None,
        final_sequence_scorer: FinalSequenceScorer = None,
        constraints: Optional[List[Constraint]] = None,
    ) -> None:
        if not max_steps > 0:
            raise ValueError("max_steps must be positive")
        if not beam_size > 0:
            raise ValueError("beam_size must be positive")
        if per_node_beam_size is not None and not per_node_beam_size > 0:
            raise ValueError("per_node_beam_size must be positive")
        if min_steps is not None:
            if not min_steps >= 0:
                raise ValueError("min_steps must be non-negative")
            if not min_steps <= max_steps:
                raise ValueError("min_steps must be less than or equal to max_steps")

        self._end_index = end_index
        self.max_steps = max_steps
        self.beam_size = beam_size
        self.per_node_beam_size = per_node_beam_size or beam_size
        self.sampler = sampler or DeterministicSampler()
        self.min_steps = min_steps or 0
        self.final_sequence_scorer = final_sequence_scorer or SequenceLogProbabilityScorer()
        self.constraints = constraints or []

    @staticmethod
    def _reconstruct_sequences(predictions, backpointers):
        # Reconstruct the sequences.
        # shape: [(batch_size, beam_size, 1)]
        reconstructed_predictions = [predictions[-1].unsqueeze(2)]

        if not backpointers:
            return reconstructed_predictions

        # shape: (batch_size, beam_size)
        cur_backpointers = backpointers[-1]

        for timestep in range(len(predictions) - 2, 0, -1):
            # shape: (batch_size, beam_size, 1)
            cur_preds = predictions[timestep].gather(1, cur_backpointers).unsqueeze(2)

            reconstructed_predictions.append(cur_preds)

            # shape: (batch_size, beam_size)
            cur_backpointers = backpointers[timestep - 1].gather(1, cur_backpointers)

        # shape: (batch_size, beam_size, 1)
        final_preds = predictions[0].gather(1, cur_backpointers).unsqueeze(2)

        reconstructed_predictions.append(final_preds)

        return reconstructed_predictions

    @torch.no_grad()
    def search(
        self,
        start_predictions: torch.Tensor,
        start_state: StateType,
        step: StepFunctionType,
    ) -> Tuple[torch.Tensor, torch.Tensor]:
        """
        Given a starting state and a step function, apply beam search to find the
        most likely target sequences.

        !!! Note
            If your step function returns `-inf` for some log probabilities
            (like if you're using a masked log-softmax) then some of the "best"
            sequences returned may also have `-inf` log probability. Specifically
            this happens when the beam size is smaller than the number of actions
            with finite log probability (non-zero probability) returned by the step function.
            Therefore if you're using a mask you may want to check the results from `search`
            and potentially discard sequences with non-finite log probability.

        # Parameters

        start_predictions : `torch.Tensor`
            A tensor containing the initial predictions with shape `(batch_size,)`.
            Usually the initial predictions are just the index of the "start" token
            in the target vocabulary.

        start_state : `StateType`
            The initial state passed to the `step` function. Each value of the state dict
            should be a tensor of shape `(batch_size, *)`, where `*` means any other
            number of dimensions.

        step : `StepFunctionType`
            A function that is responsible for computing the next most likely tokens,
            given the current state and the predictions from the last time step.
            The function should accept two or three arguments:

            - a tensor of shape `(group_size,)` representing the index of the predicted
            tokens from the last time step,
            - the current state, a `StateType`, and
            - optionally, the timestep, an `int`.

            The `group_size` will be `batch_size * beam_size`, except in the initial
            step, for which it will just be `batch_size`.

            The function is expected to return a tuple, where the first element
            is a tensor of shape `(group_size, target_vocab_size)` containing
            the log probabilities of the tokens for the next step, and the second
            element is the updated state. The tensor in the state should have shape
            `(group_size, *)`, where `*` means any other number of dimensions.

