Normalisation makes weights scale-invariant

ProofAdvanced⏱ 30 minDeep Learningbatch normscale invarianceweight decayeffective learning rate

Theory: Convolution, Recurrence and Normalisation →

Problem

Show that a normalised layer satisfies $\hat z(cW) = \hat z(W)$ for $c>0$ , deduce $\nabla_W\mathcal J(cW) = \tfrac1c\nabla_W\mathcal J(W)$ , and explain the consequence for weight decay and the effective learning rate.

Write a complete argument. Every inequality you use should be one you could state precisely.

Solution

Opening this before you have a real attempt on paper is the fastest way to feel like you understand something you cannot reproduce.

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