文章目录
概述
DM beat GANs作者改进了DDPM模型，提出了三个改进点，目的是提高在生成图像上的对数似然
第一个改进点方差改成了可学习的，预测方差线性加权的权重
第二个改进点将噪声方案的线性变化变成了非线性变换
第三个改进点将loss做了改进，Lhybrid = Lsimple+λLvlb（MSE loss+KL loss），采用了loss平滑的方法，基于loss算出重要性来采样t（不再是均匀采样t），Lvlb不直接采用Lt，而是Lt除以归一化的值pt（∑pt=1），pt是Lt平方的期望值的平方根，基于Lt最近的十个值，更少的采样步骤实现同样的效果
Lvlb，变分下界，L0加到Lt可拆解为3部分
L0 x1预测x0
0到t-1之间的，后验分布，神经网络预测的KL散度
Lt，由于一开始是一个先验的标准分布，不含参的，不参与神经网络优化
论文地址：
https://arxiv.org/abs/2102.09672
https://arxiv.org/pdf/2102.09672.pdf
项目地址：
https://github.com/openai/improved-diffusion
那么εθ的NN模型输入xt和t，输出的量和xt是保持一致的，
这里的NN模型用的是attention-based Unet，但不是本篇的重点，可以看另一篇博客
代码实现
项目地址：
https://github.com/openai/improved-diffusion
image_trian.py
image_train.py编写了大体的训练结构框架，只有短短的几行代码
def main()中
首先create_argparser
args = create_argparser().parse_args()
dist_util.setup_dist()
logger.configure()
logger.log("creating model and diffusion...")
create_argparser函数中定义了字典，数据目录，学习率一些默认的超参数，dict会更新，来源于model_and_diffusion_defaults函数，其返回也是一个字典，但是其键值对和模型和扩散相关的参数，创建argumentParser，遍历字典添加到argparser中，这样省的我们一个个去写手写add_argument，是一个很好的学习的简洁写法
def create_argparser():
defaults = dict(
data_dir="",
schedule_sampler="uniform",
lr=1e-4,
weight_decay=0.0,
lr_anneal_steps=0,
batch_size=1,
microbatch=-1, # -1 disables microbatches
ema_rate="0.9999", # comma-separated list of EMA values
log_interval=10,
save_interval=10000,
resume_checkpoint="",
use_fp16=False,
fp16_scale_growth=1e-3,
)
defaults.update(model_and_diffusion_defaults())
parser = argparse.ArgumentParser()
add_dict_to_argparser(parser, defaults)
return parser
def add_dict_to_argparser(parser, default_dict):
for k, v in default_dict.items():
v_type = type(v)
if v is None:
v_type = str
elif isinstance(v, bool):
v_type = str2bool
parser.add_argument(f"--{k}", default=v, type=v_type)
回到main函数，create_model_and_diffusion，得到unet model和diffusion框架，传入的参数是args_to_dict函数的**，args很大超参数，key只需要model和diffusion的部分
model, diffusion = create_model_and_diffusion(
**args_to_dict(args, model_and_diffusion_defaults().keys())
)
model.to(dist_util.dev())
schedule_sampler = create_named_schedule_sampler(args.schedule_sampler, diffusion)
schedule_sampler = create_named_schedule_sampler(args.schedule_sampler, diffusion)
返回的是一个采样器，可以是均匀采样，uniform，或者是基于loss重要性采样，二阶动量平滑loss，loss-second-moment
logger.log("creating data loader...")
data = load_data(
data_dir=args.data_dir,
batch_size=args.batch_size,
image_size=args.image_size,
class_cond=args.class_cond,
)
load_data函数， 返回的图片，list image files recursively，递归的找到所有图片文件，对data dir下的都遍历一遍，class_cond，类别判断，找到图片的每个类别，假设文件名的下划线的第一部分就是类别，用split做分割，将class排序设置索引，最终模型输出的还是索引
def load_data(
*, data_dir, batch_size, image_size, class_cond=False, deterministic=False
):
"""
For a dataset, create a generator over (images, kwargs) pairs.
Each images is an NCHW float tensor, and the kwargs dict contains zero or
more keys, each of which map to a batched Tensor of their own.
The kwargs dict can be used for class labels, in which case the key is "y"
and the values are integer tensors of class labels.