        # Returns

        `Tuple[torch.Tensor, torch.Tensor]`
            Tuple of `(predictions, final_scores)`, where `predictions`
            has shape `(batch_size, beam_size, max_steps)` and `final_scores`
            has shape `(batch_size, beam_size)`.
        """
        step_signature = signature(step)
        if len(step_signature.parameters) < 3:
            # If the step function we're given does not take the time step argument, wrap it
            # in one that does.
            old_step = cast(StepFunctionTypeNoTimestep, step)

            def new_step(
                last_predictions: torch.Tensor, state: Dict[str, torch.Tensor], time_step: int
            ):
                return old_step(last_predictions, state)

            return self._search(start_predictions, start_state, new_step)
        else:
            return self._search(
                start_predictions, start_state, cast(StepFunctionTypeWithTimestep, step)
            )

    def _search(
        self,
        start_predictions: torch.Tensor,
        start_state: StateType,
        step: StepFunctionTypeWithTimestep,
    ) -> Tuple[torch.Tensor, torch.Tensor]:
        batch_size = start_predictions.size()[0]

        # List of (batch_size, beam_size) tensors. One for each time step. Does not
        # include the start symbols, which are implicit.
        predictions: List[torch.Tensor] = []

        # List of (batch_size, beam_size) tensors. One for each time step. None for
        # the first.  Stores the index n for the parent prediction, i.e.
        # predictions[t-1][i][n], that it came from.
        backpointers: List[torch.Tensor] = []

        constraint_states = [constraint.init_state(batch_size) for constraint in self.constraints]

        # Calculate the first timestep. This is done outside the main loop
        # because we are going from a single decoder input (the output from the
        # encoder) to the top `beam_size` decoder outputs. On the other hand,
        # within the main loop we are going from the `beam_size` elements of the
        # beam to `beam_size`^2 candidates from which we will select the top
        # `beam_size` elements for the next iteration.
        # shape: (batch_size, num_classes)
        start_class_log_probabilities, state = step(start_predictions, start_state, 0)

        num_classes = start_class_log_probabilities.size()[1]

        # Make sure `per_node_beam_size` is not larger than `num_classes`.
        if self.per_node_beam_size > num_classes:
            raise ConfigurationError(
                f"Target vocab size ({num_classes:d}) too small "
                f"relative to per_node_beam_size ({self.per_node_beam_size:d}).\n"
                f"Please decrease beam_size or per_node_beam_size."
            )

        sampler_state = self.sampler.init_state(
            start_class_log_probabilities, batch_size, num_classes
        )

        # Apply all constraints.
        if self.constraints:
            # shape: (batch_size, 1, num_classes)
            expanded_start_class_log_probabilities = start_class_log_probabilities.unsqueeze(1)
            for constraint, constraint_state in zip(self.constraints, constraint_states):
                expanded_start_class_log_probabilities = constraint.apply(
                    constraint_state, expanded_start_class_log_probabilities
                )
            start_class_log_probabilities = expanded_start_class_log_probabilities.squeeze(1)

        # Prevent selecting the end symbol if there is any min_steps constraint
        if self.min_steps >= 1:
            start_class_log_probabilities[:, self._end_index] = min_value_of_dtype(
                start_class_log_probabilities.dtype
            )

        # Get the initial predicted classed and their log probabilities.
        # shape: (batch_size, beam_size), (batch_size, beam_size)
        (
            start_top_log_probabilities,
            start_predicted_classes,
            sampler_state,
        ) = self.sampler.sample_beams(start_class_log_probabilities, self.beam_size, sampler_state)

        if self.beam_size == 1 and (start_predicted_classes == self._end_index).all():
            warnings.warn(
                "Empty sequences predicted. You may want to increase the beam size or ensure "
                "your step function is working properly.",
                RuntimeWarning,
            )
            return start_predicted_classes.unsqueeze(-1), start_top_log_probabilities

        # The log probabilities for the last time step.
        # shape: (batch_size, beam_size)
        last_log_probabilities = start_top_log_probabilities

        # shape: [(batch_size, beam_size)]
        predictions.append(start_predicted_classes)

        # Log probability tensor that mandates that the end token is selected.
        # shape: (batch_size * beam_size, num_classes)
        log_probs_after_end = start_class_log_probabilities.new_full(
            (batch_size * self.beam_size, num_classes),
            min_value_of_dtype(start_class_log_probabilities.dtype),
        )
        log_probs_after_end[:, self._end_index] = 0.0

        # Set the same state for each element in the beam.
        self._update_initial_state(state, batch_size)

        for i, constraint in enumerate(self.constraints):
            constraint_states[i] = constraint.update_state(
                constraint_states[i], start_predicted_classes
            )

        for timestep in range(self.max_steps - 1):
            # shape: (batch_size * beam_size,)
            last_predictions = predictions[-1].reshape(batch_size * self.beam_size)

