# Copyright 2024 Shuchen Xue, etc. in University of Chinese Academy of Sciences 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: check https://arxiv.org/abs/2309.05019
# 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 Callable, List, Optional, Tuple, Union
import numpy as np
import torch
from ..configuration_utils import ConfigMixin, register_to_config
from ..utils import deprecate, is_scipy_available
from ..utils.torch_utils import randn_tensor
from .scheduling_utils import KarrasDiffusionSchedulers, SchedulerMixin, SchedulerOutput
if is_scipy_available():
import scipy.stats
# 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)
class SASolverScheduler(SchedulerMixin, ConfigMixin):
"""
`SASolverScheduler` is a fast dedicated high-order solver for diffusion SDEs.
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`.
predictor_order (`int`, defaults to 2):
The predictor order which can be `1` or `2` or `3` or '4'. It is recommended to use `predictor_order=2` for
guided sampling, and `predictor_order=3` for unconditional sampling.
corrector_order (`int`, defaults to 2):
The corrector order which can be `1` or `2` or `3` or '4'. It is recommended to use `corrector_order=2` for
guided sampling, and `corrector_order=3` for unconditional sampling.
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).
tau_func (`Callable`, *optional*):
Stochasticity during the sampling. Default in init is `lambda t: 1 if t >= 200 and t <= 800 else 0`.
SA-Solver will sample from vanilla diffusion ODE if tau_func is set to `lambda t: 0`. SA-Solver will sample
from vanilla diffusion SDE if tau_func is set to `lambda t: 1`. For more details, please check
https://arxiv.org/abs/2309.05019
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` and
`algorithm_type="dpmsolver++"`.
algorithm_type (`str`, defaults to `data_prediction`):
Algorithm type for the solver; can be `data_prediction` or `noise_prediction`. It is recommended to use
`data_prediction` with `solver_order=2` for guided sampling like in Stable Diffusion.
lower_order_final (`bool`, defaults to `True`):
Whether to use lower-order solvers in the final steps. Default = True.
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}.
use_exponential_sigmas (`bool`, *optional*, defaults to `False`):
Whether to use exponential sigmas for step sizes in the noise schedule during the sampling process.
use_beta_sigmas (`bool`, *optional*, defaults to `False`):
Whether to use beta sigmas for step sizes in the noise schedule during the sampling process. Refer to [Beta
Sampling is All You Need](https://huggingface.co/papers/2407.12173) for more information.
lambda_min_clipped (`float`, defaults to `-inf`):
Clipping threshold for the minimum value of `lambda(t)` for numerical stability. This is critical for the
cosine (`squaredcos_cap_v2`) noise schedule.
variance_type (`str`, *optional*):
Set to "learned" or "learned_range" for diffusion models that predict variance. If set, the model's output
contains the predicted Gaussian variance.
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, as required by some model families.
"""
_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[Union[np.ndarray, List[float]]] = None,
predictor_order: int = 2,
corrector_order: int = 2,
prediction_type: str = "epsilon",
tau_func: Optional[Callable] = None,
thresholding: bool = False,
dynamic_thresholding_ratio: float = 0.995,
sample_max_value: float = 1.0,
algorithm_type: str = "data_prediction",
lower_order_final: bool = True,
use_karras_sigmas: Optional[bool] = False,
use_exponential_sigmas: Optional[bool] = False,
use_beta_sigmas: Optional[bool] = False,
use_flow_sigmas: Optional[bool] = False,
flow_shift: Optional[float] = 1.0,
lambda_min_clipped: float = -float("inf"),
variance_type: Optional[str] = None,
timestep_spacing: str = "linspace",
steps_offset: int = 0,
):
if self.config.use_beta_sigmas and not is_scipy_available():
raise ImportError("Make sure to install scipy if you want to use beta sigmas.")
if sum([self.config.use_beta_sigmas, self.config.use_exponential_sigmas, self.config.use_karras_sigmas]) > 1:
raise ValueError(
"Only one of `config.use_beta_sigmas`, `config.use_exponential_sigmas`, `config.use_karras_sigmas` can be used."
