使用 Pytorch 分布与手动计算 KL 散度的不同结果

Question

我注意到使用

时 KL-Divergence 项 KL(Q(x)||P(x)) 的计算方式不同

mean(Q(x)*(log Q(x) - log P(x)))

对

torch.distributions.kl_divergence(Q, P)

哪里

Q = torch.distributions.Normal(some mean, some sigma)
P = torch.distributions.Normal(0, 1)

当我绘制 KL 散度损失时，我得到了这两个相似但不同的图： here

谁能指出造成这种差异的原因？

完整代码如下：

import numpy as np
import torch
import torch.distributions as dist
import matplotlib.pyplot as plt

def kl_1(log_qx, log_px):
    """
    inputs: [B, z_dim] torch
    """
    return (log_qx.exp() * (log_qx-log_px)).mean()

# ground-truth (target) P(x)
P = dist.Normal(0, 1)


mus = np.arange(-5, 5, 0.1)
sigma = 1
N = 100
kls = {"1": [], "2": []}
for mu in mus:
    # prediction (current) Q(x)
    Q = dist.Normal(mu, sigma)
    
    # sample from Q
    qx = Q.sample((N,))
        
    # log prob
    log_qx = Q.log_prob(qx)
    log_px = P.log_prob(qx)
    
    # kl 1
    kl1 = kl_1(log_qx, log_px)
    kls['1'].append(kl1.numpy())
    
    # kl 2
    kl2 = dist.kl_divergence(Q, P)
    kls['2'].append(kl2.numpy())
    
plt.figure()
plt.scatter(mus, kls['1'], label="Q*(logQ-logP)")
plt.scatter(mus, kls['2'], label="kl_divergence")
plt.xlabel("mean of Q(x)")
plt.ylabel("computed KL Divergence")
plt.legend()
plt.show()

Answer 1

如果要根据 dx 上的积分计算期望值，则样本按概率密度加权。如果您使用给定分布的样本，那么您可以直接将期望值近似为平均值，这对应于 d cq(x) 上的积分，因此 d cq(x) = q(x) dx，其中 cq(x) 是累积概率函数， q(x) id 变量的概率密度函数 Q.

import numpy as np
import torch
import torch.distributions as dist
import matplotlib.pyplot as plt

def kl_1(log_qx, log_px):
    """
    inputs: [B, z_dim] torch
    """
    return (log_qx-log_px).mean()

# ground-truth (target) P(x)
P = dist.Normal(0, 1)


mus = np.arange(-5, 5, 0.1)
sigma = 1
N = 100
kls = {"1": [], "2": []}
for mu in mus:
    # prediction (current) Q(x)
    Q = dist.Normal(mu, sigma)
    
    # sample from Q
    qx = Q.sample((N,))
        
    # log prob
    log_qx = Q.log_prob(qx)
    log_px = P.log_prob(qx)
    
    # kl 1
    kl1 = kl_1(log_qx, log_px)
    kls['1'].append(kl1.numpy())
    
    # kl 2
    kl2 = dist.kl_divergence(Q, P)
    kls['2'].append(kl2.numpy())
    
plt.figure()
plt.scatter(mus, kls['1'], label="Q*(logQ-logP)")
plt.scatter(mus, kls['2'], label="kl_divergence")
plt.xlabel("mean of Q(x)")
plt.ylabel("computed KL Divergence")
plt.legend()

使用 Pytorch 分布与手动计算 KL 散度的不同结果

Different results in computing KL Divergence using Pytorch Distributions vs manually

python

machine-learning

pytorch