机器学习常见的sampling策略 附PyTorch实现

简单的采样策略

首先介绍三种简单采样策略：

1. Instance-balanced sampling, 实例平衡采样。
2. Class-balanced sampling, 类平衡采样。
3. Square-root sampling, 平方根采样。

它们可抽象为：

$$p_j=\frac{n_j^q}{\sum _{i=1}^C{n}_i^q},$$

$p_j$表示从j类采样数据的概率；$C$表示类别数量；$n_j$表示j类样本数；$q\in \{1,0,\frac{1}{2}\}$
Instance-balanced sampling
最常见的数据采样方式，其中每个训练样本被选择的概率相等（$q=1$）。j类被采样的概率$p_j^{\mathbf{IB}}$与j类样本数$n_j$成正比，即$p_j^{\mathbf{IB}}=\frac{n_j}{\sum _{i=1}^C{n}_i}$。

Class-balanced sampling
实例平衡采样在不平衡的数据集中往往表现不佳，类平衡采样让所有的类有相同的被采样概率：$p_j^{\mathbf{CB}}=\frac{1}{C}$。采样可分为两个阶段：1. 从类集中统一选择一个类；2. 对该类中的实例进行统一采样。
Square-root sampling
平方根采样最常见的变体，$q=\frac{1}{2}$

由于这三种采样策略都是调整类别的采样概率（权重），因此可用PyTorch提供的WeightedRandomSampler实现：

import numpy as np
from torch.utils.data.sampler import WeightedRandomSampler
def get_sampler(sampling_type, targets):
    cls_counts = np.bincount(targets)
    if sampling_type == 'instance-balanced':
        cls_weights = cls_counts / np.sum(cls_counts)
        
    elif sampling_type == 'class-balanced':
        cls_num = len(cls_counts)
        cls_weights = [1. / cls_num] * cls_num
        
    elif sampling_type == 'square-root':
        sqrt_and_sum = np.sum([num**0.5 for num in cls_counts])
        cls_weights = [num**0.5 / sqrt_and_sum for num in cls_counts]
    else:
        raise ValueError('sampling_type should be instance-balanced, class-balanced or square-root')
    
    cls_weights = np.array(cls_weights)
    return WeightedRandomSampler(cls_weights[targets], len(targets), replacement=True)

WeightedRandomSampler，第一个参数表示每个样本的权重，第二个参数表示采样的样本数，第三个参数表示是否有放回采样。

在模拟的长尾数据集测试下：

import torch
from torch.utils.data import Dataset, DataLoader
torch.manual_seed(0)
np.random.seed(0)
class LongTailDataset(Dataset):
    def __init__(self, num_classes, max_samples_per_class):
        self.num_classes = num_classes
        self.max_samples_per_class = max_samples_per_class
 
        # Generate number of samples for each class inversely proportional to class index
        self.samples_per_class = [self.max_samples_per_class // (i + 1) for i in range(self.num_classes)]
        self.total_samples = sum(self.samples_per_class)
 
        # Generate targets for the dataset
        self.targets = torch.cat([torch.full((samples,), i, dtype=torch.long) for i, samples in enumerate(self.samples_per_class)])
 
    def __len__(self):
        return self.total_samples
 
    def __getitem__(self, idx):
        # For simplicity, just return the index as the data
        return idx, self.targets[idx]
 
# Parameters
num_classes = 25
max_samples_per_class = 1000
 
# Create dataset
dataset = LongTailDataset(num_classes, max_samples_per_class)
 
# Create dataloader
batch_size = 64
sampler1 = get_sampler('instance-balanced', dataset.targets.numpy())
sampler2 = get_sampler('class-balanced', dataset.targets.numpy())
sampler3 = get_sampler('square-root', dataset.targets.numpy())
dataloader1 = DataLoader(dataset, batch_size=64, sampler=sampler1)
dataloader2 = DataLoader(dataset, batch_size=64, sampler=sampler2)
dataloader3 = DataLoader(dataset, batch_size=64, sampler=sampler3)
 
for (_, target1), (_, target2), (_, target3)  in zip(dataloader1, dataloader2, dataloader3):
    print('Instance-balanced:')
    cls_idx, cls_counts = np.unique(target1.numpy(), return_counts=True)
    print(f'Class indices: {cls_idx}')
    print(f'Class counts: {cls_counts}')
    print('-'*20)
    print('Class-balanced:')
    cls_idx, cls_counts = np.unique(target2.numpy(), return_counts=True)
    print(f'Class indices: {cls_idx}')
    print(f'Class counts: {cls_counts}')
    print('-'*20)
    print('Square-root:')
    cls_idx, cls_counts = np.unique(target3.numpy(), return_counts=True)
    print(f'Class indices: {cls_idx}')
    print(f'Class counts: {cls_counts}')
    break # just show one batch 

Output:

Instance-balanced:
Class indices: [ 0  1  2  3  5 16 22 23]
Class counts: [43  9  5  2  2  1  1  1]
--------------------
Class-balanced:
Class indices: [ 0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 16 17 20 21 23]
Class counts: [21  8  6  4  2  1  2  2  3  3  1  2  1  1  1  1  2  1  1  1]
--------------------
Square-root:
Class indices: [ 0  1  2  3  4  5  6  9 10 21 22 23]
Class counts: [37  8  3  6  3  1  1  1  1  1  1  1]

混合采样策略

最早的混合采样是在 $0\le epoch\le t$时采用Instance-balanced采样，$t\le epoch\le T$时采用Class-balanced采样，这需要设置合适的超参数t。在[1]中，作者提出了soft版本的混合采样策略：Progressively-balanced sampling。随着epoch的增加每个类的采样概率（权重）$p_j$也发生变化：

$$p_j^{\mathbf{PB}}(t)=(1-\frac{t}{T})p_j^{\mathbf{IB}}+\frac{t}{T}{p}_j^{\mathbf{CB}}$$

t表示当前epoch，T表示总epoch数。

不平衡数据集下的采样策略

不平衡的数据集，特别是长尾数据集，为了照顾尾部类，通常设置每个类的采样概率（权重）为样本数的倒数，即$p_j=\frac{1}{n_j}$。

...
elif sampling_type == 'inverse':
    cls_weights = 1. / cls_counts
...

在[3]中提出了有效数（effective number）的概念，分母的位置不是简单的样本数，而是经过一定计算得到的，这里直接给出结果，证明请详见原论文。关于effective number的计算方式：

$$E_n=(1-{\beta }^n)/(1-\beta ),\mathrm{where}\beta =(N-1)/N.$$

这里N表示数据集样本总数。

相关代码：

...
elif sampling_type == 'effective':
    beta = (len(targets) - 1) / len(targets)
    cls_weights = (1.0 - beta) / (1.0 - np.power(beta, cls_counts))
...

Output

Effective number:
Class indices: [ 0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 16 17 18 20 21 22 23 24]
Class counts: [2 1 2 3 1 1 4 2 3 4 4 2 3 5 2 4 1 3 1 4 5 6 1]

在和上面一样的模拟长尾数据集上，采样的结果更加均衡。

参考文献

1. Kang, Bingyi, et al. "Decoupling Representation and Classifier for Long-Tailed Recognition." International Conference on Learning Representations. 2019.
2. torch.utils.data.WeightedRandomSampler
3. Cui, Yin, et al. "Class-balanced loss based on effective number of samples." Proceedings of the IEEE/CVF conference on computer vision and pattern recognition. 2019.