            # If every predicted token from the last step is `self._end_index`,
            # then we can stop early.
            if (last_predictions == self._end_index).all():
                break
            # Take a step. This get the predicted log probs of the next classes
            # and updates the state.
            # shape: (batch_size * beam_size, num_classes)
            class_log_probabilities, state = step(last_predictions, state, timestep + 1)

            # Apply all constraints.
            if self.constraints:
                # shape: (batch_size, beam_size, num_classes)
                reshaped_class_log_probabilities = class_log_probabilities.view(
                    batch_size, self.beam_size, -1
                )
                for constraint, constraint_state in zip(self.constraints, constraint_states):
                    reshaped_class_log_probabilities = constraint.apply(
                        constraint_state, reshaped_class_log_probabilities
                    )
                # shape: (batch_size * beam_size, num_classes)
                class_log_probabilities = reshaped_class_log_probabilities.view(
                    batch_size * self.beam_size, -1
                )

            # The `timestep`-th iteration of the for loop is generating the `timestep + 2`-th token
            # of the sequence (because `timestep` is 0-indexed and we generated the first token
            # before the for loop). Here we block the end index if the search is not allowed to
            # terminate on this iteration.
            if timestep + 2 <= self.min_steps:
                class_log_probabilities[:, self._end_index] = min_value_of_dtype(
                    class_log_probabilities.dtype
                )

            # shape: (batch_size * beam_size, num_classes)
            last_predictions_expanded = last_predictions.unsqueeze(-1).expand(
                batch_size * self.beam_size, num_classes
            )

            # Here we are finding any beams where we predicted the end token in
            # the previous timestep and replacing the distribution with a
            # one-hot distribution, forcing the beam to predict the end token
            # this timestep as well.
            # shape: (batch_size * beam_size, num_classes)
            cleaned_log_probabilities = torch.where(
                last_predictions_expanded == self._end_index,
                log_probs_after_end,
                class_log_probabilities,
            )

            # shape (both): (batch_size * beam_size, per_node_beam_size)
            top_log_probabilities, predicted_classes, sampler_state = self.sampler.sample_nodes(
                cleaned_log_probabilities, self.per_node_beam_size, sampler_state
            )

            # Here we expand the last log probabilities to (batch_size * beam_size, per_node_beam_size)
            # so that we can add them to the current log probs for this timestep.
            # This lets us maintain the log probability of each element on the beam.
            # shape: (batch_size * beam_size, per_node_beam_size)
            expanded_last_log_probabilities = (
                last_log_probabilities.unsqueeze(2)
                .expand(batch_size, self.beam_size, self.per_node_beam_size)
                .reshape(batch_size * self.beam_size, self.per_node_beam_size)
            )

            # shape: (batch_size * beam_size, per_node_beam_size)
            summed_top_log_probabilities = top_log_probabilities + expanded_last_log_probabilities

            # shape: (batch_size, beam_size * per_node_beam_size)
            reshaped_summed = summed_top_log_probabilities.reshape(
                batch_size, self.beam_size * self.per_node_beam_size
            )

            # shape: (batch_size, beam_size * per_node_beam_size)
            reshaped_predicted_classes = predicted_classes.reshape(
                batch_size, self.beam_size * self.per_node_beam_size
            )

            # Keep only the top `beam_size` beam indices.
            # shape (both): (batch_size, beam_size)
            (
                restricted_beam_log_probs,
                restricted_beam_indices,
                sampler_state,
            ) = self.sampler.sample_beams(reshaped_summed, self.beam_size, sampler_state)

            # Use the beam indices to extract the corresponding classes.
            # shape: (batch_size, beam_size)
            restricted_predicted_classes = reshaped_predicted_classes.gather(
                1, restricted_beam_indices
            )

            predictions.append(restricted_predicted_classes)

            # shape: (batch_size, beam_size)
            last_log_probabilities = restricted_beam_log_probs

            # The beam indices come from a `beam_size * per_node_beam_size` dimension where the
            # indices with a common ancestor are grouped together. Hence
            # dividing by per_node_beam_size gives the ancestor. (Note that this is integer
            # division as the tensor is a LongTensor.)
            # shape: (batch_size, beam_size)
            backpointer = restricted_beam_indices // self.per_node_beam_size
            backpointers.append(backpointer)

            # Keep only the pieces of the state tensors corresponding to the
            # ancestors created this iteration.
            self._update_state(state, backpointer)

            for i, constraint in enumerate(self.constraints):
                constraint_states[i] = constraint.update_state(
                    constraint_states[i], restricted_predicted_classes
                )