)
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__}")
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
if algorithm_type not in ["data_prediction", "noise_prediction"]:
raise NotImplementedError(f"{algorithm_type} 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.timestep_list = [None] * max(predictor_order, corrector_order - 1)
self.model_outputs = [None] * max(predictor_order, corrector_order - 1)
if tau_func is None:
self.tau_func = lambda t: 1 if t >= 200 and t <= 800 else 0
else:
self.tau_func = tau_func
self.predict_x0 = algorithm_type == "data_prediction"
self.lower_order_nums = 0
self.last_sample = None
self._step_index = None
self._begin_index = None
self.sigmas = self.sigmas.to("cpu") # to avoid too much CPU/GPU communication
@property
def step_index(self):
"""
The index counter for current timestep. It will increase 1 after each scheduler step.
"""
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 set_timesteps(self, num_inference_steps: int = None, 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.
"""
# Clipping the minimum of all lambda(t) for numerical stability.
# This is critical for cosine (squaredcos_cap_v2) noise schedule.
clipped_idx = torch.searchsorted(torch.flip(self.lambda_t, [0]), self.config.lambda_min_clipped)
last_timestep = ((self.config.num_train_timesteps - clipped_idx).numpy()).item()
# "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, last_timestep - 1, num_inference_steps + 1).round()[::-1][:-1].copy().astype(np.int64)
)
elif self.config.timestep_spacing == "leading":
step_ratio = last_timestep // (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(last_timestep, 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)
log_sigmas = np.log(sigmas)
if self.config.use_karras_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)
elif self.config.use_exponential_sigmas:
sigmas = np.flip(sigmas).copy()
sigmas = self._convert_to_exponential(in_sigmas=sigmas, num_inference_steps=num_inference_steps)
timesteps = np.array([self._sigma_to_t(sigma, log_sigmas) for sigma in sigmas])
sigmas = np.concatenate([sigmas, sigmas[-1:]]).astype(np.float32)
elif self.config.use_beta_sigmas:
sigmas = np.flip(sigmas).copy()
sigmas = self._convert_to_beta(in_sigmas=sigmas, num_inference_steps=num_inference_steps)
timesteps = np.array([self._sigma_to_t(sigma, log_sigmas) for sigma in sigmas])
sigmas = np.concatenate([sigmas, sigmas[-1:]]).astype(np.float32)
elif self.config.use_flow_sigmas:
alphas = np.linspace(1, 1 / self.config.num_train_timesteps, num_inference_steps + 1)
sigmas = 1.0 - alphas
sigmas = np.flip(self.config.flow_shift * sigmas / (1 + (self.config.flow_shift - 1) * sigmas))[:-1]
timesteps = (sigmas * self.config.num_train_timesteps).copy()
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,
] * max(self.config.predictor_order, self.config.corrector_order - 1)
self.lower_order_nums = 0
self.last_sample = None
# add an index counter for schedulers that allow duplicated timesteps
self._step_index = None
self._begin_index = None
self.sigmas = self.sigmas.to("cpu") # to avoid too much CPU/GPU communication
# 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
# 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):
if self.config.use_flow_sigmas:
alpha_t = 1 - sigma
sigma_t = sigma
else:
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.Tensor, num_inference_steps) -> torch.Tensor:
"""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
# Copied from diffusers.schedulers.scheduling_euler_discrete.EulerDiscreteScheduler._convert_to_exponential
def _convert_to_exponential(self, in_sigmas: torch.Tensor, num_inference_steps: int) -> torch.Tensor:
"""Constructs an exponential noise schedule."""
# 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()
sigmas = np.exp(np.linspace(math.log(sigma_max), math.log(sigma_min), num_inference_steps))
return sigmas
# Copied from diffusers.schedulers.scheduling_euler_discrete.EulerDiscreteScheduler._convert_to_beta
def _convert_to_beta(
self, in_sigmas: torch.Tensor, num_inference_steps: int, alpha: float = 0.6, beta: float = 0.6
) -> torch.Tensor:
"""From "Beta Sampling is All You Need" [arXiv:2407.12173] (Lee et. al, 2024)"""
# 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()
sigmas = np.array(
[
sigma_min + (ppf * (sigma_max - sigma_min))
for ppf in [
scipy.stats.beta.ppf(timestep, alpha, beta)
for timestep in 1 - np.linspace(0, 1, num_inference_steps)
]
]
)
return sigmas
def convert_model_output(
self,
model_output: torch.Tensor,
*args,
sample: torch.Tensor = None,
**kwargs,
) -> torch.Tensor:
"""
Convert the model output to the corresponding type the data_prediction/noise_prediction algorithm needs.
