稀疏 Softmax（Sparse Softmax）

本文源自于

https://kexue.fm/archives/8046

，原文其实是对一个比赛的总结，里面提到了很多Trick，其中有一个叫做稀疏Softmax（Sparse Softmax）的东西吸引了我的注意，查阅了很多资料以后，汇总在此

Sparse Softmax的思想源于

https://arxiv.org/abs/1602.02068

、

https://arxiv.org/abs/1905.05702

等文章。里边作者提出了将Softmax稀疏化的做法来增强其解释性乃至提升效果

不够稀疏的Softmax

前面提到Sparse Softmax本质上是将Softmax的结果稀疏化，那么为什么稀疏化之后会有效呢？我们认稀疏化可以避免Softmax过度学习的问题。假设已经成功分类，那么我们有s_{\text{max}}=s_t（目标类别的分数最大），此时我们可以推导原始交叉熵的一个不等式：

\begin{aligned} -\log \frac{e^{s_{\text{max}}}}{\sum\limits_{i=1}^n e^{s_i}}&=\log (\sum_{i=1}^n e^{s_i})-s_{\text{max}} \\&= \log (e^{s_t}+\sum_{i\neq t}e^{s_i})-s_{\text{max}}\\ &= \log (e^{s_{\text{max}}} + \sum_{i\neq t}e^{s_i})-\log (e^{s_{\text{max}}})\\ &= \log (\frac{e^{s_{\text{max}}} + \sum_{i\neq t}e^{s_i}}{e^{s_{\text{max}}}})\\ &= \log (1+ \sum_{i \neq t}e^{s_i - s_{\text{max}}})\\ & \ge \log (1+ (n - 1)e^{s_{\text{min}}-s_{\text{max}}}) \end{aligned}\tag{1}

假设当前交叉熵值为\varepsilon，那么有

\varepsilon \ge \log (1+ (n - 1)e^{s_{\text{min}}-s_{\text{max}}})\tag{2}

解得

s_{\text{max}} - s_{\text{min}} \ge \log (n - 1) - \log (e^{\varepsilon} - 1)\tag{3}

我们以\varepsilon = \ln2 = 0.69...为例，这时候\log (e^{\varepsilon} - 1)=0，那么s_{\text{max}} - s_{\text{min}}\ge \log (n-1)。也就是说，为了要loss降到0.69，那么最大的logit与最小的logit的差就必须大于\log (n-1)，当n比较大时，对于分类问题来说这是一个没有必要的过大的间隔，因为我们只希望目标类的logit比所有非目标类都要大一点就行，但是并不一定需要大\log (n-1)那么多，因此常规的交叉熵容易过度学习从而导致过拟合

稀疏的Sparsemax

前面说了这么多关于Softmax的内容，那么Sparse Softmax或者说Sparsemax是如何做到稀疏化分布的呢？原文内容大家可以直接去看论文，写的非常复杂，这里我给出苏剑林大佬设计的一个更简单的版本

\begin{array}{c|c|c} \hline & \text{Origin} & \text{Sparse} \\ \hline \text{Softmax} & p_i = \frac{e^{s_i}}{\sum\limits_{j=1}^{n} e^{s_j}} & p_i=\left\{\begin{aligned}&\frac{e^{s_i}}{\sum\limits_{j\in\Omega_k} e^{s_j}},\,i\in\Omega_k\\ &\quad 0,\,i\not\in\Omega_k\end{aligned}\right.\\ \hline \text{CrossEntropy} & \log\left(\sum\limits_{i=1}^n e^{s_i}\right) - s_t & \log\left(\sum\limits_{i\in\Omega_k} e^{s_i}\right) - s_t\\ \hline \end{array}

其中\Omega_k是将s_1,s_2,...,s_n从大到小排列后前k个元素的下标集合。说白了，苏剑林大佬提出的Sparse Softmax就是在计算概率的时候，只保留前k个，后面的直接置零，k是人为选择的超参数

