Understanding the Disharmony between Weight Normalization Family and Weight Decay: $\epsilon-$shifted $L_2$ Regularizer

Understand

The merits of fast convergence and potentially better performance of the weight normalization family have drawn increasing attention in recent years.

• These methods use standardization or normalization that changes the weight $\boldsymbol{W}$ to $\boldsymbol{W}'$, which makes $\boldsymbol{W}'$ independent to the magnitude of $\boldsymbol{W}$.
• Surprisingly, $\boldsymbol{W}$ must be decayed during gradient descent, otherwise we will observe a severe under-fitting problem, which is very counter-intuitive since weight decay is widely known to prevent deep networks from over-fitting.
• In this paper, we \emph{theoretically} prove that the weight decay term $\frac{1}{2}\lambda||{\boldsymbol{W}}||^2$ merely modulates the effective learning rate for improving objective optimization, and has no influence on generalization when the weight normalization family is compositely employed.