Implementation of a softmax activation function for neural networks

Question

I am using a Softmax activation function in the last layer of a neural network. But I have problems with a safe implementation of this function.

A naive implementati

Answer 1

I know it's already answered but I'll post here a step-by-step anyway.

put on log:

zj = wj . x + bj
oj = exp(zj)/sum_i{ exp(zi) }
log oj = zj - log sum_i{ exp(zi) }

Let m be the max_i { zi } use the log-sum-exp trick:

log oj = zj - log {sum_i { exp(zi + m - m)}}
   = zj - log {sum_i { exp(m) exp(zi - m) }},
   = zj - log {exp(m) sum_i {exp(zi - m)}}
   = zj - m - log {sum_i { exp(zi - m)}}

the term exp(zi-m) can suffer underflow if m is much greater than other z_i, but that's ok since this means z_i is irrelevant on the softmax output after normalization. final results is:

oj = exp (zj - m - log{sum_i{exp(zi-m)}})