Temperature monotonically controls entropy

ProofAdvanced⏱ 30 minLLMstemperatureentropyexponential familysampling

Theory: Language Modelling, Cross Entropy and Perplexity →

Problem

Prove that $H(\operatorname{softmax}(z/\tau))$ is non-decreasing in $\tau>0$ , using the exponential-family identity $H = \psi(\beta)-\beta\psi'(\beta)$ with $\beta = 1/\tau$ . Explain why truncated sampling is a different operation from lowering the temperature.

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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