Noise_prediction is designed to discretize an integral of the noise prediction model, and data_prediction is
designed to discretize an integral of the data prediction model.
<Tip>
The algorithm and model type are decoupled. You can use either data_prediction or noise_prediction for both
noise prediction and data prediction models.
</Tip>
Args:
model_output (`torch.Tensor`):
The direct output from the learned diffusion model.
sample (`torch.Tensor`):
A current instance of a sample created by the diffusion process.
Returns:
`torch.Tensor`:
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)
# SA-Solver_data_prediction needs to solve an integral of the data prediction model.
if self.config.algorithm_type in ["data_prediction"]:
if self.config.prediction_type == "epsilon":
# SA-Solver only needs the "mean" output.
if self.config.variance_type in ["learned", "learned_range"]:
model_output = model_output[:, :3]
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
elif self.config.prediction_type == "flow_prediction":
sigma_t = self.sigmas[self.step_index]
x0_pred = sample - sigma_t * model_output
else:
raise ValueError(
f"prediction_type given as {self.config.prediction_type} must be one of `epsilon`, `sample`, "
"`v_prediction`, or `flow_prediction` for the SASolverScheduler."
)
if self.config.thresholding:
x0_pred = self._threshold_sample(x0_pred)
return x0_pred
# SA-Solver_noise_prediction needs to solve an integral of the noise prediction model.
elif self.config.algorithm_type in ["noise_prediction"]:
if self.config.prediction_type == "epsilon":
# SA-Solver only needs the "mean" output.
if self.config.variance_type in ["learned", "learned_range"]:
epsilon = model_output[:, :3]
else:
epsilon = model_output
elif self.config.prediction_type == "sample":
epsilon = (sample - alpha_t * model_output) / sigma_t
elif self.config.prediction_type == "v_prediction":
epsilon = alpha_t * model_output + sigma_t * sample
else:
raise ValueError(
f"prediction_type given as {self.config.prediction_type} must be one of `epsilon`, `sample`, or"
" `v_prediction` for the SASolverScheduler."
)
if self.config.thresholding:
alpha_t, sigma_t = self.alpha_t[timestep], self.sigma_t[timestep]
x0_pred = (sample - sigma_t * epsilon) / alpha_t
x0_pred = self._threshold_sample(x0_pred)
epsilon = (sample - alpha_t * x0_pred) / sigma_t
return epsilon
def get_coefficients_exponential_negative(self, order, interval_start, interval_end):
"""
Calculate the integral of exp(-x) * x^order dx from interval_start to interval_end
"""
assert order in [0, 1, 2, 3], "order is only supported for 0, 1, 2 and 3"
if order == 0:
return torch.exp(-interval_end) * (torch.exp(interval_end - interval_start) - 1)
elif order == 1:
return torch.exp(-interval_end) * (
(interval_start + 1) * torch.exp(interval_end - interval_start) - (interval_end + 1)
)
elif order == 2:
return torch.exp(-interval_end) * (
(interval_start**2 + 2 * interval_start + 2) * torch.exp(interval_end - interval_start)
- (interval_end**2 + 2 * interval_end + 2)
)
elif order == 3:
return torch.exp(-interval_end) * (
(interval_start**3 + 3 * interval_start**2 + 6 * interval_start + 6)
* torch.exp(interval_end - interval_start)
- (interval_end**3 + 3 * interval_end**2 + 6 * interval_end + 6)
)
def get_coefficients_exponential_positive(self, order, interval_start, interval_end, tau):
"""
Calculate the integral of exp(x(1+tau^2)) * x^order dx from interval_start to interval_end
"""
assert order in [0, 1, 2, 3], "order is only supported for 0, 1, 2 and 3"
# after change of variable(cov)
interval_end_cov = (1 + tau**2) * interval_end
interval_start_cov = (1 + tau**2) * interval_start
if order == 0:
return (
torch.exp(interval_end_cov) * (1 - torch.exp(-(interval_end_cov - interval_start_cov))) / (1 + tau**2)
)
elif order == 1:
return (
torch.exp(interval_end_cov)
* (
(interval_end_cov - 1)
- (interval_start_cov - 1) * torch.exp(-(interval_end_cov - interval_start_cov))
)
/ ((1 + tau**2) ** 2)
)
elif order == 2:
return (
torch.exp(interval_end_cov)
* (
(interval_end_cov**2 - 2 * interval_end_cov + 2)
- (interval_start_cov**2 - 2 * interval_start_cov + 2)
