Choosing λ

$\lambda$ controls the strength of the penalty. $\lambda =0$: no regularisation, risk of overfitting. Very large $\lambda$: all weights near zero, underfitting. The best value is somewhere in between and depends on the data.

Choose it with cross-validation: try values on a logarithmic grid ($10^{-4},10^{-3},\dots ,10^2$), compute the cross-validated error for each, and pick the minimum. Plotting error against $\log \lambda$ gives the U-shaped validation curve.

The one-standard-error rule picks the largest $\lambda$ whose error is within one standard error of the best — a simpler model that is statistically just as good.