rejection_sampler.py

# SPDX-License-Identifier: Apache-2.0
# SPDX-FileCopyrightText: Copyright contributors to the vLLM project

import torch
import torch.nn as nn

from vllm.logger import init_logger
from vllm.triton_utils import tl, triton
from vllm.v1.sample.metadata import SamplingMetadata
from vllm.v1.sample.ops.bad_words import apply_bad_words_with_drafts
from vllm.v1.sample.ops.penalties import apply_all_penalties
from vllm.v1.sample.ops.topk_topp_sampler import apply_top_k_top_p
from vllm.v1.spec_decode.metadata import SpecDecodeMetadata

logger = init_logger(__name__)

PLACEHOLDER_TOKEN_ID: tl.constexpr = -1
GREEDY_TEMPERATURE: tl.constexpr = -1
# Maximum number of speculative draft tokens allowed per request in a single
# step. This value is chosen to be large enough to handle typical use cases.
MAX_SPEC_LEN = 128


class RejectionSampler(nn.Module):
    """
    The implementation strictly follows the algorithm described in
        https://arxiv.org/abs/2211.17192.
    However, we want to clarify the terminology used in the implementation:
    accepted tokens: tokens that are accepted based on the relationship
            between the "raw" draft and target probabilities.
    recovered tokens: tokens that are sampled based on the adjusted probability
        distribution, which is derived from both the draft and target
        probabilities.
    bonus tokens:
        If all proposed tokens are accepted, the bonus token is added to the
        end of the sequence. The bonus token is only sampled from the target
        probabilities. We pass in the bonus tokens instead of sampling them
        in the rejection sampler to allow for more flexibility in the
        sampling process. For example, we can use top_p, top_k sampling for
        bonus tokens, while spec decode does not support these sampling
        strategies.
    output tokens:
        Tokens are finally generated with the rejection sampler.
        output tokens = accepted tokens + recovered tokens + bonus tokens
    """

    def forward(
        self,
        metadata: SpecDecodeMetadata,
        # [num_tokens, vocab_size]
        draft_probs: torch.Tensor | None,
        # [num_tokens, vocab_size]
        target_logits: torch.Tensor,
        # [batch_size, 1]
        bonus_token_ids: torch.Tensor,
        sampling_metadata: SamplingMetadata,
    ) -> torch.Tensor:
        """
        Args:
            metadata:
                Metadata for spec decoding.
            draft_probs (Optional[torch.Tensor]):
                Probability distribution for the draft tokens. Shape is
                [num_tokens, vocab_size]. Can be None if probabilities are
                not provided, which is the case for ngram spec decode.
            target_logits (torch.Tensor):
                Target model's logits probability distribution.
                Shape is [num_tokens, vocab_size]. Here, probabilities from
                different requests are flattened into a single tensor because
                this is the shape of the output logits.
                NOTE: `target_logits` can be updated in place to save memory.
            bonus_token_ids (torch.Tensor):
                A tensor containing bonus tokens. Shape is [batch_size, 1].
                Bonus tokens are added to the end of the sequence if all
                proposed tokens are accepted. We generate the bonus tokens
                outside of the rejection sampler with the default sampling
                strategy. It allows for more flexibility in the sampling
                process such as top_p, top_k sampling.
            sampling_metadata (vllm.v1.sample.metadata.SamplingMetadata):
                Additional metadata needed for sampling, such as temperature,
                top-k/top-p parameters, or other relevant information.
        Returns:
            output_token_ids (torch.Tensor):
                A tensor containing the final output token IDs.
        """
        assert metadata.max_spec_len <= MAX_SPEC_LEN

        # Use float32 for the target_logits.
        target_logits = target_logits.to(torch.float32)

        target_logits = self.apply_logits_processors(
            target_logits, sampling_metadata, metadata
        )

        # [num_tokens, vocab_size]
        # NOTE(woosuk): `target_logits` can be updated in place inside the
        # `compute_probs` function.
        target_probs = compute_probs(
            target_logits,
            metadata.cu_num_draft_tokens,
            sampling_metadata,
        )

        output_token_ids = rejection_sample(
            metadata.draft_token_ids,
            metadata.num_draft_tokens,
            metadata.max_spec_len,
            metadata.cu_num_draft_tokens,
            draft_probs,
            target_probs,
            bonus_token_ids,
            sampling_metadata,
        )
        return output_token_ids

    @staticmethod
    def parse_output(
        output_token_ids: torch.Tensor,
        vocab_size: int,
    ) -> list[list[int]]:
        """Parse the output of the rejection sampler.

