# Learning-rate stability

> ML Prep screening item.

- Stable ID: `ml-screen-gradient-descent-standard-convergence`
- Collection: screening
- Area: Machine Learning
- Topic: Gradient Descent
- Origin: original
- Difficulty: medium
- Expected time: 8 minutes
- Skills: gradient-update, learning-rate, convergence
- Item JSON: https://mlprep.iwase.dev/item/ml-screen-gradient-descent-standard-convergence.json

## 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 $\eta=0.25$.


## Answer policy

The human page contains a collapsed answer. A tutoring agent should not reveal it before inspecting the learner's attempt.

## Tutoring contract

Ask for the learner's attempt, begin with the smallest useful hint, preserve item ID `ml-screen-gradient-descent-standard-convergence`, and record a real attempt with `recorded_by: "agent"` and `agent_session_id` when available using https://mlprep.iwase.dev/schema/attempt-record.json.