:param data_dir: a dataset directory.
:param batch_size: the batch size of each returned pair.
:param image_size: the size to which images are resized.
:param class_cond: if True, include a "y" key in returned dicts for class
label. If classes are not available and this is true, an
exception will be raised.
:param deterministic: if True, yield results in a deterministic order.
"""
if not data_dir:
raise ValueError("unspecified data directory")
all_files = _list_image_files_recursively(data_dir)
classes = None
if class_cond:
# Assume classes are the first part of the filename,
# before an underscore.
class_names = [bf.basename(path).split("_")[0] for path in all_files]
sorted_classes = {x: i for i, x in enumerate(sorted(set(class_names)))}
classes = [sorted_classes[x] for x in class_names]
dataset = ImageDataset(
image_size,
all_files,
classes=classes,
shard=MPI.COMM_WORLD.Get_rank(),
num_shards=MPI.COMM_WORLD.Get_size(),
)
if deterministic:
loader = DataLoader(
dataset, batch_size=batch_size, shuffle=False, num_workers=1, drop_last=True
)
else:
loader = DataLoader(
dataset, batch_size=batch_size, shuffle=True, num_workers=1, drop_last=True
)
while True:
yield from loader
ImageDataset类自定义了dataset，getitem传入index获取每张图片，进行处理获取单张的训练样本，图像处理进行resize，转换RGB格式，归一化到-1到1之间的浮点型
class ImageDataset(Dataset):
def __init__(self, resolution, image_paths, classes=None, shard=0, num_shards=1):
super().__init__()
self.resolution = resolution
self.local_images = image_paths[shard:][::num_shards]
self.local_classes = None if classes is None else classes[shard:][::num_shards]
def __len__(self):
return len(self.local_images)
def __getitem__(self, idx):
path = self.local_images[idx]
with bf.BlobFile(path, "rb") as f:
pil_image = Image.open(f)
pil_image.load()
# We are not on a new enough PIL to support the `reducing_gap`
# argument, which uses BOX downsampling at powers of two first.
# Thus, we do it by hand to improve downsample quality.
while min(*pil_image.size) >= 2 * self.resolution:
pil_image = pil_image.resize(
tuple(x // 2 for x in pil_image.size), resample=Image.BOX
)
scale = self.resolution / min(*pil_image.size)
pil_image = pil_image.resize(
tuple(round(x * scale) for x in pil_image.size), resample=Image.BICUBIC
)
arr = np.array(pil_image.convert("RGB"))
crop_y = (arr.shape[0] - self.resolution) // 2
crop_x = (arr.shape[1] - self.resolution) // 2
arr = arr[crop_y : crop_y + self.resolution, crop_x : crop_x + self.resolution]
arr = arr.astype(np.float32) / 127.5 - 1
out_dict = {}
if self.local_classes is not None:
out_dict["y"] = np.array(self.local_classes[idx], dtype=np.int64)
return np.transpose(arr, [2, 0, 1]), out_dict
main的最后代码的部分是实例化TrainLoop类，调用其run_loop函数，就可以开始训练了
logger.log("training...")
