scheduling_deis_multistep.py

# Copyright 2023 FLAIR Lab 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,
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# See the License for the specific language governing permissions and
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# DISCLAIMER: check https://arxiv.org/abs/2204.13902 and https://github.com/qsh-zh/deis for more info
# The codebase is modified based on https://github.com/huggingface/diffusers/blob/main/src/diffusers/schedulers/scheduling_dpmsolver_multistep.py

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

import numpy as np
import torch

from ..configuration_utils import ConfigMixin, register_to_config
from ..utils import deprecate
from .scheduling_utils import KarrasDiffusionSchedulers, SchedulerMixin, SchedulerOutput


# 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_tranform_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)


class DEISMultistepScheduler(SchedulerMixin, ConfigMixin):
    """
    `DEISMultistepScheduler` is a fast high order solver for diffusion ordinary differential equations (ODEs).

    This model inherits from [`SchedulerMixin`] and [`ConfigMixin`]. Check the superclass documentation for the generic
    methods the library implements for all schedulers such as loading and saving.

    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`.
        solver_order (`int`, defaults to 2):
            The DEIS order which can be `1` or `2` or `3`. It is recommended to use `solver_order=2` for guided
            sampling, and `solver_order=3` for unconditional sampling.
        prediction_type (`str`, defaults to `epsilon`):
            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`.
        algorithm_type (`str`, defaults to `deis`):
            The algorithm type for the solver.
        lower_order_final (`bool`, defaults to `True`):
            Whether to use lower-order solvers in the final steps. Only valid for < 15 inference steps.
        use_karras_sigmas (`bool`, *optional*, defaults to `False`):
             Whether to use Karras sigmas for step sizes in the noise schedule during the sampling process. If `True`,
             the sigmas are determined according to a sequence of noise levels {σi}.
        timestep_spacing (`str`, defaults to `"linspace"`):
            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.
        steps_offset (`int`, defaults to 0):
            An offset added to the inference steps. You can use a combination of `offset=1` and
            `set_alpha_to_one=False` to make the last step use step 0 for the previous alpha product like in Stable
            Diffusion.
    """

    _compatibles = [e.name for e in KarrasDiffusionSchedulers]
    order = 1

    @register_to_config
    def __init__(
        self,
        num_train_timesteps: int = 1000,
        beta_start: float = 0.0001,
        beta_end: float = 0.02,
        beta_schedule: str = "linear",
        trained_betas: Optional[np.ndarray] = None,
        solver_order: int = 2,
        prediction_type: str = "epsilon",
        thresholding: bool = False,
        dynamic_thresholding_ratio: float = 0.995,
        sample_max_value: float = 1.0,
        algorithm_type: str = "deis",
        solver_type: str = "logrho",
        lower_order_final: bool = True,
        use_karras_sigmas: Optional[bool] = False,
        timestep_spacing: str = "linspace",
        steps_offset: int = 0,
    ):
        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} does is not implemented for {self.__class__}")

        self.alphas = 1.0 - self.betas
        self.alphas_cumprod = torch.cumprod(self.alphas, dim=0)
        # Currently we only support VP-type noise schedule
        self.alpha_t = torch.sqrt(self.alphas_cumprod)
        self.sigma_t = torch.sqrt(1 - self.alphas_cumprod)
        self.lambda_t = torch.log(self.alpha_t) - torch.log(self.sigma_t)
        self.sigmas = ((1 - self.alphas_cumprod) / self.alphas_cumprod) ** 0.5

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

        # settings for DEIS
        if algorithm_type not in ["deis"]:
            if algorithm_type in ["dpmsolver", "dpmsolver++"]:
                self.register_to_config(algorithm_type="deis")
            else:
                raise NotImplementedError(f"{algorithm_type} does is not implemented for {self.__class__}")

        if solver_type not in ["logrho"]:
            if solver_type in ["midpoint", "heun", "bh1", "bh2"]:
                self.register_to_config(solver_type="logrho")
            else:
                raise NotImplementedError(f"solver type {solver_type} does is not implemented for {self.__class__}")

        # setable values
        self.num_inference_steps = None
        timesteps = np.linspace(0, num_train_timesteps - 1, num_train_timesteps, dtype=np.float32)[::-1].copy()
        self.timesteps = torch.from_numpy(timesteps)
        self.model_outputs = [None] * solver_order
        self.lower_order_nums = 0
        self._step_index = None

