kl 散度计算

　　KL 散度，又叫相对熵，用于衡量两个分布之间的距离。设 $p(x)$, $q(x)$ 是关于随机变量 $x$ 的两个分布，则 $p$ 相对于 $q$ 的 KL 散度为:

　　　　$D_{K L}(p \| q)=E_{p(x)} \log \frac{p(x)}{q(x)}$

　　信息论中，熵 $H(P)$ 表示对来自 $P$ 的随机变量进行编码所需的最小字节数，而交叉熵 $H(P, Q)$ 则表示使用基于 $Q$ 的编码对来自 $P$ 的变量进行编码所需的字节数，它也可以作为损失函数，即交叉熵损失函数：

　　　　$H(p, q)=-\sum_{i=1}^{n} p\left(x_{i}\right) \log \left(q\left(x_{i}\right)\right)$

　　因此，KL散度可以认为是使用基于 $Q$ 的编码对来自 $P$ 的变量进行编码所需的“额外"字节数；显然，额外字节数必然非负，当且仅当 $P=Q$ 时，额外字节数为 $0$：

　　　　$\begin{aligned}D_{K L}(p \| q) & =\sum_{i=1}^{n} p\left(x_{i}\right) \log \left(p\left(x_{i}\right)\right)-\sum_{i=1}^{n} p\left(x_{i}\right) \log \left(q\left(x_{i}\right)\right) \\& =-H(p(x))+\left[-\sum_{i=1}^{n} p\left(x_{i}\right) \log \left(q\left(x_{i}\right)\right)\right]\end{aligned}$

　　CLASS torch.nn.KLDivLoss(size_average=None, reduce=None, reduction='mean', log_target=False)

　　　　$L\left(y_{\text {pred }}, y_{\text {true }}\right)=y_{\text {true }} \cdot \log \frac{y_{\text {true }}}{y_{\text {pred }}}=y_{\text {true }} \cdot\left(\log y_{\text {true }}-\log y_{\text {pred }}\right)$

　　To avoid underflow(下溢) issues when computing this quantity, this loss expects the argument input in the log-space. The argument target may also be provided in the log-space

　　To summarise, this function is roughly equivalent to computing

if not log_target: # default
    loss_pointwise = target * (target.log() - input)
else:
    loss_pointwise = target.exp() * (target - input)

　　and then reducing this result depending on the argument reduction as

if reduction == "mean":  # default
    loss = loss_pointwise.mean()
elif reduction == "batchmean":  # mathematically correct
    loss = loss_pointwise.sum() / input.size(0)
elif reduction == "sum":
    loss = loss_pointwise.sum()
else:  # reduction == "none"
    loss = loss_pointwise

　　reduction= “mean” doesn’t return the true KL divergence value, please use reduction= “batchmean” which aligns with the mathematical definition. In a future release, “mean” will be changed to be the same as “batchmean”.

　　使用：

kl_loss = nn.KLDivLoss(reduction="batchmean")
# input should be a distribution in the log space
input = F.log_softmax(torch.randn(3, 5, requires_grad=True))
# Sample a batch of distributions. Usually this would come from the dataset
target = F.softmax(torch.rand(3, 5))
output = kl_loss(input, target)
 
kl_loss = nn.KLDivLoss(reduction="batchmean", log_target=True)
log_target = F.log_softmax(torch.rand(3, 5))
output = kl_loss(input, log_target)