Let $x_1,\ldots,x_n$ be i.i.d. random variables taking values in $\mathbb{R}^p$, and $m_\theta,\psi_\theta:\mathbb{R}^p\rightarrow\mathbb{R}$ for all $\theta\in\Theta$. An $M$-estimator is any element of: $$ \arg\min_{\theta\in\Theta}\sum_{i=1}^nm_\theta(x_i) $$ A $Z$-estimator is any zero of: $$ \sum_{i=1}^n\psi_\theta(x_i)=0 $$ In some cases, those two concepts are equivalent: e.g. given $m_\theta$, one can find $\psi_\theta$, such that an $M$-estimator is also a $Z$-estimator if $\theta\mapsto m_\theta$ convex and differentiable.
I am wondering whether one can always write a $Z$-estimator as an $M$-estimator. More precisely: Fix any $n\in\mathbb{N}$ and any collection of functions $\{\psi_\theta,\theta\in\Theta\}$, such that a $Z$-estimator exists (almost surely). Are there functions $m_\theta$, such that for (almost) all $x_1,\ldots,x_n\in\mathbb{R}^p$, $$ \left\{\theta\in\Theta:\sum_{i=1}^n\psi_\theta(x_i)=0\right\}=\arg\min_{\theta\in\Theta}\sum_{i=1}^n m_\theta(x_i) $$