    @property
    def step_index(self):
        """
        The index counter for current timestep. It will increae 1 after each scheduler step.
        """
        return self._step_index

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

        Args:
            num_inference_steps (`int`):
                The number of diffusion steps used when generating samples with a pre-trained model.
            device (`str` or `torch.device`, *optional*):
                The device to which the timesteps should be moved to. If `None`, the timesteps are not moved.
        """
        # "linspace", "leading", "trailing" corresponds to annotation of Table 2. of https://arxiv.org/abs/2305.08891
        if self.config.timestep_spacing == "linspace":
            timesteps = (
                np.linspace(0, self.config.num_train_timesteps - 1, num_inference_steps + 1)
                .round()[::-1][:-1]
                .copy()
                .astype(np.int64)
            )
        elif self.config.timestep_spacing == "leading":
            step_ratio = self.config.num_train_timesteps // (num_inference_steps + 1)
            # creates integer timesteps by multiplying by ratio
            # casting to int to avoid issues when num_inference_step is power of 3
            timesteps = (np.arange(0, num_inference_steps + 1) * step_ratio).round()[::-1][:-1].copy().astype(np.int64)
            timesteps += self.config.steps_offset
        elif self.config.timestep_spacing == "trailing":
            step_ratio = self.config.num_train_timesteps / num_inference_steps
            # creates integer timesteps by multiplying by ratio
            # casting to int to avoid issues when num_inference_step is power of 3
            timesteps = np.arange(self.config.num_train_timesteps, 0, -step_ratio).round().copy().astype(np.int64)
            timesteps -= 1
        else:
            raise ValueError(
                f"{self.config.timestep_spacing} is not supported. Please make sure to choose one of 'linspace', 'leading' or 'trailing'."
            )

        sigmas = np.array(((1 - self.alphas_cumprod) / self.alphas_cumprod) ** 0.5)
        if self.config.use_karras_sigmas:
            log_sigmas = np.log(sigmas)
            sigmas = np.flip(sigmas).copy()
            sigmas = self._convert_to_karras(in_sigmas=sigmas, num_inference_steps=num_inference_steps)
            timesteps = np.array([self._sigma_to_t(sigma, log_sigmas) for sigma in sigmas]).round()
            sigmas = np.concatenate([sigmas, sigmas[-1:]]).astype(np.float32)
        else:
            sigmas = np.interp(timesteps, np.arange(0, len(sigmas)), sigmas)
            sigma_last = ((1 - self.alphas_cumprod[0]) / self.alphas_cumprod[0]) ** 0.5
            sigmas = np.concatenate([sigmas, [sigma_last]]).astype(np.float32)

        self.sigmas = torch.from_numpy(sigmas)
        self.timesteps = torch.from_numpy(timesteps).to(device=device, dtype=torch.int64)

        self.num_inference_steps = len(timesteps)

        self.model_outputs = [
            None,
        ] * self.config.solver_order
        self.lower_order_nums = 0

        # add an index counter for schedulers that allow duplicated timesteps
        self._step_index = None

    # Copied from diffusers.schedulers.scheduling_ddpm.DDPMScheduler._threshold_sample
    def _threshold_sample(self, sample: torch.FloatTensor) -> torch.FloatTensor:
        """
        "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

    # Copied from diffusers.schedulers.scheduling_euler_discrete.EulerDiscreteScheduler._sigma_to_t
    def _sigma_to_t(self, sigma, log_sigmas):
        # get log sigma
        log_sigma = np.log(np.maximum(sigma, 1e-10))

        # get distribution
        dists = log_sigma - log_sigmas[:, np.newaxis]

        # get sigmas range
        low_idx = np.cumsum((dists >= 0), axis=0).argmax(axis=0).clip(max=log_sigmas.shape[0] - 2)
        high_idx = low_idx + 1

        low = log_sigmas[low_idx]
        high = log_sigmas[high_idx]

        # interpolate sigmas
        w = (low - log_sigma) / (low - high)
        w = np.clip(w, 0, 1)

        # transform interpolation to time range
        t = (1 - w) * low_idx + w * high_idx
        t = t.reshape(sigma.shape)
        return t

