The cross-entropy gradient is the prediction error

ProofBeginner⏱ 15 minLLMssoftmaxcross entropygradientfused kernel

Theory: Language Modelling, Cross Entropy and Perplexity →

Problem

Show $\partial\ell/\partial z = q - e_{v^\star}$ for $\ell = -\log q_{v^\star}$ with $q = \operatorname{softmax}(z)$ , and explain why softmax and cross-entropy are implemented as a single fused operation.

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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