The Dial the Search Is Not Allowed to Turn

A straight-line model $\hat{y} = w x + b$ is trained on three examples:

To discourage the model from choosing a steep slope, a penalty term is bolted onto the mean squared error, giving the training objective

$J(w,\,b) \;=\; \frac{1}{3}\sum_{i=1}^{3}\big(w x_i + b - y_i\big)^2 \;+\; \lambda\,w^2$

Before training starts, an engineer sits down and fixes

$\lambda = 0.6$

by hand. It stays at $0.6$ for the entire run — the search never touches it.

Training moves only the quantities the search is actually allowed to move, and drives their partial derivatives to zero.

Find the value of $w$ at the minimum of $J$.

Round your answer to 3 decimal places.