    # Copied from diffusers.schedulers.scheduling_dpmsolver_multistep.DPMSolverMultistepScheduler._sigma_to_alpha_sigma_t
    def _sigma_to_alpha_sigma_t(self, sigma):
        alpha_t = 1 / ((sigma**2 + 1) ** 0.5)
        sigma_t = sigma * alpha_t

        return alpha_t, sigma_t

    # Copied from diffusers.schedulers.scheduling_euler_discrete.EulerDiscreteScheduler._convert_to_karras
    def _convert_to_karras(self, in_sigmas: torch.FloatTensor, num_inference_steps) -> torch.FloatTensor:
        """Constructs the noise schedule of Karras et al. (2022)."""

        # Hack to make sure that other schedulers which copy this function don't break
        # TODO: Add this logic to the other schedulers
        if hasattr(self.config, "sigma_min"):
            sigma_min = self.config.sigma_min
        else:
            sigma_min = None

        if hasattr(self.config, "sigma_max"):
            sigma_max = self.config.sigma_max
        else:
            sigma_max = None

        sigma_min = sigma_min if sigma_min is not None else in_sigmas[-1].item()
        sigma_max = sigma_max if sigma_max is not None else in_sigmas[0].item()

        rho = 7.0  # 7.0 is the value used in the paper
        ramp = np.linspace(0, 1, num_inference_steps)
        min_inv_rho = sigma_min ** (1 / rho)
        max_inv_rho = sigma_max ** (1 / rho)
        sigmas = (max_inv_rho + ramp * (min_inv_rho - max_inv_rho)) ** rho
        return sigmas

    def convert_model_output(
        self,
        model_output: torch.FloatTensor,
        *args,
        sample: torch.FloatTensor = None,
        **kwargs,
    ) -> torch.FloatTensor:
        """
        Convert the model output to the corresponding type the DEIS algorithm needs.

        Args:
            model_output (`torch.FloatTensor`):
                The direct output from the learned diffusion model.
            timestep (`int`):
                The current discrete timestep in the diffusion chain.
            sample (`torch.FloatTensor`):
                A current instance of a sample created by the diffusion process.

        Returns:
            `torch.FloatTensor`:
                The converted model output.
        """
        timestep = args[0] if len(args) > 0 else kwargs.pop("timestep", None)
        if sample is None:
            if len(args) > 1:
                sample = args[1]
            else:
                raise ValueError("missing `sample` as a required keyward argument")
        if timestep is not None:
            deprecate(
                "timesteps",
                "1.0.0",
                "Passing `timesteps` is deprecated and has no effect as model output conversion is now handled via an internal counter `self.step_index`",
            )

        sigma = self.sigmas[self.step_index]
        alpha_t, sigma_t = self._sigma_to_alpha_sigma_t(sigma)
        if self.config.prediction_type == "epsilon":
            x0_pred = (sample - sigma_t * model_output) / alpha_t
        elif self.config.prediction_type == "sample":
            x0_pred = model_output
        elif self.config.prediction_type == "v_prediction":
            x0_pred = alpha_t * sample - sigma_t * model_output
        else:
            raise ValueError(
                f"prediction_type given as {self.config.prediction_type} must be one of `epsilon`, `sample`, or"
                " `v_prediction` for the DEISMultistepScheduler."
            )

        if self.config.thresholding:
            x0_pred = self._threshold_sample(x0_pred)

        if self.config.algorithm_type == "deis":
            return (sample - alpha_t * x0_pred) / sigma_t
        else:
            raise NotImplementedError("only support log-rho multistep deis now")

    def deis_first_order_update(
        self,
        model_output: torch.FloatTensor,
        *args,
        sample: torch.FloatTensor = None,
        **kwargs,
    ) -> torch.FloatTensor:
        """
        One step for the first-order DEIS (equivalent to DDIM).

        Args:
            model_output (`torch.FloatTensor`):
                The direct output from the learned diffusion model.
            timestep (`int`):
                The current discrete timestep in the diffusion chain.
            prev_timestep (`int`):
                The previous discrete timestep in the diffusion chain.
            sample (`torch.FloatTensor`):
                A current instance of a sample created by the diffusion process.

