背景

Focal Loss由何凯明提出，最初用于图像领域解决数据不平衡造成的模型性能问题。

交叉熵损失函数

$$Loss=L(y,\hat{p})=-ylog(\hat{p})-(1-y)log(1-\hat{p})$$

其中，$\hat{p}$为预测概率大小。y为label，二分类中对应0和1。
$$L_{ce}(y,\hat{p})=\{\begin{array}{ll}-log(\hat{p}),& \text{if }y=1\\ -log(1-\hat{p}),& \text{if }y=0\end{array}$$
对于所有样本，需要求平均作为最终的结果：
$$L=\frac{1}{N}\sum _{i=1}^{N}l(y_i,{\hat{p}}_i)$$
对于二分类问题，可以改写成：
$$L=\frac{1}{N}(\sum _{y_i=1}^m-log(\hat{p})+\sum _{y_i=0}^n-log(1-\hat{p}))$$
其中，N为样本总数，m和n为正、负样本数，$m+n=N$

当样本分布不平衡时，损失函数L的分布也会发生倾斜，若m>>n时，正样本就会在损失函数中占据主导地位，由于损失函数的倾斜，训练的模型会倾向于样本较多的类别，导致对较少样本类别的性能较差。

平衡交叉熵函数

对于样本不平衡造成的损失函数倾斜，最直接的方法就是添加权重因子，提高少数类别在损失函数中的权重，从而平衡损失函数的分布。还是以之前的二分类问题为例，我们添加权重参数$\alpha \in [0,1]$
$$L=\frac{1}{N}(\sum _{y_i=1}^m-\alpha log(\hat{p})+\sum _{y_i=0}^n-(1-\alpha )log(1-\hat{p}))$$
其中，$\frac{\alpha }{1-\alpha }=\frac{n}{m}$，权重大小由正负样本数量比来设置。

Focal Loss损失函数

Focal Loss从loss角度提供了一种样本不均衡的解决方案：
$$L_{focal}(y,\hat{p})=\{\begin{array}{ll}-(1-\hat{p}{)}^{\gamma }log(\hat{p}),& \text{if }y=1\\ -{\hat{p}}^{\gamma }log(1-\hat{p}),& \text{if }y=0\end{array}$$
令$p_t=\{\begin{array}{ll}\hat{p},& \text{if }y=1\\ 1-\hat{p},& \text{otherwise. }\end{array}$

则表达式统一为：
$$L_{focal}=-(1-p_t{)}^{\gamma }log(p_t)$$
与交叉熵表达式对照：$L_{ce}=-log(p_t)$，仅仅多了一个可变系数$(1-p_t{)}^{\gamma }$.

其中，$p_t$反应了与ground truth的接近程度，越大表示分类越准。$\gamma >0$为调节因子。

对于分类不准确的样本，$p_t\to 0$，$(1-p_t{)}^{\gamma }\to 1$，$L_{focal}\to L_{ce}$；对于分类准确的样本，$p_t\to 1$，$(1-p_t{)}^{\gamma }\to 0$，$L_{focal}\to 0$；因此，Focal Loss对于分类不准确的样本，损失没有改变；对于分类准确的样本，损失会变小。整体来看，Focal Loss增加了分类不准确样本在损失函数中的权重。

如下是不同调节因子$\gamma$对应的Loss-proba分布图，可以看出Cross Entropy(CE)和Focal Loss(FL)之间的区别，Focal Loss使损失函数更倾向于难分的样本。

Focal Loss vs Balanced Cross Entropy

• Focal Loss是从样本分类难易程度出发，让Loss聚焦于难分类的样本；
• Balanced Cross Entropy是从样本分布角度对Loss添加权重因子。
  • 缺点：仅仅考虑样本分布，有些难以区分的类别的样本数可能也比较多，此时被BCE赋予了较低的权重，会导致模型很难识别该类别！

Why does Focal Loss work?

