# Introduction #

The softmax function isn’t supposed to output zeros or ones, but sometimes it happens due to floating-point precision when the input vector contains numbers too big or too small for the exponential inside the softmax.

Exponential growth seems slow at the beginning, but it becomes stepper in a very short time.

For example, $$e^{-114}\approx3.1\cdot10^{-50}$$ is essentially evaluated as zero.
When softmax is used with cross-entropy loss function, a zero in the former’s output becomes ±$$\infin$$ as a result of the logarithm in latter, which is theoretically correct since the adjustments to make the network adapt are infinite, but it is of no use in practice as the resulting loss could be NaN.

A zero or a one in the softmax output means that the model is very confident about the prediction, therefore the solution is to decrease that confidence to produce a softer probability distribution.

# Softmax temperature #

$q_i = \dfrac{e^{(z_i/T)}}{\sum_j{e^{(z_j/T)}}}$

T is the temperature parameter, usually set to 1.
Increasing the temperature parameter will penalize bigger $$z_i$$ values more than the smaller $$z_i$$ values, owing to the amplification effect of the exponential, which leads to a decrease in the confidence of the model.

T = 1
exp(-8/T) ~ 0.0003
exp(8/T)  ~ 2981
exp(3/T)  ~ 20

T = 1.2
exp(-8/T) ~ 0.01
exp(8/T)  ~ 786
exp(3/T)  ~ 3


In % terms, the bigger the exponent is, the more it shrinks when a temperature >1 is applied, which implies that the softmax function will assign more probability mass to the smaller samples.

Beware! A high temperature makes the NN “easily excited”, resulting in more mistakes.

# PyTorch example #

Let’s write the softmax function and the cross-entropy loss function.

def softmax(input, t=1.0):
print("input", input)
ex = torch.exp(input/t)
print("exp", ex)
sum = torch.sum(ex, axis=0)
return ex / sum

def cross_entropy(distribution):
target = torch.tensor([0, 0, 1, 0, 0])
print("loss", -torch.sum(target * torch.log(distribution)))


Define a sample containing some large absolute values and apply the softmax function, then the cross-entropy loss.

input = torch.tensor([55.8906, -114.5621, 6.3440, -30.2473, -44.1440])
cross_entropy(softmax(input))

input tensor([  55.8906, -114.5621,    6.3440,  -30.2473,  -44.1440])
exp tensor([1.8749e+24, 0.0000e+00, 5.6907e+02, 7.3074e-14, 6.7376e-20])
loss tensor(nan)


The resulting probability distribution contains a zero, the loss value is NaN.
Let’s see what happens by setting the temperature to 10.

input = torch.tensor([55.8906, -114.5621, 6.3440, -30.2473, -44.1440])
cross_entropy(softmax(input, t=10))

input tensor([  55.8906, -114.5621,    6.3440,  -30.2473,  -44.1440])
exp tensor([2.6748e+02, 1.0584e-05, 1.8859e+00, 4.8571e-02, 1.2102e-02])
loss tensor(4.9619)


The probability is more equally distributed, the softmax function has assigned more probability mass to the smallest sample, from 0 to 1.0584e-05, and less probability mass to the largest sample, from 1.8749e+24 to 2.6748e+02.
Finally, the loss has changed from NaN to a valid value.