        Returns:
            `torch.FloatTensor`:
                The sample tensor at the previous timestep.
        """
        timestep = args[0] if len(args) > 0 else kwargs.pop("timestep", None)
        prev_timestep = args[1] if len(args) > 1 else kwargs.pop("prev_timestep", None)
        if sample is None:
            if len(args) > 2:
                sample = args[2]
            else:
                raise ValueError(" missing `sample` as a required keyward argument")
        if timestep is not None:
            deprecate(
                "timesteps",
                "1.0.0",
                "Passing `timesteps` is deprecated and has no effect as model output conversion is now handled via an internal counter `self.step_index`",
            )

        if prev_timestep is not None:
            deprecate(
                "prev_timestep",
                "1.0.0",
                "Passing `prev_timestep` is deprecated and has no effect as model output conversion is now handled via an internal counter `self.step_index`",
            )

        sigma_t, sigma_s = self.sigmas[self.step_index + 1], self.sigmas[self.step_index]
        alpha_t, sigma_t = self._sigma_to_alpha_sigma_t(sigma_t)
        alpha_s, sigma_s = self._sigma_to_alpha_sigma_t(sigma_s)
        lambda_t = torch.log(alpha_t) - torch.log(sigma_t)
        lambda_s = torch.log(alpha_s) - torch.log(sigma_s)

        h = lambda_t - lambda_s
        if self.config.algorithm_type == "deis":
            x_t = (alpha_t / alpha_s) * sample - (sigma_t * (torch.exp(h) - 1.0)) * model_output
        else:
            raise NotImplementedError("only support log-rho multistep deis now")
        return x_t

    def multistep_deis_second_order_update(
        self,
        model_output_list: List[torch.FloatTensor],
        *args,
        sample: torch.FloatTensor = None,
        **kwargs,
    ) -> torch.FloatTensor:
        """
        One step for the second-order multistep DEIS.

        Args:
            model_output_list (`List[torch.FloatTensor]`):
                The direct outputs from learned diffusion model at current and latter timesteps.
            sample (`torch.FloatTensor`):
                A current instance of a sample created by the diffusion process.

        Returns:
            `torch.FloatTensor`:
                The sample tensor at the previous timestep.
        """
        timestep_list = args[0] if len(args) > 0 else kwargs.pop("timestep_list", None)
        prev_timestep = args[1] if len(args) > 1 else kwargs.pop("prev_timestep", None)
        if sample is None:
            if len(args) > 2:
                sample = args[2]
            else:
                raise ValueError(" missing `sample` as a required keyward argument")
        if timestep_list is not None:
            deprecate(
                "timestep_list",
                "1.0.0",
                "Passing `timestep_list` is deprecated and has no effect as model output conversion is now handled via an internal counter `self.step_index`",
            )

        if prev_timestep is not None:
            deprecate(
                "prev_timestep",
                "1.0.0",
                "Passing `prev_timestep` is deprecated and has no effect as model output conversion is now handled via an internal counter `self.step_index`",
            )

        sigma_t, sigma_s0, sigma_s1 = (
            self.sigmas[self.step_index + 1],
            self.sigmas[self.step_index],
            self.sigmas[self.step_index - 1],
        )

        alpha_t, sigma_t = self._sigma_to_alpha_sigma_t(sigma_t)
        alpha_s0, sigma_s0 = self._sigma_to_alpha_sigma_t(sigma_s0)
        alpha_s1, sigma_s1 = self._sigma_to_alpha_sigma_t(sigma_s1)

        m0, m1 = model_output_list[-1], model_output_list[-2]

        rho_t, rho_s0, rho_s1 = sigma_t / alpha_t, sigma_s0 / alpha_s0, sigma_s1 / alpha_s1

        if self.config.algorithm_type == "deis":

            def ind_fn(t, b, c):
                # Integrate[(log(t) - log(c)) / (log(b) - log(c)), {t}]
                return t * (-np.log(c) + np.log(t) - 1) / (np.log(b) - np.log(c))

            coef1 = ind_fn(rho_t, rho_s0, rho_s1) - ind_fn(rho_s0, rho_s0, rho_s1)
            coef2 = ind_fn(rho_t, rho_s1, rho_s0) - ind_fn(rho_s0, rho_s1, rho_s0)

            x_t = alpha_t * (sample / alpha_s0 + coef1 * m0 + coef2 * m1)
            return x_t
        else:
            raise NotImplementedError("only support log-rho multistep deis now")

    def multistep_deis_third_order_update(
        self,
        model_output_list: List[torch.FloatTensor],
        *args,
        sample: torch.FloatTensor = None,
        **kwargs,
    ) -> torch.FloatTensor:
        """
        One step for the third-order multistep DEIS.