        Args:
            output_token_ids: The sampled token IDs in shape
                [batch_size, max_spec_len + 1]. The rejected tokens are
                replaced with `PLACEHOLDER_TOKEN_ID` by the rejection sampler
                and will be filtered out in this function.
            vocab_size: The size of the vocabulary.

        Returns:
            A list of lists of token IDs.
        """
        output_token_ids_np = output_token_ids.cpu().numpy()
        # Create mask for valid tokens.
        valid_mask = (output_token_ids_np != PLACEHOLDER_TOKEN_ID) & (
            output_token_ids_np < vocab_size
        )
        outputs = [
            row[valid_mask[i]].tolist() for i, row in enumerate(output_token_ids_np)
        ]
        return outputs

    def apply_logits_processors(
        self,
        logits: torch.Tensor,
        sampling_metadata: SamplingMetadata,
        metadata: SpecDecodeMetadata,
    ) -> torch.Tensor:
        has_penalties = not sampling_metadata.no_penalties
        any_penalties_or_bad_words = (
            sampling_metadata.bad_words_token_ids or has_penalties
        )

        output_token_ids = sampling_metadata.output_token_ids
        if any_penalties_or_bad_words:
            output_token_ids = self._combine_outputs_with_spec_tokens(
                output_token_ids,
                sampling_metadata.spec_token_ids,
            )

        # Calculate indices of target logits.
        if sampling_metadata.allowed_token_ids_mask is not None or has_penalties:
            num_requests = len(sampling_metadata.output_token_ids)
            num_draft_tokens = torch.tensor(metadata.num_draft_tokens, device="cpu")
            original_indices = torch.arange(num_requests, device="cpu")
            repeat_indices_cpu = original_indices.repeat_interleave(num_draft_tokens)
            repeat_indices = repeat_indices_cpu.to(
                device=logits.device, non_blocking=True
            )
            logits = self.apply_penalties(
                logits, sampling_metadata, metadata, repeat_indices, output_token_ids
            )

            # Apply allowed token ids.
            if sampling_metadata.allowed_token_ids_mask is not None:
                token_mask = sampling_metadata.allowed_token_ids_mask[repeat_indices]
                logits.masked_fill_(token_mask, float("-inf"))

        # Apply bad words exclusion.
        if bad_words_token_ids := sampling_metadata.bad_words_token_ids:
            apply_bad_words_with_drafts(
                logits, bad_words_token_ids, output_token_ids, metadata.num_draft_tokens
            )

        return logits

    @staticmethod
    def apply_penalties(
        logits: torch.Tensor,
        sampling_metadata: SamplingMetadata,
        metadata: SpecDecodeMetadata,
        repeat_indices: torch.Tensor,
        output_token_ids: list[list[int]],
    ) -> torch.Tensor:
        if sampling_metadata.no_penalties:
            return logits

        assert sampling_metadata.prompt_token_ids is not None

        prompt_token_ids = sampling_metadata.prompt_token_ids[repeat_indices]
        presence_penalties = sampling_metadata.presence_penalties[repeat_indices]
        frequency_penalties = sampling_metadata.frequency_penalties[repeat_indices]
        repetition_penalties = sampling_metadata.repetition_penalties[repeat_indices]

        logits = apply_all_penalties(
            logits,
            prompt_token_ids,
            presence_penalties,
            frequency_penalties,
            repetition_penalties,
            output_token_ids,
        )
        return logits

    @staticmethod
    def _combine_outputs_with_spec_tokens(
        output_token_ids: list[list[int]],
        spec_token_ids: list[list[int]] | None = None,
    ) -> list[list[int]]:
        if spec_token_ids is None:
            return output_token_ids

        result = []
        for out, spec in zip(output_token_ids, spec_token_ids):
            if len(spec) == 0:
                continue
            result.append(out)
            for i in range(len(spec) - 1):
                result.append([*result[-1], spec[i]])
        return result