TrainLoop(
model=model,
diffusion=diffusion,
data=data,
batch_size=args.batch_size,
microbatch=args.microbatch,
lr=args.lr,
ema_rate=args.ema_rate,
log_interval=args.log_interval,
save_interval=args.save_interval,
resume_checkpoint=args.resume_checkpoint,
use_fp16=args.use_fp16,
fp16_scale_growth=args.fp16_scale_growth,
schedule_sampler=schedule_sampler,
weight_decay=args.weight_decay,
lr_anneal_steps=args.lr_anneal_steps,
).run_loop()
总体来说：
整个训练框架分为三步，第一步超参数汇总生成argparser，第二步create model and diffusion，第三步trainloop开始训练
这是总体的训练框架，下面看看细节create model and diffusion部分，下面只介绍diffusion的实现，model部分自己随意替换成任意模型网络
def create_model_and_diffusion()
只是一个很顶层的封装函数，没有具体的实现
def create_model_and_diffusion(
image_size,
class_cond,
learn_sigma,
sigma_small,
num_channels,
num_res_blocks,
num_heads,
num_heads_upsample,
attention_resolutions,
dropout,
diffusion_steps,
noise_schedule,
timestep_respacing,
use_kl,
predict_xstart,
rescale_timesteps,
rescale_learned_sigmas,
use_checkpoint,
use_scale_shift_norm,
):
model = create_model(
image_size,
num_channels,
num_res_blocks,
learn_sigma=learn_sigma,
class_cond=class_cond,
use_checkpoint=use_checkpoint,
attention_resolutions=attention_resolutions,
num_heads=num_heads,
num_heads_upsample=num_heads_upsample,
use_scale_shift_norm=use_scale_shift_norm,
dropout=dropout,
)
diffusion = create_gaussian_diffusion(
steps=diffusion_steps,
learn_sigma=learn_sigma,
sigma_small=sigma_small,
noise_schedule=noise_schedule,
use_kl=use_kl,
predict_xstart=predict_xstart,
rescale_timesteps=rescale_timesteps,
rescale_learned_sigmas=rescale_learned_sigmas,
timestep_respacing=timestep_respacing,
)
return model, diffusion
这篇博客主要讲diffusion实现部分，那么我们可以看到diffusion由create_gaussian_diffusion()函数创建
diffusion = create_gaussian_diffusion(
steps=diffusion_steps,
learn_sigma=learn_sigma,
sigma_small=sigma_small,
noise_schedule=noise_schedule,
use_kl=use_kl,
predict_xstart=predict_xstart,
rescale_timesteps=rescale_timesteps,
rescale_learned_sigmas=rescale_learned_sigmas,
timestep_respacing=timestep_respacing,
)
def create_gaussian_diffusion()
create_gaussian_diffusion生成一个扩散过程的框架，这是一个diffusion的顶层封装函数，
def create_gaussian_diffusion(
*,
steps=1000,
learn_sigma=False,
sigma_small=False,
noise_schedule="linear",
use_kl=False,
predict_xstart=False,
rescale_timesteps=False,
rescale_learned_sigmas=False,
timestep_respacing="",
):
betas = gd.get_named_beta_schedule(noise_schedule, steps)
if use_kl:
loss_type = gd.LossType.RESCALED_KL
elif rescale_learned_sigmas:
loss_type = gd.LossType.RESCALED_MSE
else:
loss_type = gd.LossType.MSE
if not timestep_respacing:
timestep_respacing = [steps]
return SpacedDiffusion(
use_timesteps=space_timesteps(steps, timestep_respacing),
betas=betas,
model_mean_type=(
gd.ModelMeanType.EPSILON if not predict_xstart else gd.ModelMeanType.START_X
),
model_var_type=(
(
gd.ModelVarType.FIXED_LARGE
if not sigma_small
else gd.ModelVarType.FIXED_SMALL
)
if not learn_sigma
else gd.ModelVarType.LEARNED_RANGE
),
loss_type=loss_type,
rescale_timesteps=rescale_timesteps,
)
第一步确定加噪的方案，get_named_beta_schedule，生成一个加噪的方案
获得了beta schedule
betas = gd.get_named_beta_schedule(noise_schedule, steps)
然后确定loss type，取决于从命令行传来的超参数是什么，use_kl的话使用rescaled_kl，rescale_learned_sigmas超参数使用rescaled_mse，不设置超参数启动普通的mse
if use_kl:
loss_type = gd.LossType.RESCALED_KL
elif rescale_learned_sigmas:
loss_type = gd.LossType.RESCALED_MSE
else:
loss_type = gd.LossType.MSE
create_gaussian_diffusion类最后return了一个实例化
调用了SpacedDiffusion的实例化
return SpacedDiffusion( # 下略
SpacedDiffusion就是Diffusion的实现类嘛？还是一个顶层的封装函数，封装的是一种可以跳过基本扩散过程中的步骤的扩散过程
SpacedDiffusion类
SpacedDiffusion类就是创建扩散模型的框架
timestep_respacing，对timestep做改进
将参数都传入 SpaceDiffusion类中进行实例化，所以这个代码的深度很深
下面看看SpacedDiffusion，这个类继承自GaussianDiffusion类
类的注释：
A diffusion process which can skip steps in a base diffusion process
一种可以跳过基本扩散过程的步骤（skip steps）的扩散过程。
扩散过程类，init函数定义了加噪方案的β，timestep哪些时刻要保留，numstep加噪次数
p_mean_variance函数，p就是神经网络所预测的分布，故p_mean_variance就是神经网络预测的均值和方差，这里调用的是父类的方法super().
training_loss函数，根据传入的超参数不同得到不同目标函数的公式，最简单的就是MSE loss，我们也可以加上kl loss联合起来作为目标函数
_wrap_model函数，对timestep进行后处理，比如对timestep进行scale，对timestep进行一定的优化
class SpacedDiffusion(GaussianDiffusion):
"""
A diffusion process which can skip steps in a base diffusion process.