        Args:
            model_output_list (`List[torch.FloatTensor]`):
                The direct outputs from learned diffusion model at current and latter timesteps.
            sample (`torch.FloatTensor`):
                A current instance of a sample created by diffusion process.

        Returns:
            `torch.FloatTensor`:
                The sample tensor at the previous timestep.
        """

        timestep_list = args[0] if len(args) > 0 else kwargs.pop("timestep_list", None)
        prev_timestep = args[1] if len(args) > 1 else kwargs.pop("prev_timestep", None)
        if sample is None:
            if len(args) > 2:
                sample = args[2]
            else:
                raise ValueError(" missing`sample` as a required keyward argument")
        if timestep_list is not None:
            deprecate(
                "timestep_list",
                "1.0.0",
                "Passing `timestep_list` is deprecated and has no effect as model output conversion is now handled via an internal counter `self.step_index`",
            )

        if prev_timestep is not None:
            deprecate(
                "prev_timestep",
                "1.0.0",
                "Passing `prev_timestep` is deprecated and has no effect as model output conversion is now handled via an internal counter `self.step_index`",
            )

        sigma_t, sigma_s0, sigma_s1, sigma_s2 = (
            self.sigmas[self.step_index + 1],
            self.sigmas[self.step_index],
            self.sigmas[self.step_index - 1],
            self.sigmas[self.step_index - 2],
        )

        alpha_t, sigma_t = self._sigma_to_alpha_sigma_t(sigma_t)
        alpha_s0, sigma_s0 = self._sigma_to_alpha_sigma_t(sigma_s0)
        alpha_s1, sigma_s1 = self._sigma_to_alpha_sigma_t(sigma_s1)
        alpha_s2, sigma_s2 = self._sigma_to_alpha_sigma_t(sigma_s2)

        m0, m1, m2 = model_output_list[-1], model_output_list[-2], model_output_list[-3]

        rho_t, rho_s0, rho_s1, rho_s2 = (
            sigma_t / alpha_t,
            sigma_s0 / alpha_s0,
            sigma_s1 / alpha_s1,
            sigma_s2 / alpha_s2,
        )

        if self.config.algorithm_type == "deis":

            def ind_fn(t, b, c, d):
                # Integrate[(log(t) - log(c))(log(t) - log(d)) / (log(b) - log(c))(log(b) - log(d)), {t}]
                numerator = t * (
                    np.log(c) * (np.log(d) - np.log(t) + 1)
                    - np.log(d) * np.log(t)
                    + np.log(d)
                    + np.log(t) ** 2
                    - 2 * np.log(t)
                    + 2
                )
                denominator = (np.log(b) - np.log(c)) * (np.log(b) - np.log(d))
                return numerator / denominator

            coef1 = ind_fn(rho_t, rho_s0, rho_s1, rho_s2) - ind_fn(rho_s0, rho_s0, rho_s1, rho_s2)
            coef2 = ind_fn(rho_t, rho_s1, rho_s2, rho_s0) - ind_fn(rho_s0, rho_s1, rho_s2, rho_s0)
            coef3 = ind_fn(rho_t, rho_s2, rho_s0, rho_s1) - ind_fn(rho_s0, rho_s2, rho_s0, rho_s1)

            x_t = alpha_t * (sample / alpha_s0 + coef1 * m0 + coef2 * m1 + coef3 * m2)

            return x_t
        else:
            raise NotImplementedError("only support log-rho multistep deis now")

    def _init_step_index(self, timestep):
        if isinstance(timestep, torch.Tensor):
            timestep = timestep.to(self.timesteps.device)

        index_candidates = (self.timesteps == timestep).nonzero()

        if len(index_candidates) == 0:
            step_index = len(self.timesteps) - 1
        # 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)
        elif len(index_candidates) > 1:
            step_index = index_candidates[1].item()
        else:
            step_index = index_candidates[0].item()

        self._step_index = step_index

    def step(
        self,
        model_output: torch.FloatTensor,
        timestep: int,
        sample: torch.FloatTensor,
        return_dict: bool = True,
    ) -> Union[SchedulerOutput, Tuple]:
        """
        Predict the sample from the previous timestep by reversing the SDE. This function propagates the sample with
        the multistep DEIS.