* torch.exp(-(interval_end_cov - interval_start_cov))
)
/ ((1 + tau**2) ** 3)
)
elif order == 3:
return (
torch.exp(interval_end_cov)
* (
(interval_end_cov**3 - 3 * interval_end_cov**2 + 6 * interval_end_cov - 6)
- (interval_start_cov**3 - 3 * interval_start_cov**2 + 6 * interval_start_cov - 6)
* torch.exp(-(interval_end_cov - interval_start_cov))
)
/ ((1 + tau**2) ** 4)
)
def lagrange_polynomial_coefficient(self, order, lambda_list):
"""
Calculate the coefficient of lagrange polynomial
"""
assert order in [0, 1, 2, 3]
assert order == len(lambda_list) - 1
if order == 0:
return [[1]]
elif order == 1:
return [
[
1 / (lambda_list[0] - lambda_list[1]),
-lambda_list[1] / (lambda_list[0] - lambda_list[1]),
],
[
1 / (lambda_list[1] - lambda_list[0]),
-lambda_list[0] / (lambda_list[1] - lambda_list[0]),
],
]
elif order == 2:
denominator1 = (lambda_list[0] - lambda_list[1]) * (lambda_list[0] - lambda_list[2])
denominator2 = (lambda_list[1] - lambda_list[0]) * (lambda_list[1] - lambda_list[2])
denominator3 = (lambda_list[2] - lambda_list[0]) * (lambda_list[2] - lambda_list[1])
return [
[
1 / denominator1,
(-lambda_list[1] - lambda_list[2]) / denominator1,
lambda_list[1] * lambda_list[2] / denominator1,
],
[
1 / denominator2,
(-lambda_list[0] - lambda_list[2]) / denominator2,
lambda_list[0] * lambda_list[2] / denominator2,
],
[
1 / denominator3,
(-lambda_list[0] - lambda_list[1]) / denominator3,
lambda_list[0] * lambda_list[1] / denominator3,
],
]
elif order == 3:
denominator1 = (
(lambda_list[0] - lambda_list[1])
* (lambda_list[0] - lambda_list[2])
* (lambda_list[0] - lambda_list[3])
)
denominator2 = (
(lambda_list[1] - lambda_list[0])
* (lambda_list[1] - lambda_list[2])
* (lambda_list[1] - lambda_list[3])
)
denominator3 = (
(lambda_list[2] - lambda_list[0])
* (lambda_list[2] - lambda_list[1])
* (lambda_list[2] - lambda_list[3])
)
denominator4 = (
(lambda_list[3] - lambda_list[0])
* (lambda_list[3] - lambda_list[1])
* (lambda_list[3] - lambda_list[2])
)
return [
[
1 / denominator1,
(-lambda_list[1] - lambda_list[2] - lambda_list[3]) / denominator1,
(
lambda_list[1] * lambda_list[2]
+ lambda_list[1] * lambda_list[3]
+ lambda_list[2] * lambda_list[3]
)
/ denominator1,
(-lambda_list[1] * lambda_list[2] * lambda_list[3]) / denominator1,
],
[
1 / denominator2,
(-lambda_list[0] - lambda_list[2] - lambda_list[3]) / denominator2,
(
lambda_list[0] * lambda_list[2]
+ lambda_list[0] * lambda_list[3]
+ lambda_list[2] * lambda_list[3]
)
/ denominator2,
(-lambda_list[0] * lambda_list[2] * lambda_list[3]) / denominator2,
],
[
1 / denominator3,
(-lambda_list[0] - lambda_list[1] - lambda_list[3]) / denominator3,
(
lambda_list[0] * lambda_list[1]
+ lambda_list[0] * lambda_list[3]
+ lambda_list[1] * lambda_list[3]
)
/ denominator3,
(-lambda_list[0] * lambda_list[1] * lambda_list[3]) / denominator3,
],
[
1 / denominator4,
(-lambda_list[0] - lambda_list[1] - lambda_list[2]) / denominator4,
(
lambda_list[0] * lambda_list[1]
+ lambda_list[0] * lambda_list[2]
+ lambda_list[1] * lambda_list[2]
)
/ denominator4,
(-lambda_list[0] * lambda_list[1] * lambda_list[2]) / denominator4,
],
]
def get_coefficients_fn(self, order, interval_start, interval_end, lambda_list, tau):
assert order in [1, 2, 3, 4]
assert order == len(lambda_list), "the length of lambda list must be equal to the order"
coefficients = []
lagrange_coefficient = self.lagrange_polynomial_coefficient(order - 1, lambda_list)
for i in range(order):
coefficient = 0
for j in range(order):
if self.predict_x0:
coefficient += lagrange_coefficient[i][j] * self.get_coefficients_exponential_positive(
order - 1 - j, interval_start, interval_end, tau
)
else:
coefficient += lagrange_coefficient[i][j] * self.get_coefficients_exponential_negative(
order - 1 - j, interval_start, interval_end
)
coefficients.append(coefficient)
assert len(coefficients) == order, "the length of coefficients does not match the order"
return coefficients
def stochastic_adams_bashforth_update(
self,
model_output: torch.Tensor,
*args,
sample: torch.Tensor,
noise: torch.Tensor,
order: int,
tau: torch.Tensor,
**kwargs,
) -> torch.Tensor:
"""
One step for the SA-Predictor.