def rejection_sample(
    # [num_tokens]
    draft_token_ids: torch.Tensor,
    # [batch_size]
    num_draft_tokens: list[int],
    max_spec_len: int,
    # [batch_size]
    cu_num_draft_tokens: torch.Tensor,
    # [num_tokens, vocab_size]
    draft_probs: torch.Tensor | None,
    # [num_tokens, vocab_size]
    target_probs: torch.Tensor,
    # [batch_size, 1]
    bonus_token_ids: torch.Tensor,
    sampling_metadata: SamplingMetadata,
) -> torch.Tensor:
    assert draft_token_ids.ndim == 1
    assert draft_probs is None or draft_probs.ndim == 2
    assert cu_num_draft_tokens.ndim == 1
    assert target_probs.ndim == 2

    batch_size = len(num_draft_tokens)
    num_tokens = draft_token_ids.shape[0]
    vocab_size = target_probs.shape[-1]
    device = target_probs.device
    assert draft_token_ids.is_contiguous()
    assert draft_probs is None or draft_probs.is_contiguous()
    assert target_probs.is_contiguous()
    assert bonus_token_ids.is_contiguous()
    assert target_probs.shape == (num_tokens, vocab_size)

    # Create output buffer.
    output_token_ids = torch.full(
        (batch_size, max_spec_len + 1),
        PLACEHOLDER_TOKEN_ID,
        dtype=torch.int32,  # Consistent with SamplerOutput.sampled_token_ids.
        device=device,
    )

    if sampling_metadata.all_greedy:
        is_greedy = None
    else:
        is_greedy = sampling_metadata.temperature == GREEDY_TEMPERATURE
    if not sampling_metadata.all_random:
        # Rejection sampling for greedy sampling requests.
        target_argmax = target_probs.argmax(dim=-1)
        rejection_greedy_sample_kernel[(batch_size,)](
            output_token_ids,
            cu_num_draft_tokens,
            draft_token_ids,
            target_argmax,
            bonus_token_ids,
            is_greedy,
            max_spec_len,
        )
        if sampling_metadata.all_greedy:
            return output_token_ids

    # Generate uniform probabilities for rejection sampling.
    # [num_tokens]
    uniform_probs = generate_uniform_probs(
        num_tokens,
        num_draft_tokens,
        sampling_metadata.generators,
        device,
    )

    # Sample recovered tokens for each position.
    # [num_tokens]
    recovered_token_ids = sample_recovered_tokens(
        max_spec_len,
        num_draft_tokens,
        cu_num_draft_tokens,
        draft_token_ids,
        draft_probs,
        target_probs,
        sampling_metadata,
        device,
    )

    # Rejection sampling for random sampling requests.
    rejection_random_sample_kernel[(batch_size,)](
        output_token_ids,
        cu_num_draft_tokens,
        draft_token_ids,
        draft_probs,
        target_probs,
        bonus_token_ids,
        recovered_token_ids,
        uniform_probs,
        is_greedy,
        max_spec_len,
        vocab_size,
        NO_DRAFT_PROBS=draft_probs is None,
    )
    return output_token_ids


def compute_probs(
    logits: torch.Tensor,  # [num_tokens, vocab_size]
    cu_num_draft_tokens: torch.Tensor,  # [batch_size]
    sampling_metadata: SamplingMetadata,
) -> torch.Tensor:
    """Compute probability distribution from logits based on sampling metadata.

    This function applies temperature scaling to the logits and converts
    them to probabilities using softmax. For greedy decoding, it returns
    the original logits.

    Args:
        logits: Input logits tensor to be converted to probabilities.
        cu_num_draft_tokens: Cumulative number of draft tokens.
        sampling_metadata: Metadata containing sampling parameters such as
            temperature and whether greedy sampling is used.

    Returns:
        torch.Tensor: Probability distribution (softmax of scaled logits)
            if non-greedy sampling is used, otherwise returns the
            original logits.
    """
    assert logits.ndim == 2
    assert cu_num_draft_tokens.ndim == 1
    if sampling_metadata.all_greedy:
        return logits

    num_tokens = logits.shape[0]
    temperature = expand_batch_to_tokens(
        sampling_metadata.temperature,
        cu_num_draft_tokens,
        num_tokens,
        replace_from=GREEDY_TEMPERATURE,
        replace_to=1,
    )
    # NOTE(woosuk): Update `logits` in place to avoid allocating a new tensor.
    logits.div_(temperature.unsqueeze(-1))

    # Get expanded top_k and top_p tensors.
    top_k = None
    if sampling_metadata.top_k is not None:
        top_k = expand_batch_to_tokens(
            sampling_metadata.top_k,
            cu_num_draft_tokens,
            num_tokens,
        )
    top_p = None
    if sampling_metadata.top_p is not None:
        top_p = expand_batch_to_tokens(
            sampling_metadata.top_p,
            cu_num_draft_tokens,
            num_tokens,
        )