:param use_timesteps: a collection (sequence or set) of timesteps from the
original diffusion process to retain.
:param kwargs: the kwargs to create the base diffusion process.
"""
def __init__(self, use_timesteps, **kwargs):
self.use_timesteps = set(use_timesteps)
self.timestep_map = []
self.original_num_steps = len(kwargs["betas"])
base_diffusion = GaussianDiffusion(**kwargs) # pylint: disable=missing-kwoa
last_alpha_cumprod = 1.0
new_betas = []
for i, alpha_cumprod in enumerate(base_diffusion.alphas_cumprod):
if i in self.use_timesteps:
new_betas.append(1 - alpha_cumprod / last_alpha_cumprod)
last_alpha_cumprod = alpha_cumprod
self.timestep_map.append(i)
kwargs["betas"] = np.array(new_betas)
super().__init__(**kwargs)
def p_mean_variance(
self, model, *args, **kwargs
): # pylint: disable=signature-differs
return super().p_mean_variance(self._wrap_model(model), *args, **kwargs)
def training_losses(
self, model, *args, **kwargs
): # pylint: disable=signature-differs
return super().training_losses(self._wrap_model(model), *args, **kwargs)
def _wrap_model(self, model):
if isinstance(model, _WrappedModel):
return model
return _WrappedModel(
model, self.timestep_map, self.rescale_timesteps, self.original_num_steps
)
def _scale_timesteps(self, t):
# Scaling is done by the wrapped model.
return t
class _WrappedModel:
def __init__(self, model, timestep_map, rescale_timesteps, original_num_steps):
self.model = model
self.timestep_map = timestep_map
self.rescale_timesteps = rescale_timesteps
self.original_num_steps = original_num_steps
def __call__(self, x, ts, **kwargs):
map_tensor = th.tensor(self.timestep_map, device=ts.device, dtype=ts.dtype)
new_ts = map_tensor[ts]
if self.rescale_timesteps:
new_ts = new_ts.float() * (1000.0 / self.original_num_steps)
return self.model(x, new_ts, **kwargs)
GaussianDiffusion类 ⭐ LOOK HERE ⭐
下面来看SpacedDiffusion的父类GaussianDiffusion类
位置：improved_diffusion/gaussian_diffusion.py
先看注释：Utilities for training and sampling diffusion models.
训练和抽样扩散模型的实用程序，找了半天，原来这里才是真正的实现类
init函数
model_mean_type，知道这个模型要预测什么，预测的是方差还是噪声还是x0，
model_var_type，方差是固定还是可学习的，还是预测学习线性加权的权重
self.model_mean_type = model_mean_type
self.model_var_type = model_var_type
loss_type，是预测mse还是加kl
self.loss_type = loss_type
rescale-timesteps，对时间进行scale，使得timestep永远缩放到在0到1000之间
self.rescale_timesteps = rescale_timesteps
传入betas，论文中有提到一个扩散的超参数，1维的向量，在0到1之间
betas = np.array(betas, dtype=np.float64)
self.betas = betas
assert len(betas.shape) == 1, "betas must be 1-D"
assert (betas > 0).all() and (betas <= 1).all()
self.num_timesteps = int(betas.shape[0])
后面得到一些变量α=1-β，α-bar（α连乘），α-bar-prev（αt-1-bar），α-bar-next（αt+1-bar)，根号下的等等α，根号下1-αt-bar，sqrt-recip，倒数根号下alpha等等，用于论文中计算的公式
alphas = 1.0 - betas
self.alphas_cumprod = np.cumprod(alphas, axis=0)
self.alphas_cumprod_prev = np.append(1.0, self.alphas_cumprod[:-1])
self.alphas_cumprod_next = np.append(self.alphas_cumprod[1:], 0.0)
assert self.alphas_cumprod_prev.shape == (self.num_timesteps,)
# calculations for diffusion q(x_t | x_{t-1}) and others
self.sqrt_alphas_cumprod = np.sqrt(self.alphas_cumprod)
self.sqrt_one_minus_alphas_cumprod = np.sqrt(1.0 - self.alphas_cumprod)
self.log_one_minus_alphas_cumprod = np.log(1.0 - self.alphas_cumprod)
self.sqrt_recip_alphas_cumprod = np.sqrt(1.0 / self.alphas_cumprod)
self.sqrt_recipm1_alphas_cumprod = np.sqrt(1.0 / self.alphas_cumprod - 1)
接下来计算扩散过程中后验分布的真实的方差和均值，方差是一个常数可以直接计算，均值和xt有关，但是均值的两个系数是可以先确定的
# calculations for posterior q(x_{t-1} | x_t, x_0)
self.posterior_variance = (
betas * (1.0 - self.alphas_cumprod_prev) / (1.0 - self.alphas_cumprod)
)