Args:
model_output (`torch.Tensor`):
The direct output from the learned diffusion model at the current timestep.
prev_timestep (`int`):
The previous discrete timestep in the diffusion chain.
sample (`torch.Tensor`):
A current instance of a sample created by the diffusion process.
order (`int`):
The order of SA-Predictor at this timestep.
Returns:
`torch.Tensor`:
The sample tensor at the previous timestep.
"""
prev_timestep = args[0] if len(args) > 0 else kwargs.pop("prev_timestep", None)
if sample is None:
if len(args) > 1:
sample = args[1]
else:
raise ValueError(" missing `sample` as a required keyward argument")
if noise is None:
if len(args) > 2:
noise = args[2]
else:
raise ValueError(" missing `noise` as a required keyward argument")
if order is None:
if len(args) > 3:
order = args[3]
else:
raise ValueError(" missing `order` as a required keyward argument")
if tau is None:
if len(args) > 4:
tau = args[4]
else:
raise ValueError(" missing `tau` as a required keyward argument")
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`",
)
model_output_list = self.model_outputs
sigma_t, sigma_s0 = (
self.sigmas[self.step_index + 1],
self.sigmas[self.step_index],
)
alpha_t, sigma_t = self._sigma_to_alpha_sigma_t(sigma_t)
alpha_s0, sigma_s0 = self._sigma_to_alpha_sigma_t(sigma_s0)
lambda_t = torch.log(alpha_t) - torch.log(sigma_t)
lambda_s0 = torch.log(alpha_s0) - torch.log(sigma_s0)
gradient_part = torch.zeros_like(sample)
h = lambda_t - lambda_s0
lambda_list = []
for i in range(order):
si = self.step_index - i
alpha_si, sigma_si = self._sigma_to_alpha_sigma_t(self.sigmas[si])
lambda_si = torch.log(alpha_si) - torch.log(sigma_si)
lambda_list.append(lambda_si)
gradient_coefficients = self.get_coefficients_fn(order, lambda_s0, lambda_t, lambda_list, tau)
x = sample
if self.predict_x0:
if (
order == 2
): ## if order = 2 we do a modification that does not influence the convergence order similar to unipc. Note: This is used only for few steps sampling.