    # NOTE(woosuk): `apply_top_k_top_p` uses sorting to calculate the mask,
    # which is slow for large vocab sizes. This may cause performance issues.
    logits = apply_top_k_top_p(logits, top_k, top_p)
    output_prob = logits.softmax(dim=-1, dtype=torch.float32)
    return output_prob


def expand_batch_to_tokens(
    x: torch.Tensor,  # [batch_size]
    cu_num_tokens: torch.Tensor,  # [batch_size]
    num_tokens: int,
    replace_from: int = 0,
    replace_to: int = 0,
) -> torch.Tensor:
    """Expand [batch_size] tensor to [num_tokens] tensor based on the number of
    tokens per batch in cu_num_tokens.

    For example, if x = [a, b, c] and cu_num_tokens = [2, 5, 6], then
    num_tokens = 6, and expanded_x = [a, a, b, b, b, c].

    Args:
        x: [batch_size] tensor to expand.
        cu_num_tokens: [batch_size] tensor containing the cumulative number of
            tokens per batch. Each element represents the total number of
            tokens up to and including that batch.
        num_tokens: Total number of tokens.
        replace_from: int = 0
            Value to be replaced if it is found in x.
        replace_to: int = 0
            Value to replace with when replace_from is found.
    Returns:
        expanded_x: [num_tokens] tensor.
    """
    batch_size = x.shape[0]
    assert cu_num_tokens.shape[0] == batch_size
    expanded_x = x.new_empty(num_tokens)
    expand_kernel[(batch_size,)](
        expanded_x,
        x,
        cu_num_tokens,
        replace_from,
        replace_to,
        MAX_NUM_TOKENS=MAX_SPEC_LEN,  # To avoid recompilation.
    )
    return expanded_x


def generate_uniform_probs(
    num_tokens: int,
    num_draft_tokens: list[int],
    generators: dict[int, torch.Generator],
    device: torch.device,
) -> torch.Tensor:
    """
    Generates a batch of uniform random samples, with optional seeding
    if available.

    This method creates a tensor of shape `(num_tokens, )` filled
    with uniform random values in the range [0, 1). If `generators` is provided,
    the requests with their own seeds will use the provided `torch.Generator`
    for reproducibility. The samples for the other requests will be generated
    without a seed.

    Args:
        num_tokens: int
            Total number of tokens.
        num_draft_tokens: List[List[int]]
            Number of draft tokens per request.
        generators: Optional[Dict[int, torch.Generator]]
            A dictionary mapping indices in the batch to
            `torch.Generator` objects.
        device: torch.device
            The device on which to allocate the tensor.
    Returns:
        uniform_rand: torch.Tensor
            A tensor of shape `(num_tokens, )` containing uniform
            random values in the range [0, 1).
    """
    # NOTE(woosuk): We deliberately use float64 instead of float32 here
    # because when using float32, there's a non-negligible chance that
    # uniform_prob is sampled to be exact 0.0 as reported in
    # https://github.com/pytorch/pytorch/issues/16706. Using float64
    # mitigates the issue.
    uniform_probs = torch.rand(
        (num_tokens,),
        dtype=torch.float64,
        device=device,
    )
    start_idx = 0
    for req_idx, n in enumerate(num_draft_tokens):
        # Do not generate random numbers for requests with no draft tokens.
        # This can be important for reproducibility.
        if n == 0:
            continue
        end_idx = start_idx + n
        generator = generators.get(req_idx)
        if generator is not None:
            uniform_probs[start_idx:end_idx].uniform_(generator=generator)
        start_idx = end_idx
    return uniform_probs


def sample_recovered_tokens(
    max_spec_len: int,
    num_draft_tokens: list[int],
    # [batch_size]
    cu_num_draft_tokens: torch.Tensor,
    # [num_tokens]
    draft_token_ids: torch.Tensor,
    # [num_tokens, vocab_size]
    draft_probs: torch.Tensor | None,
    # [num_tokens, vocab_size]
    target_probs: torch.Tensor,
    sampling_metadata: SamplingMetadata,
    device: torch.device,
) -> torch.Tensor:
    # NOTE(woosuk): Create only one distribution for each request.
    batch_size = len(num_draft_tokens)
    vocab_size = target_probs.shape[-1]
    q = torch.empty(
        (batch_size, vocab_size),
        dtype=torch.float32,
        device=device,
    )
    q.exponential_()
    for i, generator in sampling_metadata.generators.items():
        # Do not generate random numbers for requests with no draft tokens.
        # This can be important for reproducibility.
        if num_draft_tokens[i] > 0:
            q[i].exponential_(generator=generator)