# log calculation clipped because the posterior variance is 0 at the
# beginning of the diffusion chain.
self.posterior_log_variance_clipped = np.log(
np.append(self.posterior_variance[1], self.posterior_variance[1:])
)
self.posterior_mean_coef1 = (
betas * np.sqrt(self.alphas_cumprod_prev) / (1.0 - self.alphas_cumprod)
)
self.posterior_mean_coef2 = (
(1.0 - self.alphas_cumprod_prev)
* np.sqrt(alphas)
/ (1.0 - self.alphas_cumprod)
)
接着看看类中的其他一些函数，q_mean_variance，基于下面的公式8生成均值和方差，中间的是均值，后面是标准差
def q_mean_variance(self, x_start, t):
"""
Get the distribution q(x_t | x_0).
:param x_start: the [N x C x ...] tensor of noiseless inputs.
:param t: the number of diffusion steps (minus 1). Here, 0 means one step.
:return: A tuple (mean, variance, log_variance), all of x_start's shape.
"""
mean = (
_extract_into_tensor(self.sqrt_alphas_cumprod, t, x_start.shape) * x_start
)
variance = _extract_into_tensor(1.0 - self.alphas_cumprod, t, x_start.shape)
log_variance = _extract_into_tensor(
self.log_one_minus_alphas_cumprod, t, x_start.shape
)
return mean, variance, log_variance
q_sample函数，对上面q-mean-variance进行采样，给定x0和t的情况下采样出xt，这个过程就是重参数的过程
def q_sample(self, x_start, t, noise=None):
"""
Diffuse the data for a given number of diffusion steps.
In other words, sample from q(x_t | x_0).
:param x_start: the initial data batch.
:param t: the number of diffusion steps (minus 1). Here, 0 means one step.
:param noise: if specified, the split-out normal noise.
:return: A noisy version of x_start.
"""
if noise is None:
noise = th.randn_like(x_start)
assert noise.shape == x_start.shape
return (
_extract_into_tensor(self.sqrt_alphas_cumprod, t, x_start.shape) * x_start
+ _extract_into_tensor(self.sqrt_one_minus_alphas_cumprod, t, x_start.shape)
* noise
)
q-posterior-mean-variance，基于x0，xt和t计算出公式9和公式10真实分布的均值和方差
def q_posterior_mean_variance(self, x_start, x_t, t):
"""
Compute the mean and variance of the diffusion posterior:
q(x_{t-1} | x_t, x_0)
"""
assert x_start.shape == x_t.shape
posterior_mean = (
_extract_into_tensor(self.posterior_mean_coef1, t, x_t.shape) * x_start
+ _extract_into_tensor(self.posterior_mean_coef2, t, x_t.shape) * x_t
)
posterior_variance = _extract_into_tensor(self.posterior_variance, t, x_t.shape)
posterior_log_variance_clipped = _extract_into_tensor(
self.posterior_log_variance_clipped, t, x_t.shape
)
assert (
posterior_mean.shape[0]
== posterior_variance.shape[0]
== posterior_log_variance_clipped.shape[0]
== x_start.shape[0]
)
return posterior_mean, posterior_variance, posterior_log_variance_clipped
p_mean_variance，p分布是神经网络的分布，去建模拟合的分布，得到前一时刻（逆扩散过程）的均值和方差，也包括x0的预测
def p_mean_variance(
self, model, x, t, clip_denoised=True, denoised_fn=None, model_kwargs=None
):
"""
Apply the model to get p(x_{t-1} | x_t), as well as a prediction of
the initial x, x_0.
