Algorithm 6 Weighted Softmax Loss Function—Variant 1

Require: y, $\widehat{y}$ $\widehat{y}\leftarrow exp(\widehat{y}-max\left(\widehat{y}\right))$ $\widehat{y}\leftarrow \widehat{y}/\sum \left(\widehat{y}\right)$ $grad\leftarrow \mathrm{zero}\mathrm{matrix}\mathrm{with}\mathrm{shape}\mathrm{of}\widehat{y}$ $hess\leftarrow \mathrm{zero}\mathrm{matrix}\mathrm{with}\mathrm{shape}\mathrm{of}\widehat{y}$ $weights\leftarrow \left\{\right(0,1):0.1,(1,0):0.1,(0,2):0.17,(2,0):0.17,(1,2):0.1,(2,1):0.1\}$ for $i=0$ to $\mathrm{length}\mathrm{of}y-1$ do     for $j=0$ to $\mathrm{number}\mathrm{of}\mathrm{columns}\mathrm{in}\widehat{y}-1$ do         $weight\leftarrow weights[min(y\left[i\right],j),max(y\left[i\right],j\left)\right]$         if $weight$ is not set then            $weight\leftarrow 0$         end if         $prob\leftarrow \widehat{y}[i,j]$         $grad[i,j]\leftarrow weight\times (prob-(y\left[i\right]==j\left)\right)$         $hess[i,j]\leftarrow weight\times prob\times (1-prob)+0.02$     end for end forreturn  $grad,hess$