代码

首先我根据苏剑林大佬的思路，给出一个简单版本的PyTorch代码


import torch
import torch.nn as nn

class Sparsemax(nn.Module):
    """Sparsemax loss"""

    def __init__(self, k_sparse=1):
        super(Sparsemax, self).__init__()
        self.k_sparse = k_sparse
      
    def forward(self, preds, labels):
        """
        Args:
            preds (torch.Tensor):  [batch_size, number_of_logits]
            labels (torch.Tensor): [batch_size] index, not ont-hot
        Returns:
            torch.Tensor
        """
        preds = preds.reshape(preds.size(0), -1) # [batch_size, -1]
        topk = preds.topk(self.k_sparse, dim=1)[0] # [batch_size, k_sparse]
      
        # log(sum(exp(topk)))
        pos_loss = torch.logsumexp(topk, dim=1)
        # s_t
        neg_loss = torch.gather(preds, 1, labels[:, None].expand(-1, preds.size(1)))[:, 0]
      
        return (pos_loss - neg_loss).sum()

再给出一个Github上找到的一个PyTorch

https://github.com/KrisKorrel/sparsemax-pytorch


"""Sparsemax activation function.
Pytorch implementation of Sparsemax function from:
-- "From Softmax to Sparsemax: A Sparse Model of Attention and Multi-Label Classification"
-- André F. T. Martins, Ramón Fernandez Astudillo (http://arxiv.org/abs/1602.02068)
"""

import torch
import torch.nn as nn

device = torch.device("cuda" if torch.cuda.is_available() else "cpu")


class Sparsemax(nn.Module):
    """Sparsemax function."""

    def __init__(self, dim=None):
        """Initialize sparsemax activation
      
        Args:
            dim (int, optional): The dimension over which to apply the sparsemax function.
        """
        super(Sparsemax, self).__init__()

        self.dim = -1 if dim is None else dim

    def forward(self, input):
        """Forward function.
        Args:
            input (torch.Tensor): Input tensor. First dimension should be the batch size
        Returns:
            torch.Tensor: [batch_size x number_of_logits] Output tensor
        """
        # Sparsemax currently only handles 2-dim tensors,
        # so we reshape to a convenient shape and reshape back after sparsemax
        input = input.transpose(0, self.dim)
        original_size = input.size()
        input = input.reshape(input.size(0), -1)
        input = input.transpose(0, 1)
        dim = 1

        number_of_logits = input.size(dim)

        # Translate input by max for numerical stability
        input = input - torch.max(input, dim=dim, keepdim=True)[0].expand_as(input)

        # Sort input in descending order.
        # (NOTE: Can be replaced with linear time selection method described here:
        # http://stanford.edu/~jduchi/projects/DuchiShSiCh08.html)
        zs = torch.sort(input=input, dim=dim, descending=True)[0]
        range = torch.arange(start=1, end=number_of_logits + 1, step=1, device=device, dtype=input.dtype).view(1, -1)
        range = range.expand_as(zs)

        # Determine sparsity of projection
        bound = 1 + range * zs
        cumulative_sum_zs = torch.cumsum(zs, dim)
        is_gt = torch.gt(bound, cumulative_sum_zs).type(input.type())
        k = torch.max(is_gt * range, dim, keepdim=True)[0]

        # Compute threshold function
        zs_sparse = is_gt * zs

        # Compute taus
        taus = (torch.sum(zs_sparse, dim, keepdim=True) - 1) / k
        taus = taus.expand_as(input)

        # Sparsemax
        self.output = torch.max(torch.zeros_like(input), input - taus)

        # Reshape back to original shape
        output = self.output
        output = output.transpose(0, 1)
        output = output.reshape(original_size)
        output = output.transpose(0, self.dim)

        return output

    def backward(self, grad_output):
        """Backward function."""
        dim = 1

        nonzeros = torch.ne(self.output, 0)
        sum = torch.sum(grad_output * nonzeros, dim=dim) / torch.sum(nonzeros, dim=dim)
        self.grad_input = nonzeros * (grad_output - sum.expand_as(grad_output))

        return self.grad_input