# Copyright 2023 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 from ..utils.torch_utils import randn_tensor 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 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}. 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. 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[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, lambda_min_clipped: float = -float("inf"), variance_type: Optional[str] = None, 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 if algorithm_type not in ["data_prediction", "noise_prediction"]: raise NotImplementedError(f"{algorithm_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.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.sigmas.to("cpu") # to avoid too much CPU/GPU communication @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 = 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) 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, ] * 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.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.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 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. The algorithm and model type are decoupled. You can use either data_prediction or noise_prediction for both noise prediction and data prediction models. Args: model_output (`torch.FloatTensor`): The direct output from the learned diffusion model. 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) # 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 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: 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.FloatTensor, *args, sample: torch.FloatTensor, noise: torch.FloatTensor, order: int, tau: torch.FloatTensor, **kwargs, ) -> torch.FloatTensor: """ One step for the SA-Predictor. Args: model_output (`torch.FloatTensor`): 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.FloatTensor`): A current instance of a sample created by the diffusion process. order (`int`): The order of SA-Predictor at this timestep. Returns: `torch.FloatTensor`: 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.FloatTensor, *args, last_sample: torch.FloatTensor, last_noise: torch.FloatTensor, this_sample: torch.FloatTensor, order: int, tau: torch.FloatTensor, **kwargs, ) -> torch.FloatTensor: """ One step for the SA-Corrector. Args: this_model_output (`torch.FloatTensor`): The model outputs at `x_t`. this_timestep (`int`): The current timestep `t`. last_sample (`torch.FloatTensor`): The generated sample before the last predictor `x_{t-1}`. this_sample (`torch.FloatTensor`): The generated sample after the last predictor `x_{t}`. order (`int`): The order of SA-Corrector at this step. Returns: `torch.FloatTensor`: 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 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, 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.FloatTensor`): The direct output from 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. 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.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_ddpm.DDPMScheduler.add_noise def add_noise( self, original_samples: torch.FloatTensor, noise: torch.FloatTensor, timesteps: torch.IntTensor, ) -> torch.FloatTensor: # Make sure alphas_cumprod and timestep have same device and dtype as original_samples alphas_cumprod = self.alphas_cumprod.to(device=original_samples.device, 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