    recovered_token_ids = torch.empty_like(draft_token_ids)
    sample_recovered_tokens_kernel[(batch_size, max_spec_len)](
        recovered_token_ids,
        cu_num_draft_tokens,
        draft_token_ids,
        draft_probs,
        target_probs,
        q,
        vocab_size,
        triton.next_power_of_2(vocab_size),
        NO_DRAFT_PROBS=draft_probs is None,
    )
    return recovered_token_ids


# NOTE(woosuk): Avoid specialization to prevent unnecessary recompilation.
@triton.jit(do_not_specialize=["max_spec_len"])
def rejection_greedy_sample_kernel(
    output_token_ids_ptr,  # [batch_size, max_spec_len + 1]
    cu_num_draft_tokens_ptr,  # [batch_size]
    draft_token_ids_ptr,  # [num_tokens]
    target_argmax_ptr,  # [num_tokens]
    bonus_token_ids_ptr,  # [batch_size]
    is_greedy_ptr,  # [batch_size] or None
    max_spec_len,
):
    req_idx = tl.program_id(0)
    # FIXME(woosuk): Because is_greedy_ptr is not None at profiling run,
    # re-compilation may happen during runtime when is_greedy_ptr is None.
    is_greedy = True if is_greedy_ptr is None else tl.load(is_greedy_ptr + req_idx)
    if not is_greedy:
        # Early exit for non-greedy sampling requests.
        return

    start_idx = 0 if req_idx == 0 else tl.load(cu_num_draft_tokens_ptr + req_idx - 1)
    end_idx = tl.load(cu_num_draft_tokens_ptr + req_idx)
    num_draft_tokens = end_idx - start_idx

    rejected = False
    for pos in range(num_draft_tokens):
        if not rejected:
            draft_token_id = tl.load(draft_token_ids_ptr + start_idx + pos)
            target_argmax_id = tl.load(target_argmax_ptr + start_idx + pos)
            tl.store(
                output_token_ids_ptr + req_idx * (max_spec_len + 1) + pos,
                target_argmax_id,
            )
            if draft_token_id != target_argmax_id:
                # Reject.
                rejected = True

    if not rejected:
        # If all tokens are accepted, append the bonus token.
        bonus_token_id = tl.load(bonus_token_ids_ptr + req_idx)
        tl.store(
            output_token_ids_ptr + req_idx * (max_spec_len + 1) + num_draft_tokens,
            bonus_token_id,
        )


# NOTE(woosuk): Avoid specialization to prevent unnecessary recompilation.
@triton.jit(do_not_specialize=["max_spec_len"])
def rejection_random_sample_kernel(
    output_token_ids_ptr,  # [batch_size, max_spec_len + 1]
    cu_num_draft_tokens_ptr,  # [batch_size]
    draft_token_ids_ptr,  # [num_tokens]
    draft_probs_ptr,  # [num_tokens, vocab_size] or None
    target_probs_ptr,  # [num_tokens, vocab_size]
    bonus_token_ids_ptr,  # [batch_size]
    recovered_token_ids_ptr,  # [num_tokens]
    uniform_probs_ptr,  # [num_tokens]
    is_greedy_ptr,  # [batch_size]
    max_spec_len,
    vocab_size,
    NO_DRAFT_PROBS: tl.constexpr,
):
    req_idx = tl.program_id(0)
    is_greedy = tl.load(is_greedy_ptr + req_idx)
    if is_greedy:
        # Early exit for greedy sampling requests.
        return

    start_idx = 0 if req_idx == 0 else tl.load(cu_num_draft_tokens_ptr + req_idx - 1)
    end_idx = tl.load(cu_num_draft_tokens_ptr + req_idx)
    num_draft_tokens = end_idx - start_idx