:param model: the model, which takes a signal and a batch of timesteps
as input.
:param x: the [N x C x ...] tensor at time t.
:param t: a 1-D Tensor of timesteps.
:param clip_denoised: if True, clip the denoised signal into [-1, 1].
:param denoised_fn: if not None, a function which applies to the
x_start prediction before it is used to sample. Applies before
clip_denoised.
:param model_kwargs: if not None, a dict of extra keyword arguments to
pass to the model. This can be used for conditioning.
:return: a dict with the following keys:
- 'mean': the model mean output.
- 'variance': the model variance output.
- 'log_variance': the log of 'variance'.
- 'pred_xstart': the prediction for x_0.
"""
if model_kwargs is None:
model_kwargs = {}
B, C = x.shape[:2]
assert t.shape == (B,)
model_output = model(x, self._scale_timesteps(t), **model_kwargs)
if self.model_var_type in [ModelVarType.LEARNED, ModelVarType.LEARNED_RANGE]:
assert model_output.shape == (B, C * 2, *x.shape[2:])
model_output, model_var_values = th.split(model_output, C, dim=1)
if self.model_var_type == ModelVarType.LEARNED:
model_log_variance = model_var_values
model_variance = th.exp(model_log_variance)
else:
min_log = _extract_into_tensor(
self.posterior_log_variance_clipped, t, x.shape
)
max_log = _extract_into_tensor(np.log(self.betas), t, x.shape)
# The model_var_values is [-1, 1] for [min_var, max_var].
frac = (model_var_values + 1) / 2
model_log_variance = frac * max_log + (1 - frac) * min_log
model_variance = th.exp(model_log_variance)
else:
model_variance, model_log_variance = {
# for fixedlarge, we set the initial (log-)variance like so
# to get a better decoder log likelihood.
ModelVarType.FIXED_LARGE: (
np.append(self.posterior_variance[1], self.betas[1:]),
np.log(np.append(self.posterior_variance[1], self.betas[1:])),
),
ModelVarType.FIXED_SMALL: (
self.posterior_variance,
self.posterior_log_variance_clipped,
),
}[self.model_var_type]
model_variance = _extract_into_tensor(model_variance, t, x.shape)
model_log_variance = _extract_into_tensor(model_log_variance, t, x.shape)
def process_xstart(x):
if denoised_fn is not None:
x = denoised_fn(x)
if clip_denoised:
return x.clamp(-1, 1)
return x
if self.model_mean_type == ModelMeanType.PREVIOUS_X:
pred_xstart = process_xstart(
self._predict_xstart_from_xprev(x_t=x, t=t, xprev=model_output)
)
model_mean = model_output
elif self.model_mean_type in [ModelMeanType.START_X, ModelMeanType.EPSILON]:
if self.model_mean_type == ModelMeanType.START_X:
pred_xstart = process_xstart(model_output)
else:
pred_xstart = process_xstart(
self._predict_xstart_from_eps(x_t=x, t=t, eps=model_output)
)
model_mean, _, _ = self.q_posterior_mean_variance(
x_start=pred_xstart, x_t=x, t=t
)
else:
raise NotImplementedError(self.model_mean_type)
assert (
model_mean.shape == model_log_variance.shape == pred_xstart.shape == x.shape
)
return {
"mean": model_mean,
"variance": model_variance,
"log_variance": model_log_variance,
"pred_xstart": pred_xstart,
}
_predict_xstart_from_eps，辅助函数，从预测处的噪声预测x0，对应公式12 
给定xt，t和x0到xt所加的噪声反推出x0
def _predict_xstart_from_eps(self, x_t, t, eps):
assert x_t.shape == eps.shape
return (
_extract_into_tensor(self.sqrt_recip_alphas_cumprod, t, x_t.shape) * x_t
- _extract_into_tensor(self.sqrt_recipm1_alphas_cumprod, t, x_t.shape) * eps
)
_predict_xstart_from_xprev，从xt-1中预测出x0
基于公式10，xt-1就是μ~t，有xt，反推出x0
def _predict_xstart_from_xprev(self, x_t, t, xprev):
assert x_t.shape == xprev.shape
return ( # (xprev - coef2*x_t) / coef1
_extract_into_tensor(1.0 / self.posterior_mean_coef1, t, x_t.shape) * xprev
- _extract_into_tensor(
self.posterior_mean_coef2 / self.posterior_mean_coef1, t, x_t.shape
)
* x_t
)
_predict_eps_from_xstart，从x0和xt，推导eps，对公式8的反推
def _predict_eps_from_xstart(self, x_t, t, pred_xstart):
return (
_extract_into_tensor(self.sqrt_recip_alphas_cumprod, t, x_t.shape) * x_t
- pred_xstart
) / _extract_into_tensor(self.sqrt_recipm1_alphas_cumprod, t, x_t.shape)
p_sample，从xt采样出xt-1，所有的p分布都是模型预测的，其实就是推理的函数
def p_sample(
self, model, x, t, clip_denoised=True, denoised_fn=None, model_kwargs=None
):
"""
Sample x_{t-1} from the model at the given timestep.