# The added term is O(h^3). Empirically we find it will slightly improve the image quality.
# ODE case
# gradient_coefficients[0] += 1.0 * torch.exp(lambda_t) * (h ** 2 / 2 - (h - 1 + torch.exp(-h))) / (ns.marginal_lambda(t_prev_list[-1]) - ns.marginal_lambda(t_prev_list[-2]))
# gradient_coefficients[1] -= 1.0 * torch.exp(lambda_t) * (h ** 2 / 2 - (h - 1 + torch.exp(-h))) / (ns.marginal_lambda(t_prev_list[-1]) - ns.marginal_lambda(t_prev_list[-2]))
temp_sigma = self.sigmas[self.step_index - 1]
temp_alpha_s, temp_sigma_s = self._sigma_to_alpha_sigma_t(temp_sigma)
temp_lambda_s = torch.log(temp_alpha_s) - torch.log(temp_sigma_s)
gradient_coefficients[0] += (
1.0
* torch.exp((1 + tau**2) * lambda_t)
* (h**2 / 2 - (h * (1 + tau**2) - 1 + torch.exp((1 + tau**2) * (-h))) / ((1 + tau**2) ** 2))
/ (lambda_s0 - temp_lambda_s)
)
gradient_coefficients[1] -= (
1.0
* torch.exp((1 + tau**2) * lambda_t)
* (h**2 / 2 - (h * (1 + tau**2) - 1 + torch.exp((1 + tau**2) * (-h))) / ((1 + tau**2) ** 2))
/ (lambda_s0 - temp_lambda_s)
)
for i in range(order):
if self.predict_x0:
gradient_part += (
(1 + tau**2)
* sigma_t
* torch.exp(-(tau**2) * lambda_t)
* gradient_coefficients[i]
* model_output_list[-(i + 1)]
)
else:
gradient_part += -(1 + tau**2) * alpha_t * gradient_coefficients[i] * model_output_list[-(i + 1)]
if self.predict_x0:
noise_part = sigma_t * torch.sqrt(1 - torch.exp(-2 * tau**2 * h)) * noise
else:
noise_part = tau * sigma_t * torch.sqrt(torch.exp(2 * h) - 1) * noise
if self.predict_x0:
x_t = torch.exp(-(tau**2) * h) * (sigma_t / sigma_s0) * x + gradient_part + noise_part
else:
x_t = (alpha_t / alpha_s0) * x + gradient_part + noise_part
x_t = x_t.to(x.dtype)
return x_t
def stochastic_adams_moulton_update(
self,
this_model_output: torch.Tensor,
*args,
last_sample: torch.Tensor,
last_noise: torch.Tensor,
this_sample: torch.Tensor,
order: int,
tau: torch.Tensor,
**kwargs,
) -> torch.Tensor:
"""
One step for the SA-Corrector.
Args:
this_model_output (`torch.Tensor`):
The model outputs at `x_t`.
this_timestep (`int`):
The current timestep `t`.
last_sample (`torch.Tensor`):
The generated sample before the last predictor `x_{t-1}`.
this_sample (`torch.Tensor`):
The generated sample after the last predictor `x_{t}`.
order (`int`):
The order of SA-Corrector at this step.
Returns:
`torch.Tensor`:
The corrected sample tensor at the current timestep.
"""
this_timestep = args[0] if len(args) > 0 else kwargs.pop("this_timestep", None)
if last_sample is None:
if len(args) > 1:
last_sample = args[1]
else:
raise ValueError(" missing`last_sample` as a required keyward argument")
if last_noise is None:
if len(args) > 2:
last_noise = args[2]
else:
raise ValueError(" missing`last_noise` as a required keyward argument")
if this_sample is None:
if len(args) > 3:
this_sample = args[3]
else:
raise ValueError(" missing`this_sample` as a required keyward argument")
if order is None:
if len(args) > 4:
order = args[4]
else:
raise ValueError(" missing`order` as a required keyward argument")
if tau is None:
if len(args) > 5:
tau = args[5]
else:
raise ValueError(" missing`tau` as a required keyward argument")
if this_timestep is not None:
deprecate(
"this_timestep",
"1.0.0",
"Passing `this_timestep` is deprecated and has no effect as model output conversion is now handled via an internal counter `self.step_index`",
)
model_output_list = self.model_outputs
sigma_t, sigma_s0 = (
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)
lambda_t = torch.log(alpha_t) - torch.log(sigma_t)
lambda_s0 = torch.log(alpha_s0) - torch.log(sigma_s0)
gradient_part = torch.zeros_like(this_sample)
h = lambda_t - lambda_s0
lambda_list = []
for i in range(order):
si = self.step_index - i
alpha_si, sigma_si = self._sigma_to_alpha_sigma_t(self.sigmas[si])
lambda_si = torch.log(alpha_si) - torch.log(sigma_si)
lambda_list.append(lambda_si)
model_prev_list = model_output_list + [this_model_output]
gradient_coefficients = self.get_coefficients_fn(order, lambda_s0, lambda_t, lambda_list, tau)
x = last_sample
if self.predict_x0:
if (
order == 2
): ## if order = 2 we do a modification that does not influence the convergence order similar to UniPC. Note: This is used only for few steps sampling.