Focal Loss从样本难易分类的角度出发，解决了样本不平衡导致模型性能较低的问题。

WHY？

样本不平衡造成的问题就是，样本数少的类别分类难度大，因此Focal Loss聚焦于难分样本，解决了样本少的类别分类精度不高的问题，对于难分样本中样本多的类别，也会被Focal Loss聚焦。因此，它不仅解决了样本不平衡问题，还提升了模型整体性能。

但是，要使模型训练过程中聚焦于难分类样本，仅仅将Loss倾向于难分类样本是不够的，因为模型参数更新取决于Loss的梯度：
$$w=w-\alpha \frac{\partial L}{\partial w}$$
若Loss中难分类样本的权重较高，但是难分类样本的Loss梯度为0，难分类样本就不会影响到模型的参数更新。对于梯度问题，Focal Loss中的梯度与$x_t$的关系如下所示，其中$x_t=yx$， y ∈ { − 1 , 1 } y∈\{-1,1\} y∈{−1,1}为类别，$p_t=\sigma (x_t)$，对于易分样本，$x_t>0$，即$p_t>0.5$，由下图可知，此时的导数趋于0。对于难分样本，导数数值较大，因此，学习过程中更聚焦于难分样本。

难易分类样本是动态的，$p_t$在训练的过程中，可能会在难易之间相互转换。

在Loss梯度中，难训练样本起主导作用，参数朝着优化难训练样本的方向改变，变化之后可能会导致原本易训练的样本$p_t$变化，即变成难训练样本。若发生了这种情况会导致模型收敛速度较慢。

为了防止这种难易样本的频繁变化，应该选择较小的学习率。

针对VidHOI数据集

因为VidHOI数据集中的一个人-物对会被多个交互标签同时标注，如< human,next to & watch & hold, cup >，所以会面临multi-class multi-label的分类问题。以往常常使用Binary cross-entropy，能够计算每个交互类别独立于其他类别的损失。但是，VidHOI数据集分布不均且具有长尾分布，为了解决这个不均衡问题同时避免过分强调最频繁类别的重要性，我们采用class-balanced Focal loss：
$$\begin{gathered} C{B}_{focal}(p_i,y_i)=-\frac{1-\beta }{1-{\beta }^{n_i}}(1-p_{y_i}{)}^{\gamma }log(p_{y_i}) \\ with{p}_{y_i}=\{\begin{array}{ll}{p}_i,& \text{if }{y}_i=1\\ 1-p_i,& \text{otherwise.}\end{array} \end{gathered}$$

其中的$-(1-p_{y_i}{)}^{\gamma }log(p_{y_i})$是Lin提出的Focal loss，$p_i$表示预估为第i个类别的可能性， y i ∈ { 0 , 1 } y_i∈\{0,1\} yi​∈{0,1}表示Ground Truth的label。变量$n_i$表示第i个类别在Ground Truth下的样本量，$\beta \in [0,1)$是可调节参数。所有类别的平均损失作为一个预测的损失。

import numpy as np
import torch
import torch.nn as nn
import torch.nn.functional as F
from typing import Optional


class FocalBCEWithLogitLoss(nn.modules.loss._Loss):
    """Focal Loss with binary cross-entropy
    Implement the focal loss with class-balanced loss, using binary cross-entropy as criterion
    Following paper "Class-Balanced Loss Based on Effective Number of Samples" (CVPR2019)

    Args:
        gamma (int, optional): modulation factor gamma in focal loss. Defaults to 2.
        alpha (int, optional): modulation factor alpha in focal loss. If a integer, apply to all;
            if a list or array or tensor, regard as alpha for each class; if none, no alpha. Defaults to None.
        weight (Optional[torch.Tensor], optional): weight to each class, !not the same as alpha. Defaults to None.
        size_average (_type_, optional): _description_. Defaults to None.
        reduce (_type_, optional): _description_. Defaults to None.
        reduction (str, optional): _description_. Defaults to "mean".
    """

    def __init__(
        self,
        gamma=2,
        alpha=None,
        weight: Optional[torch.Tensor] = None,
        size_average=None,
        reduce=None,
        reduction: str = "mean",
        pos_weight: Optional[torch.Tensor] = None,
    ):
        super(FocalBCEWithLogitLoss, self).__init__(size_average, reduce, reduction)
        self.gamma = gamma
        # a number for all, or a Tensor with the same num_classes as input
        if isinstance(alpha, (list, np.ndarray)):
            self.alpha = torch.Tensor(alpha)
        else:
            self.alpha = alpha
        self.register_buffer("weight", weight)
        self.register_buffer("pos_weight", pos_weight)
        self.weight: Optional[torch.Tensor]
        self.pos_weight: Optional[torch.Tensor]

