Derive the optimal weight update

ProofAdvanced⏱ 1 hourQuantizationlagrangeobsclosed formkkt

Problem

Let $H \succ 0$ and fix coordinate $q$ of $w$ to $\operatorname{quant}(w_q)$ .

Derive the error-minimising $\delta$ from the KKT conditions, stating where positive definiteness is used.
Derive the incurred error $\mathcal E_q$ .
Show that if $H$ is diagonal, the optimal update touches no other coordinate — so round-to-nearest is optimal exactly in that case.

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

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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