Paper: Neur2RO: Neural Two-Stage Robust Optimization (ICLR-2024)

Reviewer comment (Experiments):
How much of an impact does the complexity of the Neural Network have on solution time vs relative error.

Author response:
We did not compare this systematically.  However, generally, we observed that decreasing the capacity significantly resulted in a training/validation error that was significantly larger, whereas increasing capacity did not significantly improve the error.  In general, larger NNs are more challenging to optimize over in the MILP, as it requires more variables and constraints so that we would expect a significant increase in solution time.  In future work, we will aim to address this in more detail.

Reviewer comment (Experiments):
Did you evaluate the approach on any non binary problems. How was the performance on them.

Author response:
The main focus of this work is on two-stage robust optimization (2RO) problems with discrete recourse.  As we have mainly focused on benchmark instances from the 2RO literature, most of these include only binary decisions.  We note that there is nothing limiting our approach to the binary case, so we agree this would be an interesting comparison.

Reviewer comment (Experiments):
Did both the knapsack and the budgeting problems involve only one constraint? Did you try the approach on problems with more constraints?

Author response:
Both problems do indeed involve one (non-trivial) constraint. Our method would in theory be capable of handling more than this, as we predict the second-stage objective with respect to its constraints.  We are currently working on an extension of this work that addresses problems with more challenging constraints, namely, the facility location problem.

Reviewer comment (Novelty):
The contribution of the paper is not clear. See the comments below.

Author response:
We will address the first two questions in this comment.
1.  Firstly, we do acknowledge that the learning of the value function for the second-stage problem is shared.  However, the domains in which the papers focus, i.e., two-stage stochastic programming (2SP) and two-stage robust optimization (2RO), differ substantially and require specific contributions to the case of 2RO.  Below, we summarize the important differences and contributions.  
    - **Architecture**: In [R1], a major contribution is made by architecture, which leads to efficient optimization by learning to aggregate scenarios.  In 2RO, aggregation of scenario information, especially before the first-stage decision is input to the model as in [R1], is not ideal, as the worst-case scenario directly depends on the first-stage decision.  As such, 2RO essentially requires a learning model that predicts based on a single scenario.  While this can be done with a standard feed-forward style architecture, such as the per-scenario architecture in [R1], this will suffer from the same scalability issues as demonstrated in [R1] as optimizing over an entire neural network for each scenario will quickly become intractable as the number of scenarios increases in the main problem (MP).  This highlights the main contribution of the architecture in our paper, which enables efficient optimization in the MP for 2RO via only embedding the Value Network for each scenario.
    - **Generalization**: One major limitation of [R1] is that the learning models were trained and evaluated on the same instance with varying scenarios, whereas, in this paper, we propose an architecture that can generalize across families of instances representing coefficients of the optimization problems as features, and applying set-based architectures.  
    - **NN approximations within an iterative algorithm**: One major distinction between [R1] and this work is how the NN-based approximations are utilized.  In the 2SP setting, [R1] use NN-based value function approximation to directly approximate the extensive form, which results in a single-level optimization problem.  To the best of our knowledge, our work is the first to address NN-based value function approximation within the context of an iterative algorithm for mathematical optimization.  
    - **The argmax formulation**: As demonstrated in our supplementary results, the obvious choice of using a max over the NN predictions, which would be equivalent to the expectation in the per-scenario approximation of [R1], results in strictly worse performance.  This argmax formulation for the MP provides an alternative (and completely new) way to approach the CCG main problem by leveraging the actual coefficients of the underlying optimization problem, rather than purely relying on the predictive model.  While this exact formulation may be limited to 2RO, we suspect the general idea, utilizing predictions in conjunction with true parameters of the optimization model, may have extensions within various ML-based algorithms for other mathematical optimization settings.

Reviewer comment (Experiments):
To accelerat the column and constraint genration algorithm using DNN, one straightforward approach is using DNN to predict the worst-scenario for the stage-2 problem and then add the obtained scenario to the main problem. The authors are suggested to explain why this straightforwrd design will not work. Otherwise, the authors are suggested to use this straightfoward design as a baseline approach to show the superiority of the proposed approach.

Author response:
While predicting the worst-case scenario may be possible, there are a few important reasons we consider the prediction of the value function.
    - The first notable issue with predicting the scenario directly is that the scenario vector is a high-dimensional constrained vector.  Without even considering the hard constraints on the vector, the high dimensionality alone is already a significantly more challenging prediction task.  Furthermore, it is unclear if minimizing MSE on the uncertainty vector would be reasonable, or how a feasible vector can be recovered if the model predicts an infeasible scenario. 
    - Another challenge with predicting the worst-case scenario directly is that the data collection would be much more costly.  For example, capital budgeting would require solving a bilevel optimization problem to determine the worst-case scenario. In contrast, we only need to solve a single-level deterministic problem, which is generally much easier.
    - Lastly, in some 2RO problems, such as the knapsack problem, the uncertainty dimension changes across instances.  This means that if we want to train a single model across variable-sized instances as we do in this work, then we would need a learning model that is able to predict with variable-sized outputs.  While there are some models that can do this, we generally believe that a much more straightforward approach is handling the variable dimensions within the input of the model.

Reviewer comment (Experiments):
The authors are suggested to use more real-world problem as the case study. For example, the robust unit commitment problem in power system operation.

Author response:
We acknowledge that this is a great direction for future work, and we are currently exploring a facility location problem with disruptions and uncertain demands, which is definitely closer to a real-world problem, but we will consider the robust unit commitment problem as a potential case study as well.  For facility location, our preliminary experiments indicate that a similar performance boost is possible. These additional results will be included in the final version of the paper if accepted, space permitting.

Reviewer comment (Theory):
The authors are suggested to analyze the impact of the DNN prediction errors to the performance (such as convergence and optimality gap) of the proposed approach.

Author response:
We agree that this would be an excellent addition to the paper and we will consider some analysis on this in the future.  To briefly comment on this, we note that there is a trade-off between the prediction error, size, and solving time of the optimization model.  For example, networks with much less capacity often have worse prediction errors but require less optimization time, whereas very large networks may have higher prediction errors, but require significantly more optimization time.  We thank you for the comment and agree that we will aim to include this in future extensions of this work as this is an important question for the practical adoption implications of this work.