    def forward(self, input: torch.Tensor, target: torch.Tensor):
        if self.alpha is not None:
            if isinstance(self.alpha, torch.Tensor):
                alpha_t = self.alpha.repeat(input.shape[0], 1)
            else:
                alpha_t = torch.ones_like(input) * self.alpha
        else:
            alpha_t = None
		# 二元交叉熵
        ce = F.binary_cross_entropy_with_logits(input, target, reduction="none")
        # pt = torch.exp(-ce)
        # modulator = ((1 - pt) ** self.gamma)
        # following author's repo https://github.com/richardaecn/class-balanced-loss/blob/master/src/cifar_main.py#L226-L266
        # explaination https://github.com/richardaecn/class-balanced-loss/issues/1
        # A numerically stable implementation of modulator.
        if self.gamma == 0.0:
            modulator = 1.0
        else:
            # e^(-gamma*target*input - gamma*log(1+e^(-input)))
            modulator = torch.exp(-self.gamma * target * input - self.gamma * torch.log1p(torch.exp(-input)))
        # focal loss
        fl_loss = modulator * ce
        # alpha
        if alpha_t is not None:
            alpha_t = alpha_t * target + (1 - alpha_t) * (1 - target)
            fl_loss = alpha_t * fl_loss
        # pos weight
        if self.pos_weight is not None:
            fl_loss = self.pos_weight * fl_loss
        # reduction
        if self.reduction == "mean":
            return fl_loss.mean()
        elif self.reduction == "sum":
            return fl_loss.sum()
        else:
            return fl_loss

$$\begin{gathered} C{B}_{focal}(p_i,y_i)=-\frac{1-\beta }{1-{\beta }^{n_i}}(1-p_{y_i}{)}^{\gamma }log(p_{y_i}) \\ with{p}_{y_i}=\{\begin{array}{ll}{p}_i,& \text{if }{y}_i=1\\ 1-p_i,& \text{otherwise.}\end{array} \end{gathered}$$

原始版本的代码：

def focal_loss(labels, logits, alpha, gamma):
  """Compute the focal loss between `logits` and the ground truth `labels`.

  Focal loss = -alpha_t * (1-pt)^gamma * log(pt)
  where pt is the probability of being classified to the true class.
  pt = p (if true class), otherwise pt = 1 - p. p = sigmoid(logit).

  Args:
    labels: A float32 tensor of size [batch, num_classes].
    logits: A float32 tensor of size [batch, num_classes].
    alpha: A float32 tensor of size [batch_size]
      specifying per-example weight for balanced cross entropy.
    gamma: A float32 scalar modulating loss from hard and easy examples.
  Returns:
    focal_loss: A float32 scalar representing normalized total loss.
  """
  with tf.name_scope('focal_loss'):
    logits = tf.cast(logits, dtype=tf.float32)
    cross_entropy = tf.nn.sigmoid_cross_entropy_with_logits(
        labels=labels, logits=logits)

    # positive_label_mask = tf.equal(labels, 1.0)
    # probs = tf.sigmoid(logits)
    # probs_gt = tf.where(positive_label_mask, probs, 1.0 - probs)
    # # With gamma < 1, the implementation could produce NaN during back prop.
    # modulator = tf.pow(1.0 - probs_gt, gamma)

    # A numerically stable implementation of modulator.
    if gamma == 0.0:
      modulator = 1.0
    else:
      modulator = tf.exp(-gamma * labels * logits - gamma * tf.log1p(
          tf.exp(-1.0 * logits)))

    loss = modulator * cross_entropy

    weighted_loss = alpha * loss
    focal_loss = tf.reduce_sum(weighted_loss)
    # Normalize by the total number of positive samples.
    focal_loss /= tf.reduce_sum(labels)
  return focal_loss

Reference

1. https://zhuanlan.zhihu.com/p/266023273
2. https://github.com/nizhf/hoi-prediction-gaze-transformer
3. https://github.com/richardaecn/class-balanced-loss/blob/master/src/cifar_main.py#L226-L266