loss函数之KLDivLoss

KL散度KL散度，又叫相对熵，用于衡量两个连续分布之间的距离，公式如下：DKL(p∥q)=∑i=1Np(xi)⋅(log⁡p(xi)−log⁡q(xi))D_{K L}(p \| q)=\sum_{i=1}^{N} p\left(x_{i}\right) \cdot\left(\log p\left(x_{i}\right)-\log q\left(x_{i}\right)\right)DKL(

旺旺棒棒冰

8739人浏览 · 2021-07-12 15:01:51

旺旺棒棒冰 · 2021-07-12 15:01:51 发布

KL散度

KL散度，又叫相对熵，用于衡量两个分布（离散分布和连续分布）之间的距离。

设 $p (x)$ 、 $q (x)$ 是离散随机变量 $X$ 的两个概率分布，则 $p$ 对 $q$ 的KL散度是:

$D_{K L}(p \| q)=E_{p(x)} \log \frac{p(x)}{q(x)}=\sum_{i=1}^{N} p\left(x_{i}\right) \cdot\left(\log p\left(x_{i}\right)-\log q\left(x_{i}\right)\right)$

KLDivLoss

对于包含 $N$ 个样本的batch数据 $D (x, y)$ ， $x$ 是神经网络的输出，并且进行了归一化和对数化； $y$ 是真实的标签（默认为概率）， $x$ 与 $y$ 同维度。

第 $n$ 个样本的损失值 $l_{n}$ 计算如下:

$l_{n}=y_{n} \cdot\left(\log y_{n}-x_{n}\right)$

class KLDivLoss(_Loss):
    __constants__ = ['reduction']
    def __init__(self, size_average=None, reduce=None, reduction='mean'):
        super(KLDivLoss, self).__init__(size_average, reduce, reduction)
    def forward(self, input, target):
        return F.kl_div(input, target, reduction=self.reduction)

pytorch中通过torch.nn.KLDivLoss类实现，也可以直接调用F.kl_div 函数，代码中的size_average与reduce已经弃用。reduction有四种取值mean,batchmean, sum, none，对应不同的返回 $\ell(x, y)$ 。默认为mean

$L=\left\{l_{1}, \ldots, l_{N}\right\}$

$\ell(x, y)=\left\{\begin{array}{ll}\operatorname L, & \text { if reduction }=\text { 'none' } \\ \operatorname{mean}(L), & \text { if reduction }=\text { 'mean' } \\ N*\operatorname {mean}(L), & \text { if reduction }=\text { 'batchmean' } \\ \operatorname{sum}(L), & \text { if reduction }=\text { 'sum' }\end{array} \right.$

例子：

import torch
import torch.nn as nn
import math

def validate_loss(output, target):
    val = 0
    for li_x, li_y in zip(output, target):
        for i, xy in enumerate(zip(li_x, li_y)):
            x, y = xy
            loss_val = y * (math.log(y, math.e) - x)
            val += loss_val
    return val / output.nelement()

torch.manual_seed(20)
loss = nn.KLDivLoss()
input = torch.Tensor([[-2, -6, -8], [-7, -1, -2], [-1, -9, -2.3], [-1.9, -2.8, -5.4]])
target = torch.Tensor([[0.8, 0.1, 0.1], [0.1, 0.7, 0.2], [0.5, 0.2, 0.3], [0.4, 0.3, 0.3]])
output = loss(input, target)
print("default loss:", output)

output = validate_loss(input, target)
print("validate loss:", output)

loss = nn.KLDivLoss(reduction="batchmean")
output = loss(input, target)
print("batchmean loss:", output)

loss = nn.KLDivLoss(reduction="mean")
output = loss(input, target)
print("mean loss:", output)

loss = nn.KLDivLoss(reduction="none")
output = loss(input, target)
print("none loss:", output)

输出：

default loss: tensor(0.6209)
validate loss: tensor(0.6209)
batchmean loss: tensor(1.8626)
mean loss: tensor(0.6209)
none loss: tensor([[1.4215, 0.3697, 0.5697],
        [0.4697, 0.4503, 0.0781],
        [0.1534, 1.4781, 0.3288],
        [0.3935, 0.4788, 1.2588]])