scheduling_tcd.py

# Copyright 2024 Stanford University Team and The HuggingFace Team. All rights reserved.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
#     http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.

# DISCLAIMER: This code is strongly influenced by https://github.com/pesser/pytorch_diffusion
# and https://github.com/hojonathanho/diffusion

import math
from dataclasses import dataclass
from typing import List, Optional, Tuple, Union

import numpy as np
import torch

from ..configuration_utils import ConfigMixin, register_to_config
from ..schedulers.scheduling_utils import SchedulerMixin
from ..utils import BaseOutput, logging
from ..utils.torch_utils import randn_tensor


logger = logging.get_logger(__name__)  # pylint: disable=invalid-name


@dataclass
class TCDSchedulerOutput(BaseOutput):
    """
    Output class for the scheduler's `step` function output.

    Args:
        prev_sample (`torch.Tensor` of shape `(batch_size, num_channels, height, width)` for images):
            Computed sample `(x_{t-1})` of previous timestep. `prev_sample` should be used as next model input in the
            denoising loop.
        pred_noised_sample (`torch.Tensor` of shape `(batch_size, num_channels, height, width)` for images):
            The predicted noised sample `(x_{s})` based on the model output from the current timestep.
    """

    prev_sample: torch.Tensor
    pred_noised_sample: Optional[torch.Tensor] = None


# Copied from diffusers.schedulers.scheduling_ddpm.betas_for_alpha_bar
def betas_for_alpha_bar(
    num_diffusion_timesteps,
    max_beta=0.999,
    alpha_transform_type="cosine",
):
    """
    Create a beta schedule that discretizes the given alpha_t_bar function, which defines the cumulative product of
    (1-beta) over time from t = [0,1].

    Contains a function alpha_bar that takes an argument t and transforms it to the cumulative product of (1-beta) up
    to that part of the diffusion process.


    Args:
        num_diffusion_timesteps (`int`): the number of betas to produce.
        max_beta (`float`): the maximum beta to use; use values lower than 1 to
                     prevent singularities.
        alpha_transform_type (`str`, *optional*, default to `cosine`): the type of noise schedule for alpha_bar.
                     Choose from `cosine` or `exp`

    Returns:
        betas (`np.ndarray`): the betas used by the scheduler to step the model outputs
    """
    if alpha_transform_type == "cosine":

        def alpha_bar_fn(t):
            return math.cos((t + 0.008) / 1.008 * math.pi / 2) ** 2

    elif alpha_transform_type == "exp":

        def alpha_bar_fn(t):
            return math.exp(t * -12.0)

    else:
        raise ValueError(f"Unsupported alpha_transform_type: {alpha_transform_type}")

    betas = []
    for i in range(num_diffusion_timesteps):
        t1 = i / num_diffusion_timesteps
        t2 = (i + 1) / num_diffusion_timesteps
        betas.append(min(1 - alpha_bar_fn(t2) / alpha_bar_fn(t1), max_beta))
    return torch.tensor(betas, dtype=torch.float32)


# Copied from diffusers.schedulers.scheduling_ddim.rescale_zero_terminal_snr
def rescale_zero_terminal_snr(betas: torch.Tensor) -> torch.Tensor:
    """
    Rescales betas to have zero terminal SNR Based on https://arxiv.org/pdf/2305.08891.pdf (Algorithm 1)


    Args:
        betas (`torch.Tensor`):
            the betas that the scheduler is being initialized with.

    Returns:
        `torch.Tensor`: rescaled betas with zero terminal SNR
    """
    # Convert betas to alphas_bar_sqrt
    alphas = 1.0 - betas
    alphas_cumprod = torch.cumprod(alphas, dim=0)
    alphas_bar_sqrt = alphas_cumprod.sqrt()

    # Store old values.
    alphas_bar_sqrt_0 = alphas_bar_sqrt[0].clone()
    alphas_bar_sqrt_T = alphas_bar_sqrt[-1].clone()

