of the primal-dual algorithm to solve it.

Almost Zero Duality Gap {#sec_zero_duality_gap}
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By definition of the dual problem, it follows that the dual solution $D^*_\theta$ is always a lower bound for the primal solution $P^*_\theta$. That is, $P^*_\theta \geq D^*$ [@boyd2004convex]. The converse is however not true, but we can show that in the case of sufficiently accurate learning the duality gap is small. To that end we consider the following generalization of the problem $$\begin{aligned}
\label{eq:opt_problem_func}
P^* &= \min_{\phi \in \Phi} \mathbb{E}_B [l( {s},  {a}, \phi)] \\
& \text{s.t. } \mathbb{E}_B [g( {s},  {a}, \phi)] \leq 0, \nonumber\end{aligned}$$ where instead of optimizing the weights of a function approximator, the optimization is done over the space of all possible integrable functions $\Phi$. We now have the following result shown in @ribeiro2012optimal.

Given the optimization problem in , if (i) the distribution $B$, is non-atomic, (ii) the inequality constraints define a compact region within $\Phi$, and (iii) there exists a strictly feasible solution $(\phi, \lambda)$ (Slater’s condition) then the duality gap is zero.

With this result, a natural question to ask is how the parameterization of functions $\phi \in \Phi$ affects the duality gap. The following theorem from @eisen2018learning describes 