Fast online estimates on the GPU
Estimating moments is an important step of any statistical analysis of data. The mean, variance, skewness and kurtosis of a dataset can already tell a lot about the distribution of our data.
However, some datasets don’t quite fit in memory. If you have a dataset of N samples and C features where N is a lot bigger than C, you can benefit a lot by using online algorithms.
$$ \bar x = \frac 1N\sum_{i=1}^N x_i $$
$$ \sigma^2 = \frac N{N-1}\left(\sum_{i=1}^N x_i^2 - \bar x^2\right) $$
If you are using PyTorch on a GPU, it is easy to compute those moments. But since CUDA and multi-threading do not play well together, we have to optimize the batching of our samples.
import torch
from torch import Tensor
class Moments:
"""
Online estimator of moments up to 4.
>>> m = Moments(5)
>>> m.fit(torch.zeros(5,))
1
>>> m.fit(torch.ones(5,))
2
>>> m.mean()
tensor([0.5000, 0.5000, 0.5000, 0.5000, 0.5000])
>>> m.std()
tensor([0.5000, 0.5000, 0.5000, 0.5000, 0.5000])
>>> m.var(corrected=False)
tensor([0.2500, 0.2500, 0.2500, 0.2500, 0.2500])
>>> m.var(corrected=True)
tensor([0.5000, 0.5000, 0.5000, 0.5000, 0.5000])
>>> m.skewness()
tensor([0., 0., 0., 0., 0.])
>>> m.kurtosis()
tensor([-2., -2., -2., -2., -2.])
"""
def __init__(self, size: int, device="cpu") -> None:
self.n = 0
self.m = torch.zeros((size, 4), device=device) # moments
def fit(self, new_obs: Tensor) -> int:
if len(new_obs.shape) == 1: # single obs
new_obs = new_obs.unsqueeze(0)
self.n += new_obs.size(0)
y = new_obs.clone()
for i in range(4):
self.m[:, i] += y.sum(dim=0)
y = y * new_obs
return self.n
def merge(self, other):
self.m += other.m
self.n += other.n
return self
def mean(self) -> Tensor:
return self.m[:, 0] / self.n
def std(self) -> Tensor:
return self.var(corrected=False).sqrt()
def var(self, corrected: bool = False) -> Tensor:
var = self.m[:,1]/self.n - self.mean().pow(2)
if corrected:
var = var * self.n / (self.n - 1)
return var
def skewness(self) -> Tensor:
var = self.var()
mean = self.mean()
return (self.m[:,2] / self.n - 3.0 * mean * var - mean.pow(3)) / var.pow(1.5)
def kurtosis(self) -> Tensor:
m1, m2, m3, m4 = tuple(self.m[:,i] / self.n for i in range(4))
return (m4 - 4.0 * m1 * m3 + 6.0 * m1.pow(2) * m2 - 3.0 * m1.pow(4)) / self.var().pow(2) - 3.0
Example usage:
device = "cuda" if torch.cuda.is_available() else "cpu"
m = Moments(100, device)
def embed(imgs: Tensor) -> Tensor:
# transform input images into embedding vectors
embeddings = ...
return embeddings
for imgs in tqdm(dataloader):
embeddings = embed(imgs.to(device))
m.fit(embeddings)
# You can then get the moments
m.mean()
m.var(corrected=True)
m.skewness()
m.kurtosis()
Since we are embedding images, the bottleneck is likely to be the disk reads to load the image files. Make sure to use a torch.utils.data.DataLoader with new_workers set to a high enough value to leverage asynchronous loading of files.
With enough num_workers, the fitting process of 127,000 images which each produce 1024 embedding vectors takes around 3 minutes. This process could take a lot more time with a more naive approach and it could even be undoable when saving all the vectors in memory (127,000 * 1024 * 100 * 32 byte -> 416,154 Gbytes).
Computing the CIFAR10 mean and standard deviation
Let’s first instantiate the CIFAR10 dataset provided by torchvision:
import torch
from torchvision.datasets import CIFAR10
import torchvision.transforms as T
cifar10 = CIFAR10(
"./data/cifar10", train=True, download=True, transform=T.ToTensor()
)
loader = torch.utils.data.DataLoader(cifar10, batch_size=32, num_workers=16,)
Then we can use the Moments class to compute the mean and standard deviation for each color channel:
device = "cuda" if torch.cuda.is_available() else "cpu"
moments = Moments(size=3, device=device)
for x, _ in tqdm(loader):
moments.fit(x.to(device).permute(0, 2, 3, 1).reshape(-1, 3))
mean = moments.mean()
std = moments.std()
print(f">> mean = {mean},\nstd = {std}")
Which returns the following mean and std:
mean = tensor([0.4914, 0.4822, 0.4465], device='cuda:0'),
std = tensor([0.2470, 0.2435, 0.2616], device='cuda:0')
Helper Function
from tqdm import tqdm
import torch
def get_mean_and_std(dataset: torch.utils.data.Dataset):
loader = torch.utils.data.DataLoader(dataset, batch_size = 32, num_workers = 4)
device = "cuda" if torch.cuda.is_available() else "cpu"
moments = Moments(size=3, device=device)
for x, _ in tqdm(loader):
moments.fit(x.to(device).permute(0, 2, 3, 1).reshape(-1, 3))
mean = moments.mean()
std = moments.std()
return mean, std
An online multi-variate Gaussian estimator
$$
\begin{align}
\mu &= \frac 1N\sum_{i = 1}^Nx_i \
\Sigma &= \frac 1{N-1}\left[\sum_{i=1}^N(x_ix_i^T) -N\mu\mu^T\right]
\end{align}
$$
from typing import Tuple
import torch
from torch import Tensor
class OnlineGaussian:
"""
Estimates the mean and covariance matrix of a multi-variate Gaussian
of `dim` variables.
>>> gauss = OnlineGaussian(3)
>>> gauss.fit(torch.ones((12, 3,)))
12
>>> gauss.value()
(tensor([1., 1., 1.]), tensor([[0., 0., 0.],
[0., 0., 0.],
[0., 0., 0.]]))
"""
def __init__(self, dim: int, device: str="cpu"):
self.dim = dim
self.sum1 = torch.zeros((dim,), device=device)
self.sum2 = torch.zeros((dim, dim), device=device)
self.N = 0
@torch.no_grad()
def fit(self, x: Tensor) -> int:
x = x.view(-1, self.dim)
assert x.size(-1) == self.dim
self.N += x.size(0)
self.sum1 += x.sum(0)
self.sum2 += torch.einsum("ni,nj->ij", x, x)
return self.N
@torch.no_grad()
def value(self, corrected: bool=True) -> Tuple[Tensor, Tensor]:
means = self.sum1 / self.N
covs = (self.sum2 - self.N * torch.outer(means, means)) / (self.N - corrected)
return means, covs
References
- OnlineStats.jl, a cool Julia package that implements a lot of online estimators.