    # Shift so the last timestep is zero.
    alphas_bar_sqrt -= alphas_bar_sqrt_T

    # Scale so the first timestep is back to the old value.
    alphas_bar_sqrt *= alphas_bar_sqrt_0 / (alphas_bar_sqrt_0 - alphas_bar_sqrt_T)

    # Convert alphas_bar_sqrt to betas
    alphas_bar = alphas_bar_sqrt**2  # Revert sqrt
    alphas = alphas_bar[1:] / alphas_bar[:-1]  # Revert cumprod
    alphas = torch.cat([alphas_bar[0:1], alphas])
    betas = 1 - alphas

    return betas


class TCDScheduler(SchedulerMixin, ConfigMixin):
    """
    `TCDScheduler` incorporates the `Strategic Stochastic Sampling` introduced by the paper `Trajectory Consistency
    Distillation`, extending the original Multistep Consistency Sampling to enable unrestricted trajectory traversal.

    This code is based on the official repo of TCD(https://github.com/jabir-zheng/TCD).

    This model inherits from [`SchedulerMixin`] and [`ConfigMixin`]. [`~ConfigMixin`] takes care of storing all config
    attributes that are passed in the scheduler's `__init__` function, such as `num_train_timesteps`. They can be
    accessed via `scheduler.config.num_train_timesteps`. [`SchedulerMixin`] provides general loading and saving
    functionality via the [`SchedulerMixin.save_pretrained`] and [`~SchedulerMixin.from_pretrained`] functions.

    Args:
        num_train_timesteps (`int`, defaults to 1000):
            The number of diffusion steps to train the model.
        beta_start (`float`, defaults to 0.0001):
            The starting `beta` value of inference.
        beta_end (`float`, defaults to 0.02):
            The final `beta` value.
        beta_schedule (`str`, defaults to `"linear"`):
            The beta schedule, a mapping from a beta range to a sequence of betas for stepping the model. Choose from
            `linear`, `scaled_linear`, or `squaredcos_cap_v2`.
        trained_betas (`np.ndarray`, *optional*):
            Pass an array of betas directly to the constructor to bypass `beta_start` and `beta_end`.
        original_inference_steps (`int`, *optional*, defaults to 50):
            The default number of inference steps used to generate a linearly-spaced timestep schedule, from which we
            will ultimately take `num_inference_steps` evenly spaced timesteps to form the final timestep schedule.
        clip_sample (`bool`, defaults to `True`):
            Clip the predicted sample for numerical stability.
        clip_sample_range (`float`, defaults to 1.0):
            The maximum magnitude for sample clipping. Valid only when `clip_sample=True`.
        set_alpha_to_one (`bool`, defaults to `True`):
            Each diffusion step uses the alphas product value at that step and at the previous one. For the final step
            there is no previous alpha. When this option is `True` the previous alpha product is fixed to `1`,
            otherwise it uses the alpha value at step 0.
        steps_offset (`int`, defaults to 0):
            An offset added to the inference steps, as required by some model families.
        prediction_type (`str`, defaults to `epsilon`, *optional*):
            Prediction type of the scheduler function; can be `epsilon` (predicts the noise of the diffusion process),
            `sample` (directly predicts the noisy sample`) or `v_prediction` (see section 2.4 of [Imagen
            Video](https://imagen.research.google/video/paper.pdf) paper).
        thresholding (`bool`, defaults to `False`):
            Whether to use the "dynamic thresholding" method. This is unsuitable for latent-space diffusion models such
            as Stable Diffusion.
        dynamic_thresholding_ratio (`float`, defaults to 0.995):
            The ratio for the dynamic thresholding method. Valid only when `thresholding=True`.
        sample_max_value (`float`, defaults to 1.0):
            The threshold value for dynamic thresholding. Valid only when `thresholding=True`.
        timestep_spacing (`str`, defaults to `"leading"`):
            The way the timesteps should be scaled. Refer to Table 2 of the [Common Diffusion Noise Schedules and
            Sample Steps are Flawed](https://huggingface.co/papers/2305.08891) for more information.
        timestep_scaling (`float`, defaults to 10.0):
            The factor the timesteps will be multiplied by when calculating the consistency model boundary conditions
            `c_skip` and `c_out`. Increasing this will decrease the approximation error (although the approximation
            error at the default of `10.0` is already pretty small).
        rescale_betas_zero_snr (`bool`, defaults to `False`):
            Whether to rescale the betas to have zero terminal SNR. This enables the model to generate very bright and
            dark samples instead of limiting it to samples with medium brightness. Loosely related to
            [`--offset_noise`](https://github.com/huggingface/diffusers/blob/74fd735eb073eb1d774b1ab4154a0876eb82f055/examples/dreambooth/train_dreambooth.py#L506).
    """

