ImplementCore implementation

Gradient descent: predict convergence on a ravine

Implement `run(lr, x0, y0, steps)` that applies gradient descent on f(x, y) = x^2 + 10y^2 and returns the final loss value after `steps` iterations. Update rule: x <- x - lr * 2x, y <- y - lr * 20y. On this ravine, stability breaks near lr ~ 0.1 (the steep y-axis overcorrects). Return the final loss rounded to 6 decimal places.

python12 mindifficulty 2/4

From Adam: A Method for Stochastic Optimization

Sample tests

run([0.05,1,1,25])→0.005154
run([0.01,1,1,25])→0.364312

Hints

The lab models f(x, y) = x^2 + 10y^2 — a ravine that punishes big steps along the steep axis.
Gradients: df/dx = 2x, df/dy = 20y. The update is x <- x - lr*2x, y <- y - lr*20y.
Stability on this surface breaks near lr ~ 0.1; below that the step descends, above it y overcorrects.
After `steps` iterations, loss = x^2 + 10*y^2. Watch out: large lr can make y diverge to huge values.

Hints

Code runs in an isolated sandbox on the OpenSandbox judge. No model key required.