$S = \gamma G + \beta$
$M = ge(S,0)$
$O = O_p * M + O_s * (1-M)$

let $\alpha = layers*batchsize*spatialsize$
$L_1 = crossentropy$
$L_2 = \frac{1}{\alpha}\sum{\left\|M\right\|_1}$
$loss = L_1 + \lambda L_2$

$\frac{\partial{L_1}}{\partial{S}}=\frac{\partial{L_1}}{\partial{M}}\frac{\partial{M}}{\partial{S}} =(O_p - O_s) \frac{\partial{M}}{\partial{S}}$

$\frac{\partial{L_2}}{\partial{S}}=\frac{\partial{L_2}}{\partial{M}}\frac{\partial{M}}{\partial{S}} =\frac{1}{\alpha} \frac{\partial{M}}{\partial{S}}$

Case Standard

$\frac{\partial{M}}{\partial{S}}=M$

$\frac{\partial{L_1}}{\partial{S}}=M(O_p - O_s)$

$\frac{\partial{L_1}}{\partial{\gamma}}=\frac{\partial{L_1}}{\partial{S}}\frac{\partial{S}}{\partial{\gamma}} =G*M*(O_p - O_s)$

$\frac{\partial{L_1}}{\partial{\beta}}=\frac{\partial{L12}}{\partial{S}}\frac{\partial{S}}{\partial{\beta}} =M*(O_p - O_s)$

$\frac{\partial{L_2}}{\partial{S}}=\frac{M}{\alpha}$

$\frac{\partial{L_2}}{\partial{\gamma}}=\frac{\partial{L_2}}{\partial{S}}\frac{\partial{S}}{\partial{\gamma}} =\frac{M * G}{\alpha}$

$\frac{\partial{L_2}}{\partial{\beta}}=\frac{\partial{L_2}}{\partial{S}}\frac{\partial{S}}{\partial{\beta}} =\frac{M}{\alpha}$

Conclusion

$\frac{\partial{L}}{\partial{\gamma}}=\frac{\partial{L_1}}{\partial{\gamma}}+\lambda \frac{\partial{L_2}}{\partial{\gamma}} = M*G(\frac{\lambda}{\alpha} + O_p - O_s)$

$\frac{\partial{L}}{\partial{\beta}}=\frac{\partial{L_1}}{\partial{\beta}}+\lambda \frac{\partial{L_2}}{\partial{\beta}} = M(\frac{\lambda}{\alpha} + O_p - O_s)$

Case Identity

$\frac{\partial{M}}{\partial{S}}=1$

$\frac{\partial{L_1}}{\partial{S}}=O_p - O_s$

$\frac{\partial{L_1}}{\partial{\gamma}}=\frac{\partial{L_1}}{\partial{S}}\frac{\partial{S}}{\partial{\gamma}} =G*(O_p - O_s)$

$\frac{\partial{L_1}}{\partial{\beta}}=\frac{\partial{L12}}{\partial{S}}\frac{\partial{S}}{\partial{\beta}} =O_p - O_s$

$\frac{\partial{L_2}}{\partial{S}}=\frac{1}{\alpha}$

$\frac{\partial{L_2}}{\partial{\gamma}}=\frac{\partial{L_2}}{\partial{S}}\frac{\partial{S}}{\partial{\gamma}} =\frac{G}{\alpha}$

$\frac{\partial{L_2}}{\partial{\beta}}=\frac{\partial{L_2}}{\partial{S}}\frac{\partial{S}}{\partial{\beta}} =\frac{1}{\alpha}$

Conclusion

$\frac{\partial{L}}{\partial{\gamma}}=\frac{\partial{L_1}}{\partial{\gamma}}+\lambda \frac{\partial{L_2}}{\partial{\gamma}} = G*(\frac{\lambda}{\alpha} + O_p - O_s)$

$\frac{\partial{L}}{\partial{\beta}}=\frac{\partial{L_1}}{\partial{\beta}}+\lambda \frac{\partial{L_2}}{\partial{\beta}} = \frac{\lambda}{\alpha} + O_p - O_s$

Case Reverse

$S = \gamma G + \beta$
$M_1 = ge_1(S,0)$
$M_2 = ge_2(S,0)$

$M_1$ should be equal to $M_2$
$O = O_p * M_1 + O_s * (1-M_2)$

$\frac{\partial{M_1}}{\partial{S}}=M_1$
$\frac{\partial{M_2}}{\partial{S}}=1-M_2$

$\frac{\partial{L_1}}{\partial{S}}= \frac{\partial{L_1}}{\partial{O}}(\frac{\partial{O}}{\partial{M_1}}\frac{\partial{M_1}}{\partial{S}} + \frac{\partial{O}}{\partial{M_2}}\frac{\partial{M_2}}{\partial{S}}) = M_1*O_p - (1-M_2)*O_s$

$\frac{\partial{L_1}}{\partial{\gamma}}=\frac{\partial{L_1}}{\partial{S}}\frac{\partial{S}}{\partial{\gamma}} =G*(M_1*O_p - (1-M_2)*O_s)$

$\frac{\partial{L_1}}{\partial{\beta}}=\frac{\partial{L12}}{\partial{S}}\frac{\partial{S}}{\partial{\beta}} =M_1*O_p - (1-M_2)*O_s$

$\frac{\partial{L_2}}{\partial{S}}=\frac{M_1}{\alpha}$

$\frac{\partial{L_2}}{\partial{\gamma}}=\frac{\partial{L_2}}{\partial{S}}\frac{\partial{S}}{\partial{\gamma}} =\frac{M_1*G}{\alpha}$

$\frac{\partial{L_2}}{\partial{\beta}}=\frac{\partial{L_2}}{\partial{S}}\frac{\partial{S}}{\partial{\beta}} =\frac{M_1}{\alpha}$

Conclusion

$\frac{\partial{L}}{\partial{\gamma}}=\frac{\partial{L_1}}{\partial{\gamma}}+\lambda \frac{\partial{L_2}}{\partial{\gamma}} = G*(\frac{\lambda}{\alpha} + M_1*O_p - (1-M_2)*O_s)$

$\frac{\partial{L}}{\partial{\beta}}=\frac{\partial{L_1}}{\partial{\beta}}+\lambda \frac{\partial{L_2}}{\partial{\beta}} = \frac{\lambda}{\alpha} + M_1*O_p - (1-M_2)*O_s$