    order = 1

    @register_to_config
    def __init__(
        self,
        num_train_timesteps: int = 1000,
        beta_start: float = 0.00085,
        beta_end: float = 0.012,
        beta_schedule: str = "scaled_linear",
        trained_betas: Optional[Union[np.ndarray, List[float]]] = None,
        original_inference_steps: int = 50,
        clip_sample: bool = False,
        clip_sample_range: float = 1.0,
        set_alpha_to_one: bool = True,
        steps_offset: int = 0,
        prediction_type: str = "epsilon",
        thresholding: bool = False,
        dynamic_thresholding_ratio: float = 0.995,
        sample_max_value: float = 1.0,
        timestep_spacing: str = "leading",
        timestep_scaling: float = 10.0,
        rescale_betas_zero_snr: bool = False,
    ):
        if trained_betas is not None:
            self.betas = torch.tensor(trained_betas, dtype=torch.float32)
        elif beta_schedule == "linear":
            self.betas = torch.linspace(beta_start, beta_end, num_train_timesteps, dtype=torch.float32)
        elif beta_schedule == "scaled_linear":
            # this schedule is very specific to the latent diffusion model.
            self.betas = torch.linspace(beta_start**0.5, beta_end**0.5, num_train_timesteps, dtype=torch.float32) ** 2
        elif beta_schedule == "squaredcos_cap_v2":
            # Glide cosine schedule
            self.betas = betas_for_alpha_bar(num_train_timesteps)
        else:
            raise NotImplementedError(f"{beta_schedule} is not implemented for {self.__class__}")

        # Rescale for zero SNR
        if rescale_betas_zero_snr:
            self.betas = rescale_zero_terminal_snr(self.betas)

        self.alphas = 1.0 - self.betas
        self.alphas_cumprod = torch.cumprod(self.alphas, dim=0)

        # At every step in ddim, we are looking into the previous alphas_cumprod
        # For the final step, there is no previous alphas_cumprod because we are already at 0
        # `set_alpha_to_one` decides whether we set this parameter simply to one or
        # whether we use the final alpha of the "non-previous" one.
        self.final_alpha_cumprod = torch.tensor(1.0) if set_alpha_to_one else self.alphas_cumprod[0]

        # standard deviation of the initial noise distribution
        self.init_noise_sigma = 1.0

        # setable values
        self.num_inference_steps = None
        self.timesteps = torch.from_numpy(np.arange(0, num_train_timesteps)[::-1].copy().astype(np.int64))
        self.custom_timesteps = False

        self._step_index = None
        self._begin_index = None

    # Copied from diffusers.schedulers.scheduling_euler_discrete.EulerDiscreteScheduler.index_for_timestep
    def index_for_timestep(self, timestep, schedule_timesteps=None):
        if schedule_timesteps is None:
            schedule_timesteps = self.timesteps

        indices = (schedule_timesteps == timestep).nonzero()

        # The sigma index that is taken for the **very** first `step`
        # is always the second index (or the last index if there is only 1)
        # This way we can ensure we don't accidentally skip a sigma in
        # case we start in the middle of the denoising schedule (e.g. for image-to-image)
        pos = 1 if len(indices) > 1 else 0

        return indices[pos].item()

    # Copied from diffusers.schedulers.scheduling_euler_discrete.EulerDiscreteScheduler._init_step_index
    def _init_step_index(self, timestep):
        if self.begin_index is None:
            if isinstance(timestep, torch.Tensor):
                timestep = timestep.to(self.timesteps.device)
            self._step_index = self.index_for_timestep(timestep)
        else:
            self._step_index = self._begin_index

    @property
    def step_index(self):
        return self._step_index

    @property
    def begin_index(self):
        """
        The index for the first timestep. It should be set from pipeline with `set_begin_index` method.
        """
        return self._begin_index

    # Copied from diffusers.schedulers.scheduling_dpmsolver_multistep.DPMSolverMultistepScheduler.set_begin_index
    def set_begin_index(self, begin_index: int = 0):
        """
        Sets the begin index for the scheduler. This function should be run from pipeline before the inference.

