&\leq
    \mathbb{E} \left\| \frac{\mathbb{I}_{\mathcal{A}}}{| f( {s},  {a})| }\right\| \left\| | \phi_1( {s},  {a}) - f( {s},  {a}) | - | \phi_2( {s},  {a}) - f( {s},  {a}) | \right\|_\infty \label{eq:lemma_1}\\
    &\leq \mathbb{E} \left\| \frac{\mathbb{I}_{\mathcal{A}}}{| f( {s},  {a})| }\right\| \left\| | \phi_1( {s},  {a}) - f( {s},  {a}) - \phi_2( {s},  {a}) + f( {s},  {a})| \right\|_\infty \label{eq:lemma_2} \\
    &\leq   \frac{1}{\min_{( {s},  {a})} | f( {s},  {a})| }\mathbb{E} \left\| \phi_1( {s},  {a}) - \phi_2( {s},  {a}) \right\|_\infty \label{eq:lemma_3} \\
    &= \frac{1}{\epsilon} \mathbb{E} \left\| \phi_1( {s},  {a}) - \phi_2( {s},  {a}) \right\|_\infty \label{eq:lemma_4}\end{aligned}$$

Where step to comes from the triangle inequality and step to comes from the definition of the set $\mathcal{A} = \{ ( {s},  {a}) : \left\| f( {s},  {a}) \right\| \geq \delta \}$. Thus, the loss is expectation-wise Lipschitz continuous.

A strictly feasible solution, $\phi$, exists since the ground truth model, which is representable in $\Phi$, is strictly feasible.

Lastly, the parameterization used is a neural network which is a universal function approximator.

Therefore the conditions for Theorem 2 are fulfilled.

This proposition is easy to show and a proof of it along with Theorem \[thm:parameterized\_gap\] is given in the supplementary material. This theorem states that for a large 