We thank the reviewer for responding to our rebuttal.
- **Regression.** We note that in this paper we do not observe $f(\theta, x)$, we only observe $x$. So, we can not apply existing methods to regress $f(\theta, x)$ against $x$. For example, consider FO to be a shortest path problem on a road network with $d$ road segments, our context $u$ specifies an origin and a destination of a trip, decision $x$ is a $d$-dimensional binary vector representing if each road segment is selected (=1) or not (=0), and $\theta$ is a $d$-dimensional real vector representing the (perceived) travel cost on each road segment. In this setting $f(\theta, x)$ is the perceived travel cost of a trip, which we never observe. We only observe the trip ($x$) and its origin and destination ($u$) (see line 99 for the data we are dealing with). We will add this running example in Section 3 to add clarity.
Note that we had confusion about this comment in the initial rebuttal. We thought the reviewer meant we could regress $x$ against $u$. So, we use the term *decision* to highlight that $x$ is subject to constraints, so this regression idea is not applicable.
- **Relation to CP.** Our result in equation (11) is a conformal prediction result. To build a conceptual connection, we can think of our observation $(u, x)$ as features, our inverse optimization problem (5) as an ML model, and the estimation target $\theta$ as the prediction target in CP. We apply the simplest CP method to construct a set (which is the same for all $(x, u)$) that contains a $\theta$ that can explain thee next $(u, x)$ with a specified probability. We hope this analogy helps to understand the connection between our work and CP.
- **Distribution.** We apply concentration inequality to bound the difference between the *observed probability of even E* and the *real probability of event E*, where E indicates there exists a $\theta$ in our set that can explain the next $(u, x)$. Since
*probability* is bounded in $[0, 1]$, we do not need any distributional assumptions.
- **Ground-truth parameters**. Different data points are generated by different $\theta_k$ which are noisy perceptions of the ground-truth $\theta^*$. In the above shortest path example, $\theta^*$ is the ground-truth travel time on each road segment and $\theta_k$ are noisy human perception. As mentioned above, we will add this example to the paper to add clarity.
- **Parameter $\eta$**. Yes, in short, our method can provide a smaller set than the bounded support set. This is very important in our setting because our uncertainty set will be used to prescribe future decisions. If the set is too large, then the solution will be too conservative. We point the reviewer to lines 202-204 where we mention we want the smallest set that can achieve the coverage requirement. Our bounds in Theorem 3 also decreases as the size of the set $\alpha$ decreases, reflecting this intuition.
- **Selection $\eta$**. We'd first like to emphasize that we do not need to select a $\eta$ to apply our method, so the computational cost should not be a concern in practice. The purpose for calculating $\eta$ in Example 1 is simply to verify our theoretical results. Regarding the estimation method we use, as stated in Assumption 2, $\eta$ is the largest possible diameter of $\Theta(x, u)$. So calculating the diameter of $\Theta(x, u)$ for all possible $(x, u)$ should give as a valid $\eta$. In real problems, there might be a large number possible $(x, u)$, so calculating $\eta$ could be challenging. But again, we don't need to calculate this parameter in practice. We will add the description of how we calculate $\eta$ in Table 1 in the appendix and mention this in the main body when presenting Table 1.
Thank you again for your questions which helped to improve the clarity of our paper. We'd be happy to discuss further if there are other questions.