Balancing noise and approximation bias (${\alpha }_{\ast }=\sqrt{A\delta /(Bc)}$)

An error bound $A\delta /\alpha +Bc\alpha$, with positive $A,B,c$, is minimized at ${\alpha }_{\ast }=\sqrt{A\delta /(Bc)}$, with value $2\sqrt{ABc\delta }$. Differentiate the scalar bound, or apply the arithmetic-geometric mean inequality. This is an a priori regularization parameter choice for a linear regularization with inverse-parameter noise amplification and first-order approximation bias; the parameter must also lie in the method's allowed interval.