Fast online moments estimates on the GPU

← Posts · 06/08/2021 · 3 minutes

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.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 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).

References

  • OnlineStats.jl, a cool Julia package that implements a lot of online estimators.