        # Warn about "-inf" log probabilities if not using any constraints (negligible
        # log probabilities are expected when using constraints).
        if not self.constraints and (
            not torch.isfinite(last_log_probabilities).all()
            or (last_log_probabilities == min_value_of_dtype(last_log_probabilities.dtype)).any()
        ):
            warnings.warn(
                "Negligible log probabilities encountered ('-inf' or equivalent). "
                "Some final sequences may not make sense. "
                "This can happen when the beam size is larger than the number of valid (non-zero "
                "probability) transitions that the step function produces.",
                RuntimeWarning,
            )

        reconstructed_predictions = self._reconstruct_sequences(predictions, backpointers)

        # shape: (batch_size, beam_size, max_steps)
        all_predictions = torch.cat(list(reversed(reconstructed_predictions)), 2)

        # Calculate the final sequence scores
        # shape: (batch_size, beam_size)
        final_scores = self.final_sequence_scorer.score(
            all_predictions, last_log_probabilities, self._end_index
        )

        # Sort the sequences based on the final scores so the best scoring
        # sequence is at index 0
        sorted_final_scores, sorted_indices = torch.sort(final_scores, dim=1, descending=True)
        sorted_all_predictions = torch.gather(
            all_predictions, 1, sorted_indices.unsqueeze(-1).expand_as(all_predictions)
        )

        return sorted_all_predictions, sorted_final_scores

    @staticmethod
    def _is_multilayer_rnn_decoder(key: str, state_tensor: torch.Tensor) -> bool:
        return state_tensor.dim() == 3 and key in {
            "decoder_hidden",
            "decoder_context",
        }

    def _update_initial_state(self, state: StateType, batch_size: int):
        """
        Expand tensors in a state dictionary from `(batch_size, *)` to `(batch_size * beam_size, *)`.
        """
        for key, state_tensor in state.items():
            if state_tensor is None:
                continue
            multilayer_rnn_decoder = self._is_multilayer_rnn_decoder(key, state_tensor)

            if multilayer_rnn_decoder:
                # shape: (num_layers, batch_size * beam_size, *)
                num_layers, _, *last_dims = state_tensor.size()
                state[key] = (
                    state_tensor.unsqueeze(2)
                    .expand(num_layers, batch_size, self.beam_size, *last_dims)
                    .reshape(num_layers, batch_size * self.beam_size, *last_dims)
                )
            else:
                # shape: (batch_size * beam_size, *)
                _, *last_dims = state_tensor.size()
                state[key] = (
                    state_tensor.unsqueeze(1)
                    .expand(batch_size, self.beam_size, *last_dims)
                    .reshape(batch_size * self.beam_size, *last_dims)
                )

    def _update_state(self, state: StateType, backpointer: torch.Tensor):
        batch_size = backpointer.size()[0]

        for key, state_tensor in state.items():
            if state_tensor is None:
                continue
            multilayer_rnn_decoder = self._is_multilayer_rnn_decoder(key, state_tensor)

            if multilayer_rnn_decoder:
                # shape: (num_layers, batch_size * beam_size, *)
                num_layers, _, *last_dims = state_tensor.size()
                expanded_backpointer = backpointer.view(
                    batch_size, self.beam_size, *([1] * len(last_dims))
                ).expand(batch_size, self.beam_size, *last_dims)
                expanded_backpointer = expanded_backpointer.unsqueeze(0).repeat(num_layers, 1, 1, 1)
                # shape: (num_layers, batch_size * beam_size, *)
                state[key] = (
                    state_tensor.reshape(num_layers, batch_size, self.beam_size, *last_dims)
                    .gather(2, expanded_backpointer)
                    .reshape(num_layers, batch_size * self.beam_size, *last_dims)
                )
            else:
                _, *last_dims = state_tensor.size()
                # shape: (batch_size, beam_size, *)
                expanded_backpointer = backpointer.view(
                    batch_size, self.beam_size, *([1] * len(last_dims))
                ).expand(batch_size, self.beam_size, *last_dims)
                # shape: (batch_size * beam_size, *)
                state[key] = (
                    state_tensor.reshape(batch_size, self.beam_size, *last_dims)
                    .gather(1, expanded_backpointer)
                    .reshape(batch_size * self.beam_size, *last_dims)
                )


BeamSearch.register("beam_search")(BeamSearch)