Definition(Polyak-Lojasiewicz (PL) condition)
If Polyak-Lojasiewicz (PL) condition is met for a function $f: \mathbb{R}^n \to \mathbb{R}$, then the following holds for some $\mu>0$
$$\frac{1}{2}\|\nabla f(x)\|^2 \geq \mu(f(x) - f(x^*))$$
where $f(x^*)$ is the optimal value of the function.
Question:
Does PL condition hold for $f(x)=\|Ax-b\|^2$ where $A \in \mathbb{R}^{N \times n}$ where $n> N$? In other words, does the following hold: $$\frac{1}{2}\|\nabla f(x)\|^2 \geq \mu f(x)$$ because $f(x^*)=0$ and we have infinitely many $x^*$.
My try:
$$ \begin{aligned} \|Ay-b\|^2 &= \|A(y-x+x)-b\|^2 \\ &=\|Ax-b + A(y-x)\|^2 \\ &=\|Ax-b\|^2+\langle2(Ax-b),A(y-x)\rangle+\|A(y-x)\|^2 \\ &=\|Ax-b\|^2+\langle2A^{\top}(Ax-b),y-x\rangle+\|A(y-x)\|^2 \\ &=f(x)+\langle\nabla f(x),y-x\rangle+\|A(y-x)\|^2 \end{aligned} $$
How can I continue? (Prove or disprove)