    rejected = False
    for pos in range(num_draft_tokens):
        if not rejected:
            draft_token_id = tl.load(draft_token_ids_ptr + start_idx + pos)
            if NO_DRAFT_PROBS:
                draft_prob = 1
            else:
                draft_prob = tl.load(
                    draft_probs_ptr + (start_idx + pos) * vocab_size + draft_token_id
                )
            target_prob = tl.load(
                target_probs_ptr + (start_idx + pos) * vocab_size + draft_token_id
            )
            uniform_prob = tl.load(uniform_probs_ptr + start_idx + pos)
            # NOTE(woosuk): While the draft probability should never be 0,
            # we check it to avoid NaNs. If it happens to be 0, we reject.
            if draft_prob > 0 and target_prob / draft_prob >= uniform_prob:
                # Accept.
                token_id = draft_token_id
            else:
                # Reject. Use recovered token.
                rejected = True
                token_id = tl.load(recovered_token_ids_ptr + start_idx + pos)
            tl.store(
                output_token_ids_ptr + req_idx * (max_spec_len + 1) + pos, token_id
            )

    if not rejected:
        # If all tokens are accepted, append the bonus token.
        bonus_token_id = tl.load(bonus_token_ids_ptr + req_idx)
        tl.store(
            output_token_ids_ptr + req_idx * (max_spec_len + 1) + num_draft_tokens,
            bonus_token_id,
        )


# NOTE(woosuk): Avoid specialization to prevent unnecessary recompilation.
@triton.jit(do_not_specialize=["replace_from", "replace_to"])
def expand_kernel(
    output_ptr,  # [num_tokens]
    input_ptr,  # [batch_size]
    cu_num_tokens_ptr,  # [batch_size]
    replace_from,
    replace_to,
    MAX_NUM_TOKENS: tl.constexpr,
):
    req_idx = tl.program_id(0)
    if req_idx == 0:  # noqa: SIM108
        start_idx = 0
    else:
        start_idx = tl.load(cu_num_tokens_ptr + req_idx - 1)
    end_idx = tl.load(cu_num_tokens_ptr + req_idx)
    num_tokens = end_idx - start_idx

    src_val = tl.load(input_ptr + req_idx)
    src_val = tl.where(src_val == replace_from, replace_to, src_val)
    offset = tl.arange(0, MAX_NUM_TOKENS)
    tl.store(output_ptr + start_idx + offset, src_val, mask=offset < num_tokens)


@triton.jit
def sample_recovered_tokens_kernel(
    output_token_ids_ptr,  # [num_tokens]
    cu_num_draft_tokens_ptr,  # [batch_size]
    draft_token_ids_ptr,  # [num_tokens]
    draft_probs_ptr,  # [num_tokens, vocab_size] or None
    target_probs_ptr,  # [num_tokens, vocab_size]
    q_ptr,  # [batch_size, vocab_size]
    vocab_size,
    PADDED_VOCAB_SIZE: tl.constexpr,
    NO_DRAFT_PROBS: tl.constexpr,
):
    req_idx = tl.program_id(0)
    start_idx = 0 if req_idx == 0 else tl.load(cu_num_draft_tokens_ptr + req_idx - 1)
    end_idx = tl.load(cu_num_draft_tokens_ptr + req_idx)
    num_draft_tokens = end_idx - start_idx

    # Early exit for out-of-range positions.
    pos = tl.program_id(1)
    if pos >= num_draft_tokens:
        return

    vocab_offset = tl.arange(0, PADDED_VOCAB_SIZE)
    if NO_DRAFT_PROBS:
        draft_token_id = tl.load(draft_token_ids_ptr + start_idx + pos)
        prob = tl.load(
            target_probs_ptr + (start_idx + pos) * vocab_size + vocab_offset,
            mask=((vocab_offset < vocab_size) & (vocab_offset != draft_token_id)),
            other=0,
        )
    else:
        draft_prob = tl.load(
            draft_probs_ptr + (start_idx + pos) * vocab_size + vocab_offset,
            mask=vocab_offset < vocab_size,
            other=0,
        )
        target_prob = tl.load(
            target_probs_ptr + (start_idx + pos) * vocab_size + vocab_offset,
            mask=vocab_offset < vocab_size,
            other=0,
        )
        prob = tl.maximum(target_prob - draft_prob, 0)
        # NOTE(woosuk): We don't need `prob = prob / tl.sum(prob)` here because
        # `tl.argmax` will select the maximum value.

    q = tl.load(
        q_ptr + req_idx * vocab_size + vocab_offset,
        mask=vocab_offset < vocab_size,
        other=float("-inf"),
    )
    recovered_id = tl.argmax(prob / q, axis=-1)
    tl.store(output_token_ids_ptr + start_idx + pos, recovered_id)