:param model: the model to sample from.
:param x: the current tensor at x_{t-1}.
:param t: the value of t, starting at 0 for the first diffusion step.
:param clip_denoised: if True, clip the x_start prediction to [-1, 1].
:param denoised_fn: if not None, a function which applies to the
x_start prediction before it is used to sample.
:param model_kwargs: if not None, a dict of extra keyword arguments to
pass to the model. This can be used for conditioning.
:return: a dict containing the following keys:
- 'sample': a random sample from the model.
- 'pred_xstart': a prediction of x_0.
"""
out = self.p_mean_variance(
model,
x,
t,
clip_denoised=clip_denoised,
denoised_fn=denoised_fn,
model_kwargs=model_kwargs,
)
noise = th.randn_like(x)
nonzero_mask = (
(t != 0).float().view(-1, *([1] * (len(x.shape) - 1)))
) # no noise when t == 0
sample = out["mean"] + nonzero_mask * th.exp(0.5 * out["log_variance"]) * noise
return {"sample": sample, "pred_xstart": out["pred_xstart"]}
_vb_terms_bpd， 计算最终的kl散度
kl散度包括两项，当t在0到t之间，用模型预测分布计算高斯分布算一个kl散度，另一项是最后一个时刻，L0 loss，使用的是似然函数，负对数似然函数，使用的是累积分布函数的差分拟合离散的高斯分布
def _vb_terms_bpd(
self, model, x_start, x_t, t, clip_denoised=True, model_kwargs=None
):
"""
Get a term for the variational lower-bound.
The resulting units are bits (rather than nats, as one might expect).
This allows for comparison to other papers.
:return: a dict with the following keys:
- 'output': a shape [N] tensor of NLLs or KLs.
- 'pred_xstart': the x_0 predictions.
"""
true_mean, _, true_log_variance_clipped = self.q_posterior_mean_variance(
x_start=x_start, x_t=x_t, t=t
)
out = self.p_mean_variance(
model, x_t, t, clip_denoised=clip_denoised, model_kwargs=model_kwargs
)
kl = normal_kl(
true_mean, true_log_variance_clipped, out["mean"], out["log_variance"]
)
kl = mean_flat(kl) / np.log(2.0)
decoder_nll = -discretized_gaussian_log_likelihood(
x_start, means=out["mean"], log_scales=0.5 * out["log_variance"]
)
assert decoder_nll.shape == x_start.shape
decoder_nll = mean_flat(decoder_nll) / np.log(2.0)
# At the first timestep return the decoder NLL,
# otherwise return KL(q(x_{t-1}|x_t,x_0) || p(x_{t-1}|x_t))
output = th.where((t == 0), decoder_nll, kl)
return {"output": output, "pred_xstart": out["pred_xstart"]}
traning-loss，计算一个使用的loss
def training_losses(self, model, x_start, t, model_kwargs=None, noise=None):
"""
Compute training losses for a single timestep.
:param model: the model to evaluate loss on.
:param x_start: the [N x C x ...] tensor of inputs.
:param t: a batch of timestep indices.
:param model_kwargs: if not None, a dict of extra keyword arguments to
pass to the model. This can be used for conditioning.
