The RNN gradient is a repeated matrix product

ProofIntermediate⏱ 30 minDeep Learningrnnvanishing gradientbpttspectral radius

Theory: Convolution, Recurrence and Normalisation →

Problem

Derive the backpropagation-through-time gradient for a vanilla RNN and show that its norm is controlled by $(\|\rho'\|_\infty\|W\|)^{T}$ . Explain why the repeated use of the same $W$ makes the situation worse than for a feedforward network of the same depth.

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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