of the quantity being estimated. For example, 1 unit of error when the output is 100 is very different from 1 unit of error when the output is 1. A natural way to mitigate these difference is to consider a normalized error, $\left\| \phi_\theta( {s},  {a}) - f( {s},  {a}) \right\| / \left\| f( {s},  {a}) \right\|$. It is often the case that small values of $f( {s},  {a})$ are hard to even measure accurately, thus, it matters only that the error is bounded rather than minimized. The set of non-small values will be defined as $\mathcal{K} = \{ ( {s},  {a}) : \left\| f( {s},  {a}) \right\| \geq \delta \}$, where $\delta > 0$. One can then pose the following model learning objective $$\begin{aligned}
\label{eq:constrained_model_learning_example}
    \min_\theta & \mathbb{E}_{( {s},  {a}) \sim D} \frac{\left\| \phi_\theta( {s},  {a}) - f( {s},  {a}) \right\|}{ \left\| f( {s},  {a}) \right\| } \mathbb{I}_{\mathcal{K}} \\
    & \text{s.t. } \mathbb{E}_{( {s},  {a}) \sim D} \left\| \phi_\theta( {s},  {a}) - f( {s},  {a}) \right\| \mathbb{I}_{\mathcal{K}^C}  \leq \epsilon . \nonumber\end{aligned}$$ This objective simply states that for all large enough $f( {s},  {a})$, the normalized error should be minimized, and all small values should be bounded by $\epsilon$.

Primal-Dual 