sufficient conditions under which a proxy of the duality gap is bounded.

\[thm:parameterized\_gap\] For the optimization problems  and , if,

-   there exists a strictly feasible solution $(\phi, \lambda)$ to the primal problem ;

-   the parameterization $\theta$ of the function space $\Phi$ is a universal approximator within error $\delta$, i.e. $\mathbb{E}_{D} \left\| \phi - \phi_\theta \right\| \leq \delta$ for some $\theta$ for all $\phi \in \Phi$;

-   the loss function, $l( {s},  {a}, \phi)$ is expectation-wise Lipschitz continuous, i.e., there exists a $K$ such that $\mathbb{E} \left\| l( {s},  {a}, \phi_1) -  l( {s},  {a}, \phi_2)\right\|_\infty \leq K \mathbb{E} \left\| \phi_1 - \phi_2 \right\|_\infty$ for all $\phi_1, \phi_2 \in \Phi$;

then the optimal parameterized dual value $D_{\theta}^*$ is bounded by $$\label{eq:dual_gap}
        P^* \leq D_\theta^* \leq P^* + \| {\lambda}^* \|_1 K \delta,$$ where $\lambda^*$ is the solution of .

The following proposition formalizes that the problem of sufficient accurate learning in has small duality gap. **Proposition 1.** *Sufficiently Accurate Learning and its dual, defined in and respectively, satisfy the assumptions of Theorem 2. Hence holds for sufficiently accurate learning.*

First we look at the Lipschitz condition. $$\begin{aligned}
    \mathbb{E} \left\| l( {s},  {a}, \phi_1) - l( {s},  {a}, \phi_1) \right\|_\infty 