Lemma 2.4.1

Let $L:\mathbb{R}^{m_{1}}\times\dots \times \mathbb{R}^{m_{N}}\to \mathbb{R}$ be a convex (deterministic) loss function, with pbp optimal solution $\mathbf{u}^{\circ}:=(u^{1\circ},\dots,u^{N\circ})$. If $L$ is continuously differentiable at $\mathbf{u}^{\circ}$, then $\mathbf{u}^{\circ}$ is globally (team) optimal.

Intuition

So all we need is our loss function to be differentiable at our pbp solution for it to also be team optimal.