# The added term is O(h^3). Empirically we find it will slightly improve the image quality.
# ODE case
# gradient_coefficients[0] += 1.0 * torch.exp(lambda_t) * (h / 2 - (h - 1 + torch.exp(-h)) / h)
# gradient_coefficients[1] -= 1.0 * torch.exp(lambda_t) * (h / 2 - (h - 1 + torch.exp(-h)) / h)
gradient_coefficients[0] += (
1.0
* torch.exp((1 + tau**2) * lambda_t)
* (h / 2 - (h * (1 + tau**2) - 1 + torch.exp((1 + tau**2) * (-h))) / ((1 + tau**2) ** 2 * h))
)
gradient_coefficients[1] -= (
1.0
* torch.exp((1 + tau**2) * lambda_t)
* (h / 2 - (h * (1 + tau**2) - 1 + torch.exp((1 + tau**2) * (-h))) / ((1 + tau**2) ** 2 * h))
)
for i in range(order):
if self.predict_x0:
gradient_part += (
(1 + tau**2)
* sigma_t
* torch.exp(-(tau**2) * lambda_t)
* gradient_coefficients[i]
* model_prev_list[-(i + 1)]
)
else:
gradient_part += -(1 + tau**2) * alpha_t * gradient_coefficients[i] * model_prev_list[-(i + 1)]
if self.predict_x0:
noise_part = sigma_t * torch.sqrt(1 - torch.exp(-2 * tau**2 * h)) * last_noise
else:
noise_part = tau * sigma_t * torch.sqrt(torch.exp(2 * h) - 1) * last_noise
if self.predict_x0:
x_t = torch.exp(-(tau**2) * h) * (sigma_t / sigma_s0) * x + gradient_part + noise_part
else:
x_t = (alpha_t / alpha_s0) * x + gradient_part + noise_part
x_t = x_t.to(x.dtype)
return x_t
# Copied from diffusers.schedulers.scheduling_dpmsolver_multistep.DPMSolverMultistepScheduler.index_for_timestep
def index_for_timestep(self, timestep, schedule_timesteps=None):
if schedule_timesteps is None:
schedule_timesteps = self.timesteps
index_candidates = (schedule_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()
return step_index
# Copied from diffusers.schedulers.scheduling_dpmsolver_multistep.DPMSolverMultistepScheduler._init_step_index
def _init_step_index(self, timestep):
"""
Initialize the step_index counter for the scheduler.
"""
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
def step(
self,
model_output: torch.Tensor,
timestep: int,
sample: torch.Tensor,
generator=None,
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 SA-Solver.
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.
generator (`torch.Generator`, *optional*):
A random number generator.
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)
use_corrector = self.step_index > 0 and self.last_sample is not None
model_output_convert = self.convert_model_output(model_output, sample=sample)
if use_corrector:
current_tau = self.tau_func(self.timestep_list[-1])
sample = self.stochastic_adams_moulton_update(
this_model_output=model_output_convert,
last_sample=self.last_sample,
last_noise=self.last_noise,
this_sample=sample,
order=self.this_corrector_order,
tau=current_tau,
)
for i in range(max(self.config.predictor_order, self.config.corrector_order - 1) - 1):
self.model_outputs[i] = self.model_outputs[i + 1]
self.timestep_list[i] = self.timestep_list[i + 1]
self.model_outputs[-1] = model_output_convert
self.timestep_list[-1] = timestep
noise = randn_tensor(
model_output.shape,
generator=generator,
device=model_output.device,
dtype=model_output.dtype,
)
if self.config.lower_order_final:
this_predictor_order = min(self.config.predictor_order, len(self.timesteps) - self.step_index)
this_corrector_order = min(self.config.corrector_order, len(self.timesteps) - self.step_index + 1)
else:
this_predictor_order = self.config.predictor_order
this_corrector_order = self.config.corrector_order
self.this_predictor_order = min(this_predictor_order, self.lower_order_nums + 1) # warmup for multistep
self.this_corrector_order = min(this_corrector_order, self.lower_order_nums + 2) # warmup for multistep
assert self.this_predictor_order > 0
assert self.this_corrector_order > 0
self.last_sample = sample
self.last_noise = noise
current_tau = self.tau_func(self.timestep_list[-1])
prev_sample = self.stochastic_adams_bashforth_update(
model_output=model_output_convert,
sample=sample,
noise=noise,
order=self.this_predictor_order,
tau=current_tau,
)
if self.lower_order_nums < max(self.config.predictor_order, self.config.corrector_order - 1):
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.Tensor, *args, **kwargs) -> 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.
Returns:
`torch.Tensor`:
A scaled input sample.
"""
return 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
def __len__(self):
return self.config.num_train_timesteps