        Args:
            begin_index (`int`):
                The begin index for the scheduler.
        """
        self._begin_index = begin_index

    def scale_model_input(self, sample: torch.Tensor, timestep: Optional[int] = None) -> torch.Tensor:
        """
        Ensures interchangeability with schedulers that need to scale the denoising model input depending on the
        current timestep.

        Args:
            sample (`torch.Tensor`):
                The input sample.
            timestep (`int`, *optional*):
                The current timestep in the diffusion chain.

        Returns:
            `torch.Tensor`:
                A scaled input sample.
        """
        return sample

    # Copied from diffusers.schedulers.scheduling_ddim.DDIMScheduler._get_variance
    def _get_variance(self, timestep, prev_timestep):
        alpha_prod_t = self.alphas_cumprod[timestep]
        alpha_prod_t_prev = self.alphas_cumprod[prev_timestep] if prev_timestep >= 0 else self.final_alpha_cumprod
        beta_prod_t = 1 - alpha_prod_t
        beta_prod_t_prev = 1 - alpha_prod_t_prev

        variance = (beta_prod_t_prev / beta_prod_t) * (1 - alpha_prod_t / alpha_prod_t_prev)

        return variance

    # Copied from diffusers.schedulers.scheduling_ddpm.DDPMScheduler._threshold_sample
    def _threshold_sample(self, sample: torch.Tensor) -> torch.Tensor:
        """
        "Dynamic thresholding: At each sampling step we set s to a certain percentile absolute pixel value in xt0 (the
        prediction of x_0 at timestep t), and if s > 1, then we threshold xt0 to the range [-s, s] and then divide by
        s. Dynamic thresholding pushes saturated pixels (those near -1 and 1) inwards, thereby actively preventing
        pixels from saturation at each step. We find that dynamic thresholding results in significantly better
        photorealism as well as better image-text alignment, especially when using very large guidance weights."

        https://arxiv.org/abs/2205.11487
        """
        dtype = sample.dtype
        batch_size, channels, *remaining_dims = sample.shape

        if dtype not in (torch.float32, torch.float64):
            sample = sample.float()  # upcast for quantile calculation, and clamp not implemented for cpu half

        # Flatten sample for doing quantile calculation along each image
        sample = sample.reshape(batch_size, channels * np.prod(remaining_dims))

        abs_sample = sample.abs()  # "a certain percentile absolute pixel value"

        s = torch.quantile(abs_sample, self.config.dynamic_thresholding_ratio, dim=1)
        s = torch.clamp(
            s, min=1, max=self.config.sample_max_value
        )  # When clamped to min=1, equivalent to standard clipping to [-1, 1]
        s = s.unsqueeze(1)  # (batch_size, 1) because clamp will broadcast along dim=0
        sample = torch.clamp(sample, -s, s) / s  # "we threshold xt0 to the range [-s, s] and then divide by s"

        sample = sample.reshape(batch_size, channels, *remaining_dims)
        sample = sample.to(dtype)

        return sample

    def set_timesteps(
        self,
        num_inference_steps: Optional[int] = None,
        device: Union[str, torch.device] = None,
        original_inference_steps: Optional[int] = None,
        timesteps: Optional[List[int]] = None,
        strength: float = 1.0,
    ):
        """
        Sets the discrete timesteps used for the diffusion chain (to be run before inference).