        Args:
            model_output (`torch.FloatTensor`):
                The direct output from learned diffusion model.
            timestep (`float`):
                The current discrete timestep in the diffusion chain.
            sample (`torch.FloatTensor`):
                A current instance of a sample created by the diffusion process.
            return_dict (`bool`):
                Whether or not to return a [`~schedulers.scheduling_utils.SchedulerOutput`] or `tuple`.

        Returns:
            [`~schedulers.scheduling_utils.SchedulerOutput`] or `tuple`:
                If return_dict is `True`, [`~schedulers.scheduling_utils.SchedulerOutput`] 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)

        lower_order_final = (
            (self.step_index == len(self.timesteps) - 1) and self.config.lower_order_final and len(self.timesteps) < 15
        )
        lower_order_second = (
            (self.step_index == len(self.timesteps) - 2) and self.config.lower_order_final and len(self.timesteps) < 15
        )

        model_output = self.convert_model_output(model_output, sample=sample)
        for i in range(self.config.solver_order - 1):
            self.model_outputs[i] = self.model_outputs[i + 1]
        self.model_outputs[-1] = model_output

        if self.config.solver_order == 1 or self.lower_order_nums < 1 or lower_order_final:
            prev_sample = self.deis_first_order_update(model_output, sample=sample)
        elif self.config.solver_order == 2 or self.lower_order_nums < 2 or lower_order_second:
            prev_sample = self.multistep_deis_second_order_update(self.model_outputs, sample=sample)
        else:
            prev_sample = self.multistep_deis_third_order_update(self.model_outputs, sample=sample)

        if self.lower_order_nums < self.config.solver_order:
            self.lower_order_nums += 1

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

        if not return_dict:
            return (prev_sample,)

        return SchedulerOutput(prev_sample=prev_sample)

    def scale_model_input(self, sample: torch.FloatTensor, *args, **kwargs) -> torch.FloatTensor:
        """
        Ensures interchangeability with schedulers that need to scale the denoising model input depending on the
        current timestep.

        Args:
            sample (`torch.FloatTensor`):
                The input sample.

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

    # Copied from diffusers.schedulers.scheduling_dpmsolver_multistep.DPMSolverMultistepScheduler.add_noise
    def add_noise(
        self,
        original_samples: torch.FloatTensor,
        noise: torch.FloatTensor,
        timesteps: torch.IntTensor,
    ) -> torch.FloatTensor:
        # Make sure sigmas and timesteps have the same device and dtype as original_samples
        sigmas = self.sigmas.to(device=original_samples.device, dtype=original_samples.dtype)
        if original_samples.device.type == "mps" and torch.is_floating_point(timesteps):
            # mps does not support float64
            schedule_timesteps = self.timesteps.to(original_samples.device, dtype=torch.float32)
            timesteps = timesteps.to(original_samples.device, dtype=torch.float32)
        else:
            schedule_timesteps = self.timesteps.to(original_samples.device)
            timesteps = timesteps.to(original_samples.device)

        step_indices = []
        for timestep in timesteps:
            index_candidates = (schedule_timesteps == timestep).nonzero()
            if len(index_candidates) == 0:
                step_index = len(schedule_timesteps) - 1
            elif len(index_candidates) > 1:
                step_index = index_candidates[1].item()
            else:
                step_index = index_candidates[0].item()
            step_indices.append(step_index)

        sigma = sigmas[step_indices].flatten()
        while len(sigma.shape) < len(original_samples.shape):
            sigma = sigma.unsqueeze(-1)

        alpha_t, sigma_t = self._sigma_to_alpha_sigma_t(sigma)
        noisy_samples = alpha_t * original_samples + sigma_t * noise
        return noisy_samples

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