乙状结肠的Jax自动标记总是返回nan

Question

我试图区分一个函数，该函数近似于一个高斯分数，该分数包含在给定移动平均值的2个极限（一个截断高斯）内。 jnp.grad不允许我区分相加布尔过滤器（注释行），因此我不得不临时使用一个sigmoid

然而，当截断边界很高时，梯度总是nan，我不明白为什么

在下面的例子中，我正在计算一个平均值为0，std=1的高斯梯度，然后用x移动它

如果减少边界，则函数的行为与预期的一样。但这不是一个解决方案。 当边界较高时belows始终变为1。但是如果是这种情况，x对下面没有影响，那么它对梯度的贡献应该是0而不是nan。但如果我返回belows[0][0]而不是jnp.mean(filt, axis=0)，我仍然得到nan

有什么想法吗？ 提前感谢（github上还有一个问题有待解决）

import os

from tqdm import tqdm

os.environ["XLA_FLAGS"] = '--xla_force_host_platform_device_count=4' # Use 8 CPU devices
import numpy as np
from jax.config import config
config.update("jax_enable_x64", True)
import jax
import jax.numpy as jnp
from jax import vmap

from functools import reduce

def sigmoid(x, scale=100):
    return 1 / (1 + jnp.exp(-x*scale))

def above_lower(x, l, scale=100):
    return sigmoid(x - l, scale)

def below_upper(x, u, scale=100):
    return 1 - sigmoid(x - u, scale)

def combine_soft_filters(a):
    return jnp.prod(jnp.stack(a), axis=0)


def fraction_not_truncated(mu, v, limits, stdnorm_samples):
    L = jnp.linalg.cholesky(v)
    y = vmap(lambda x: jnp.dot(L, x))(stdnorm_samples) + mu
    # filt = reduce(jnp.logical_and, [(y[..., i] > l) & (y[..., i] < u) for i, (l, u) in enumerate(limits)])
    aboves = [above_lower(y[..., i], l) for i, (l, u) in enumerate(limits)]
    belows = [below_upper(y[..., i], u) for i, (l, u) in enumerate(limits)]
    filt = combine_soft_filters(aboves+belows)
    return jnp.mean(filt, axis=0)

limits = np.array([
        [0.,1000],
])

stdnorm_samples = np.random.multivariate_normal([0], np.eye(1), size=1000)

def func(x):
    return fraction_not_truncated(jnp.zeros(1)+x, jnp.eye(1), limits, stdnorm_samples)

_x = np.linspace(-2, 2, 500)
gradfunc = jax.grad(func)
vals = [func(x) for x in tqdm(_x)]
grads = [gradfunc(x) for x in tqdm(_x)]
print(vals)
print(grads)
import matplotlib.pyplot as plt
plt.plot(_x, np.asarray(vals))
plt.ylabel('f(x)')
plt.twinx()
plt.plot(_x, np.asarray(grads), c='r')
plt.ylabel("f(x)'")
plt.title('Fraction not truncated')
plt.axhline(0, color='k', alpha=0.2)
plt.xlabel('shift')
plt.tight_layout()
plt.show()

[DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64), DeviceArray(1., dtype=float64)]
[DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64), DeviceArray(nan, dtype=float64)]

Answer 1

问题在于sigmoid函数的实现方式使得自动确定的梯度对于x的大负值不稳定：

import jax.numpy as jnp
import jax

def sigmoid(x, scale=100):
    return 1 / (1 + jnp.exp(-x*scale))

print(jax.grad(sigmoid)(-1000.0))
# nan

您可以使用jax.make_jaxpr函数来内省自动确定的梯度（注释是我的注释）产生的操作，从而了解为什么会发生这种情况：

>>> jax.make_jaxpr(jax.grad(sigmoid))(-1000.0)
{ lambda  ; a.                    # a = -1000
  let b = neg a                   # b = 1000
      c = mul b 100.0             # c = 100,000
      d = exp c                   # d = inf
      e = add d 1.0
      _ = div 1.0 e
      f = integer_pow[ y=-2 ] e   # f = 0
      g = mul 1.0 f               # g = 0
      h = mul g 1.0               # h = 0
      i = neg h                   # i = 0
      j = mul i d                 # j = 0 * inf = NaN
      k = mul j 100.0             # k = NaN
      l = neg k                   # l = NaN
  in (l,) }                       # return NaN

这是64位浮点运算失败的情况之一：它没有处理像exp(100000)这样的数字的范围

那你能做什么呢？一个重要的选项是使用custom derivative rule来告诉autodiff如何以更稳定的方式处理sigmoid函数。不过，在这种情况下，一个更简单的选择是根据在autodiff转换下表现更好的内容来重新表达sigmoid函数。一个选择是：

def sigmoid(x, scale=100):
    return 0.5 * (jnp.tanh(x * scale / 2) + 1)

在脚本中使用此版本可以修复此问题