        Args:
            num_inference_steps (`int`, *optional*):
                The number of diffusion steps used when generating samples with a pre-trained model. If used,
                `timesteps` must be `None`.
            device (`str` or `torch.device`, *optional*):
                The device to which the timesteps should be moved to. If `None`, the timesteps are not moved.
            original_inference_steps (`int`, *optional*):
                The original number of inference steps, which will be used to generate a linearly-spaced timestep
                schedule (which is different from the standard `diffusers` implementation). We will then take
                `num_inference_steps` timesteps from this schedule, evenly spaced in terms of indices, and use that as
                our final timestep schedule. If not set, this will default to the `original_inference_steps` attribute.
            timesteps (`List[int]`, *optional*):
                Custom timesteps used to support arbitrary spacing between timesteps. If `None`, then the default
                timestep spacing strategy of equal spacing between timesteps on the training/distillation timestep
                schedule is used. If `timesteps` is passed, `num_inference_steps` must be `None`.
            strength (`float`, *optional*, defaults to 1.0):
                Used to determine the number of timesteps used for inference when using img2img, inpaint, etc.
        """
        # 0. Check inputs
        if num_inference_steps is None and timesteps is None:
            raise ValueError("Must pass exactly one of `num_inference_steps` or `custom_timesteps`.")

        if num_inference_steps is not None and timesteps is not None:
            raise ValueError("Can only pass one of `num_inference_steps` or `custom_timesteps`.")

        # 1. Calculate the TCD original training/distillation timestep schedule.
        original_steps = (
            original_inference_steps if original_inference_steps is not None else self.config.original_inference_steps
        )

        if original_inference_steps is None:
            # default option, timesteps align with discrete inference steps
            if original_steps > self.config.num_train_timesteps:
                raise ValueError(
                    f"`original_steps`: {original_steps} cannot be larger than `self.config.train_timesteps`:"
                    f" {self.config.num_train_timesteps} as the unet model trained with this scheduler can only handle"
                    f" maximal {self.config.num_train_timesteps} timesteps."
                )
            # TCD Timesteps Setting
            # The skipping step parameter k from the paper.
            k = self.config.num_train_timesteps // original_steps
            # TCD Training/Distillation Steps Schedule
            tcd_origin_timesteps = np.asarray(list(range(1, int(original_steps * strength) + 1))) * k - 1
        else:
            # customised option, sampled timesteps can be any arbitrary value
            tcd_origin_timesteps = np.asarray(list(range(0, int(self.config.num_train_timesteps * strength))))

        # 2. Calculate the TCD inference timestep schedule.
        if timesteps is not None:
            # 2.1 Handle custom timestep schedules.
            train_timesteps = set(tcd_origin_timesteps)
            non_train_timesteps = []
            for i in range(1, len(timesteps)):
                if timesteps[i] >= timesteps[i - 1]:
                    raise ValueError("`custom_timesteps` must be in descending order.")

                if timesteps[i] not in train_timesteps:
                    non_train_timesteps.append(timesteps[i])

            if timesteps[0] >= self.config.num_train_timesteps:
                raise ValueError(
                    f"`timesteps` must start before `self.config.train_timesteps`:"
                    f" {self.config.num_train_timesteps}."
                )

            # Raise warning if timestep schedule does not start with self.config.num_train_timesteps - 1
            if strength == 1.0 and timesteps[0] != self.config.num_train_timesteps - 1:
                logger.warning(
                    f"The first timestep on the custom timestep schedule is {timesteps[0]}, not"
                    f" `self.config.num_train_timesteps - 1`: {self.config.num_train_timesteps - 1}. You may get"
                    f" unexpected results when using this timestep schedule."
                )

            # Raise warning if custom timestep schedule contains timesteps not on original timestep schedule
            if non_train_timesteps:
                logger.warning(
                    f"The custom timestep schedule contains the following timesteps which are not on the original"
                    f" training/distillation timestep schedule: {non_train_timesteps}. You may get unexpected results"
                    f" when using this timestep schedule."
                )

