reviews: 
Comments: This paper considers a classical fidelity + regularization model for image reconstruction, where the regularizer is an Input Convex Neural Network (ICNN). This construction, together with the choice of a fidelity term as a convex function, make the objective function of the optimization problem convex. However, the nested structure of such regularizer make the problem hard to optimize, therefore the authors consider an equivalent formulation where the nested structure is removed by inserting a constraint to the optimization problem. However, due to the non-convexity of the feasible set, the problem becomes non-convex. To avoid this issue, the authors propose to consider instead the epigraph of the feasible set, obtaining an optimization problem which is equivalent to the original optimization problem, while being convex. This allows the authors to use the fast and stable Primal-Dual (PD) algorithm from [Chambolle, Pock; 2011].
Strengths: The paper is well-written and the math is mainly correct. The results shows that the proposed method is superior than the algorithms to which the authors compared, showing a small, but non-zero, improvement in the state-of-the-art to which this paper refers.
Weaknesses: I believe the idea is not original enough to be published on a big conference such as ICLR. Indeed, to my understanding, the proposed model is obtained by combining the known and widely used ICNN-based regularizer (see, e.g. Learning Convex regularizers for inverse problems; Mukherjee et al.; 2021), with a known technique to remove the nested structure (Carreira, Perpinan; 2014) and the known epigraphical projection (Chierchia et al.; 2015). The problem is then solved through the application of the classical Primal-Dual optimization algorithm, widely used to solve convex non-smooth optimization problems. All of these methods appear, to the reviewer, to be combined with no major improvements or adaptations with respect to the paper from which they are taken. While this could not be a major issue for most journals and conferences, I believe this is not enough novelty to be accepted and published on a conference such as ICLR. Clearly, I am open to change my mind if I understood it wrong.

Anyway, I also found some aspects of the paper which I suggest to improve:

-	The most relevant criticism is on the experimental section. In particular, the authors compare their ICNN-based regularization model solved by PD algorithm against the same model solved by other Subgradient Methods (SM). However, I believe that the optimization algorithm employed is the least innovative aspect of the paper, in particular when compared to the choice of the learned regularizer. I suggest the authors to compare their method with other regularization methods such as Total Variation, Wavelets, or any other learned regularization method. Moreover, I kindly ask the authors to explain their rationale or focusing on optimization algorithm comparisons rather than regularizer comparisons.
-	The method on which the authors compare (i.e. the SM-C and SM-D methods) are not introduced nor cited. In particular, I did not find any reference to which algorithms have been employed specifically, and which choice of the parameters has been employed. Could you please provide them?
-	Be more specific in the proof of Theorem 3, you say that the two formulations are equivalent, but you only show that their minimum is the same. For two optimization problems to be equivalent, they are supposed to have the same minimum points, i.e. the two minima are achieved at the same value of $x$. This does not necessarily hold if they have the same minima. For example, the functions $f(x) = x^2$ and $g(x) = (x-1)^2$, both have the same minimum (i.e. 0), but their minimum points is $x=0$ for $f(x)$, $x=1$ for $g(x)$. Therefore, they are not “equivalent”.

**Minor Comments.**

-	In line 66, “step-size selection” should be erased.
-	In line 69-70, there is a repetition of “remove”.
-	In Assumption 2, $D$ does not directly depend on $x$, while it depends on $x$ through $A$. I suggest to explicitly say that “$D(Ax, y)$ is convex in $x$”.
-	In Theorem 3, specify is the “min” in $w_{L-1}$ is a minimum or an inf, since the notation is changed through the paper.
-	In line 241, add a citation to SM-D and SM-C methods.
-	In equation (10), how do you compute the matrix norms in the algorithm?
-	In Figure 3, when the legend says “Proposed”, to which parameter choice it refers? I also suggest to add a grid on the plot background of Figure 3 to improve readability.
-	In line 314, “3% Gaussian noise”, what does it mean? You add Gaussian noise with standard deviation of 0.03? The norm of the noise is 3% of the norm of the data? 
-	In the Computed Tomography (CT) experiment, the amount of information measured is too high, making the problem “easy” to solve. I suggest testing more extreme geometries such as 60-90 angles, instead of 200, to show how the proposed method behaves in other scenarios.
Recommendation: 5
Reviewer confidence: 4
Soundness/correctness: 4
Clarity: 3