使用Pytorch分布与手动计算KL散度的结果不同

Question

我注意到KL-divergence项KL(Q(x)|| p (x))在使用

时计算方式不同

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

和

torch.distributions.kl_divergence(Q, P)

,

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:

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

如果要从dx的积分中计算期望值，则使用概率密度对样本进行加权。如果您使用来自给定分布的样本，那么您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)是变量Q的概率密度函数。

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()