            # Raise warning if custom timestep schedule is longer than original_steps
            if original_steps is not None:
                if len(timesteps) > original_steps:
                    logger.warning(
                        f"The number of timesteps in the custom timestep schedule is {len(timesteps)}, which exceeds the"
                        f" the length of the timestep schedule used for training: {original_steps}. You may get some"
                        f" unexpected results when using this timestep schedule."
                    )
            else:
                if len(timesteps) > self.config.num_train_timesteps:
                    logger.warning(
                        f"The number of timesteps in the custom timestep schedule is {len(timesteps)}, which exceeds the"
                        f" the length of the timestep schedule used for training: {self.config.num_train_timesteps}. You may get some"
                        f" unexpected results when using this timestep schedule."
                    )

            timesteps = np.array(timesteps, dtype=np.int64)
            self.num_inference_steps = len(timesteps)
            self.custom_timesteps = True

            # Apply strength (e.g. for img2img pipelines) (see StableDiffusionImg2ImgPipeline.get_timesteps)
            init_timestep = min(int(self.num_inference_steps * strength), self.num_inference_steps)
            t_start = max(self.num_inference_steps - init_timestep, 0)
            timesteps = timesteps[t_start * self.order :]
            # TODO: also reset self.num_inference_steps?
        else:
            # 2.2 Create the "standard" TCD inference timestep schedule.
            if num_inference_steps > self.config.num_train_timesteps:
                raise ValueError(
                    f"`num_inference_steps`: {num_inference_steps} cannot be larger than `self.config.train_timesteps`:"
                    f" {self.config.num_train_timesteps} as the unet model trained with this scheduler can only handle"
                    f" maximal {self.config.num_train_timesteps} timesteps."
                )

            if original_steps is not None:
                skipping_step = len(tcd_origin_timesteps) // num_inference_steps

                if skipping_step < 1:
                    raise ValueError(
                        f"The combination of `original_steps x strength`: {original_steps} x {strength} is smaller than `num_inference_steps`: {num_inference_steps}. Make sure to either reduce `num_inference_steps` to a value smaller than {int(original_steps * strength)} or increase `strength` to a value higher than {float(num_inference_steps / original_steps)}."
                    )

            self.num_inference_steps = num_inference_steps

            if original_steps is not None:
                if num_inference_steps > original_steps:
                    raise ValueError(
                        f"`num_inference_steps`: {num_inference_steps} cannot be larger than `original_inference_steps`:"
                        f" {original_steps} because the final timestep schedule will be a subset of the"
                        f" `original_inference_steps`-sized initial timestep schedule."
                    )
            else:
                if num_inference_steps > self.config.num_train_timesteps:
                    raise ValueError(
                        f"`num_inference_steps`: {num_inference_steps} cannot be larger than `num_train_timesteps`:"
                        f" {self.config.num_train_timesteps} because the final timestep schedule will be a subset of the"
                        f" `num_train_timesteps`-sized initial timestep schedule."
                    )

            # TCD Inference Steps Schedule
            tcd_origin_timesteps = tcd_origin_timesteps[::-1].copy()
            # Select (approximately) evenly spaced indices from tcd_origin_timesteps.
            inference_indices = np.linspace(0, len(tcd_origin_timesteps), num=num_inference_steps, endpoint=False)
            inference_indices = np.floor(inference_indices).astype(np.int64)
            timesteps = tcd_origin_timesteps[inference_indices]

        self.timesteps = torch.from_numpy(timesteps).to(device=device, dtype=torch.long)

        self._step_index = None
        self._begin_index = None

    def step(
        self,
        model_output: torch.Tensor,
        timestep: int,
        sample: torch.Tensor,
        eta: float = 0.3,
        generator: Optional[torch.Generator] = None,
        return_dict: bool = True,
    ) -> Union[TCDSchedulerOutput, Tuple]:
        """
        Predict the sample from the previous timestep by reversing the SDE. This function propagates the diffusion
        process from the learned model outputs (most often the predicted noise).