:param noise: if specified, the specific Gaussian noise to try to remove.
:return: a dict with the key "loss" containing a tensor of shape [N].
Some mean or variance settings may also have other keys.
"""
if model_kwargs is None:
model_kwargs = {}
if noise is None:
noise = th.randn_like(x_start)
x_t = self.q_sample(x_start, t, noise=noise)
terms = {}
if self.loss_type == LossType.KL or self.loss_type == LossType.RESCALED_KL:
terms["loss"] = self._vb_terms_bpd(
model=model,
x_start=x_start,
x_t=x_t,
t=t,
clip_denoised=False,
model_kwargs=model_kwargs,
)["output"]
if self.loss_type == LossType.RESCALED_KL:
terms["loss"] *= self.num_timesteps
elif self.loss_type == LossType.MSE or self.loss_type == LossType.RESCALED_MSE:
model_output = model(x_t, self._scale_timesteps(t), **model_kwargs)
if self.model_var_type in [
ModelVarType.LEARNED,
ModelVarType.LEARNED_RANGE,
]:
B, C = x_t.shape[:2]
assert model_output.shape == (B, C * 2, *x_t.shape[2:])
model_output, model_var_values = th.split(model_output, C, dim=1)
# Learn the variance using the variational bound, but don't let
# it affect our mean prediction.
frozen_out = th.cat([model_output.detach(), model_var_values], dim=1)
terms["vb"] = self._vb_terms_bpd(
model=lambda *args, r=frozen_out: r,
x_start=x_start,
x_t=x_t,
t=t,
clip_denoised=False,
)["output"]
if self.loss_type == LossType.RESCALED_MSE:
# Divide by 1000 for equivalence with initial implementation.
# Without a factor of 1/1000, the VB term hurts the MSE term.
terms["vb"] *= self.num_timesteps / 1000.0
target = {
ModelMeanType.PREVIOUS_X: self.q_posterior_mean_variance(
x_start=x_start, x_t=x_t, t=t
)[0],
ModelMeanType.START_X: x_start,
ModelMeanType.EPSILON: noise,
}[self.model_mean_type]
assert model_output.shape == target.shape == x_start.shape
terms["mse"] = mean_flat((target - model_output) ** 2)
if "vb" in terms:
terms["loss"] = terms["mse"] + terms["vb"]
else:
terms["loss"] = terms["mse"]
else:
raise NotImplementedError(self.loss_type)
return terms
_extract_into_tensor，辅助函数，从tensor中取出第t时刻
def _extract_into_tensor(arr, timesteps, broadcast_shape):
"""
Extract values from a 1-D numpy array for a batch of indices.
:param arr: the 1-D numpy array.
:param timesteps: a tensor of indices into the array to extract.
:param broadcast_shape: a larger shape of K dimensions with the batch
dimension equal to the length of timesteps.
:return: a tensor of shape [batch_size, 1, ...] where the shape has K dims.
"""
res = th.from_numpy(arr).to(device=timesteps.device)[timesteps].float()
while len(res.shape) < len(broadcast_shape):
res = res[..., None]
return res.expand(broadcast_shape)
边角料
一个很小很小的改动，算是技巧的noise scheduling
noise scheduling
原始的DDPM中使用的是线性的增长的β加噪方案，此处使用了余弦的方案，同时控制上界在0.999
def get_named_beta_schedule(schedule_name, num_diffusion_timesteps):
"""
Get a pre-defined beta schedule for the given name.
The beta schedule library consists of beta schedules which remain similar
in the limit of num_diffusion_timesteps.
Beta schedules may be added, but should not be removed or changed once
they are committed to maintain backwards compatibility.
"""
if schedule_name == "linear":
# Linear schedule from Ho et al, extended to work for any number of
# diffusion steps.
scale = 1000 / num_diffusion_timesteps
beta_start = scale * 0.0001
beta_end = scale * 0.02
return np.linspace(
beta_start, beta_end, num_diffusion_timesteps, dtype=np.float64
)
elif schedule_name == "cosine":
return betas_for_alpha_bar(
num_diffusion_timesteps,
lambda t: math.cos((t + 0.008) / 1.008 * math.pi / 2) ** 2,
)
else:
raise NotImplementedError(f"unknown beta schedule: {schedule_name}")
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