Learning-rate stability

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Problem

Let $f(u,v)=\frac{1}{2}(u^2+9v^2)$ . Derive the fixed learning-rate interval that converges from every start. From $(1,1)$ , compute one update for $\eta=0.2$ and explain what happens for $\eta=0.25$ .

Reveal answer or reference solution

The updates are

u_{\text{next}}=(1-\eta)u,\qquad v_{\text{next}}=(1-9\eta)v.

Both multipliers must have magnitude less than $1$ , giving

0<\eta<\frac{2}{9}.

At $\eta=0.2$ , the next point is $(0.8,-0.8)$ . At $\eta=0.25$ , the $v$ multiplier is $-1.25$ , so nonzero $v$ oscillates with growing magnitude and diverges.

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Learning-rate stability

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