        Args:
            model_output (`torch.Tensor`):
                The direct output from learned diffusion model.
            timestep (`int`):
                The current discrete timestep in the diffusion chain.
            sample (`torch.Tensor`):
                A current instance of a sample created by the diffusion process.
            eta (`float`):
                A stochastic parameter (referred to as `gamma` in the paper) used to control the stochasticity in every
                step. When eta = 0, it represents deterministic sampling, whereas eta = 1 indicates full stochastic
                sampling.
            generator (`torch.Generator`, *optional*):
                A random number generator.
            return_dict (`bool`, *optional*, defaults to `True`):
                Whether or not to return a [`~schedulers.scheduling_tcd.TCDSchedulerOutput`] or `tuple`.
        Returns:
            [`~schedulers.scheduling_utils.TCDSchedulerOutput`] or `tuple`:
                If return_dict is `True`, [`~schedulers.scheduling_tcd.TCDSchedulerOutput`] is returned, otherwise a
                tuple is returned where the first element is the sample tensor.
        """
        if self.num_inference_steps is None:
            raise ValueError(
                "Number of inference steps is 'None', you need to run 'set_timesteps' after creating the scheduler"
            )

        if self.step_index is None:
            self._init_step_index(timestep)

        assert 0 <= eta <= 1.0, "gamma must be less than or equal to 1.0"

        # 1. get previous step value
        prev_step_index = self.step_index + 1
        if prev_step_index < len(self.timesteps):
            prev_timestep = self.timesteps[prev_step_index]
        else:
            prev_timestep = torch.tensor(0)

        timestep_s = torch.floor((1 - eta) * prev_timestep).to(dtype=torch.long)

        # 2. compute alphas, betas
        alpha_prod_t = self.alphas_cumprod[timestep]
        beta_prod_t = 1 - alpha_prod_t

        alpha_prod_t_prev = self.alphas_cumprod[prev_timestep] if prev_timestep >= 0 else self.final_alpha_cumprod

        alpha_prod_s = self.alphas_cumprod[timestep_s]
        beta_prod_s = 1 - alpha_prod_s

        # 3. Compute the predicted noised sample x_s based on the model parameterization
        if self.config.prediction_type == "epsilon":  # noise-prediction
            pred_original_sample = (sample - beta_prod_t.sqrt() * model_output) / alpha_prod_t.sqrt()
            pred_epsilon = model_output
            pred_noised_sample = alpha_prod_s.sqrt() * pred_original_sample + beta_prod_s.sqrt() * pred_epsilon
        elif self.config.prediction_type == "sample":  # x-prediction
            pred_original_sample = model_output
            pred_epsilon = (sample - alpha_prod_t ** (0.5) * pred_original_sample) / beta_prod_t ** (0.5)
            pred_noised_sample = alpha_prod_s.sqrt() * pred_original_sample + beta_prod_s.sqrt() * pred_epsilon
        elif self.config.prediction_type == "v_prediction":  # v-prediction
            pred_original_sample = (alpha_prod_t**0.5) * sample - (beta_prod_t**0.5) * model_output
            pred_epsilon = (alpha_prod_t**0.5) * model_output + (beta_prod_t**0.5) * sample
            pred_noised_sample = alpha_prod_s.sqrt() * pred_original_sample + beta_prod_s.sqrt() * pred_epsilon
        else:
            raise ValueError(
                f"prediction_type given as {self.config.prediction_type} must be one of `epsilon`, `sample` or"
                " `v_prediction` for `TCDScheduler`."
            )

        # 4. Sample and inject noise z ~ N(0, I) for MultiStep Inference
        # Noise is not used on the final timestep of the timestep schedule.
        # This also means that noise is not used for one-step sampling.
        # Eta (referred to as "gamma" in the paper) was introduced to control the stochasticity in every step.
        # When eta = 0, it represents deterministic sampling, whereas eta = 1 indicates full stochastic sampling.
        if eta > 0:
            if self.step_index != self.num_inference_steps - 1:
                noise = randn_tensor(
                    model_output.shape, generator=generator, device=model_output.device, dtype=pred_noised_sample.dtype
                )
                prev_sample = (alpha_prod_t_prev / alpha_prod_s).sqrt() * pred_noised_sample + (
                    1 - alpha_prod_t_prev / alpha_prod_s
                ).sqrt() * noise
            else:
                prev_sample = pred_noised_sample
        else:
            prev_sample = pred_noised_sample

        # upon completion increase step index by one
        self._step_index += 1

        if not return_dict:
            return (prev_sample, pred_noised_sample)

        return TCDSchedulerOutput(prev_sample=prev_sample, pred_noised_sample=pred_noised_sample)

    # Copied from diffusers.schedulers.scheduling_ddpm.DDPMScheduler.add_noise
    def add_noise(
        self,
        original_samples: torch.Tensor,
        noise: torch.Tensor,
        timesteps: torch.IntTensor,
    ) -> torch.Tensor:
        # Make sure alphas_cumprod and timestep have same device and dtype as original_samples
        # Move the self.alphas_cumprod to device to avoid redundant CPU to GPU data movement
        # for the subsequent add_noise calls
        self.alphas_cumprod = self.alphas_cumprod.to(device=original_samples.device)
        alphas_cumprod = self.alphas_cumprod.to(dtype=original_samples.dtype)
        timesteps = timesteps.to(original_samples.device)

        sqrt_alpha_prod = alphas_cumprod[timesteps] ** 0.5
        sqrt_alpha_prod = sqrt_alpha_prod.flatten()
        while len(sqrt_alpha_prod.shape) < len(original_samples.shape):
            sqrt_alpha_prod = sqrt_alpha_prod.unsqueeze(-1)

        sqrt_one_minus_alpha_prod = (1 - alphas_cumprod[timesteps]) ** 0.5
        sqrt_one_minus_alpha_prod = sqrt_one_minus_alpha_prod.flatten()
        while len(sqrt_one_minus_alpha_prod.shape) < len(original_samples.shape):
            sqrt_one_minus_alpha_prod = sqrt_one_minus_alpha_prod.unsqueeze(-1)

        noisy_samples = sqrt_alpha_prod * original_samples + sqrt_one_minus_alpha_prod * noise
        return noisy_samples

    # Copied from diffusers.schedulers.scheduling_ddpm.DDPMScheduler.get_velocity
    def get_velocity(self, sample: torch.Tensor, noise: torch.Tensor, timesteps: torch.IntTensor) -> torch.Tensor:
        # Make sure alphas_cumprod and timestep have same device and dtype as sample
        self.alphas_cumprod = self.alphas_cumprod.to(device=sample.device)
        alphas_cumprod = self.alphas_cumprod.to(dtype=sample.dtype)
        timesteps = timesteps.to(sample.device)

        sqrt_alpha_prod = alphas_cumprod[timesteps] ** 0.5
        sqrt_alpha_prod = sqrt_alpha_prod.flatten()
        while len(sqrt_alpha_prod.shape) < len(sample.shape):
            sqrt_alpha_prod = sqrt_alpha_prod.unsqueeze(-1)

        sqrt_one_minus_alpha_prod = (1 - alphas_cumprod[timesteps]) ** 0.5
        sqrt_one_minus_alpha_prod = sqrt_one_minus_alpha_prod.flatten()
        while len(sqrt_one_minus_alpha_prod.shape) < len(sample.shape):
            sqrt_one_minus_alpha_prod = sqrt_one_minus_alpha_prod.unsqueeze(-1)

        velocity = sqrt_alpha_prod * noise - sqrt_one_minus_alpha_prod * sample
        return velocity

    def __len__(self):
        return self.config.num_train_timesteps

    # Copied from diffusers.schedulers.scheduling_ddpm.DDPMScheduler.previous_timestep
    def previous_timestep(self, timestep):
        if self.custom_timesteps:
            index = (self.timesteps == timestep).nonzero(as_tuple=True)[0][0]
            if index == self.timesteps.shape[0] - 1:
                prev_t = torch.tensor(-1)
            else:
                prev_t = self.timesteps[index + 1]
        else:
            num_inference_steps = (
                self.num_inference_steps if self.num_inference_steps else self.config.num_train_timesteps
            )
            prev_t = timestep - self.config.num_train_timesteps // num_inference_steps

        return prev_t