Paper: A Discretization Framework for Robust Contextual Stochastic Optimization (ICLR-2024)

Reviewer comment (Theory):
Corresponding to Part 1 in the weaknesses, I wonder whether one can learn the probability efficiently. Or, equivalently, I wonder whether the authors could elaborate more on the sample complexity for learning and reducing $\Epsilon$ in (11) for Theorem 1. Please correct me if I am wrong. For me, I am not sure whether it is possible to learn $\Epsilon$ efficiently, especially when the dimension of the feasible set is large. In other words, one may require the size of training data to be exponentially large as the dimension of the feasible set to have a low $\Epsilon$. Particularly, based on the algorithm, H_k^{\epsilon} may only cover a small region, the volume of which can decrease exponentially when the dimension is high. Then, if the size of the training samples is only polynomially large as the dimension, it is very likely that $H_k$'s are disjointed or rarely have non-empty intersections. In this case, the multi-label data set for each training sample might be a vector with only one non-zero entry.

Author response:
Thank you for the thoughtful questions and suggestions. We agree especially that the learnability of the problem  and choice of $\epsilon$ are very important. To address these issues, we have added a discussion on the effect of $\epsilon$ on each term of the bound. Indeed, you are correct, in the worst case one would need an exponential amount of data for small $\epsilon$. Nevertheless, instead, one can increase the value of $\epsilon$ so that the number of positive labels increases. Of course, in the worst case  this may require the choice of $\epsilon$ to be a constant fraction of the diameter of the feasible region. 

Primarily, the choice of $\epsilon$ will depend on the distribution of  $w^*(\nu)$. There are two factors that can reduce the dimensionality of this. (1) The $w^*({\nu})$ often belong to a lower-dimensional subspace of $\mathbb{R}^d$ since they must satisfy constraints. For example, if $\mathcal{P}$ is the set of {$ \{ w : Aw = b, w \geq 0 \}$} then the effective dimension of the problem is equal to the dimension of the null space of $A$. For instance, if $Aw = b$ defines a 1-dimensional line, we can view the problem as 1-dimensional regardless of the ambient dimension of the decisions $w \in \mathbb{R}^d$.  (2) the distribution further depends on the data itself. The distribution of ${\nu}$, and as a result of $w^*({\nu})$, does not pathologically cover the entire space when presented with real-world data. 

This dimensionality issue is not unique to our approach. The same problem is encountered for example, for the methods of Bertsimas and Kallus (2020) and more generally for methods such as KNN. That being said, we agree with you that it is important to address this issue in the paper.

Reviewer comment (Theory):
Can the algorithm still be useful when the feasible region also depends on the contextual data?

Author response:
Finally, that is an interesting extension to consider if the feasible region also depends on the contextual data and uncertainty. That is something we have been looking into and can add in the future.

Reviewer comment (Experiments):
In practice, I wonder how one should choose $\phi$ and $\epsilon$.

Author response:
Thank you for the thoughtful questions and suggestions. We agree especially that the learnability of the problem  and choice of $\epsilon$ are very important. To address these issues, we have added a discussion on the effect of $\epsilon$ on each term of the bound. Indeed, you are correct, in the worst case one would need an exponential amount of data for small $\epsilon$. Nevertheless, instead, one can increase the value of $\epsilon$ so that the number of positive labels increases. Of course, in the worst case  this may require the choice of $\epsilon$ to be a constant fraction of the diameter of the feasible region. 

Primarily, the choice of $\epsilon$ will depend on the distribution of  $w^*(\nu)$. There are two factors that can reduce the dimensionality of this. (1) The $w^*({\nu})$ often belong to a lower-dimensional subspace of $\mathbb{R}^d$ since they must satisfy constraints. For example, if $\mathcal{P}$ is the set of {$ \{ w : Aw = b, w \geq 0 \}$} then the effective dimension of the problem is equal to the dimension of the null space of $A$. For instance, if $Aw = b$ defines a 1-dimensional line, we can view the problem as 1-dimensional regardless of the ambient dimension of the decisions $w \in \mathbb{R}^d$.  (2) the distribution further depends on the data itself. The distribution of ${\nu}$, and as a result of $w^*({\nu})$, does not pathologically cover the entire space when presented with real-world data. 

This dimensionality issue is not unique to our approach. The same problem is encountered for example, for the methods of Bertsimas and Kallus (2020) and more generally for methods such as KNN. That being said, we agree with you that it is important to address this issue in the paper.

Reviewer comment (Theory):
It is difficult to appreciate the main theoretical performance guarantee on the proposed framework (given in Theorem 4.1) due to its dependence on the parameters $c$ and $\alpha$. Here, $c$ would increase as the accuracy parameter $\epsilon$ as well as the size of each subset decrease, in which case the bound gets weak. We may try to decrease the parameter $\epsilon$ to make $c$ close to 1, but this would make the mean prediction error $\mathcal{E}$ large. At the same time, no explicit dependence of $c\mathcal{E}$ on $\epsilon$ is studied. Lastly, $\alpha$ could be arbitrarily large when the objective function is not bi-Lipschitz.

Author response:
Thank you very much for your comments and review. Taking your comments into account, we hope that the new section we added regarding the impact of $\epsilon$ upon each of the terms $c \cdot \mathcal{E}$ and $c \cdot OBJ({w})$ addresses your questions and concerns. Ultimately, both $c \cdot \mathcal{E}$ and $c \cdot OBJ({w})$
are always bounded by a constant for any $\epsilon$. The only term left is the generalization term $c \cdot \sqrt{\log(1 / \delta) / N}$. From here we give a procedure on how to  choose $\epsilon$ based on the amount of data available. This gives us a guide to understand the magnitude needed for $\epsilon$, but ultimately something like cross-validation can be used to make a final choice for $\epsilon$ in practice. 
Please see the discussion following the theorem statement that contains more details on this.

Reviewer comment (Novelty):
The framework is almost equivalent to minimizing the conditional value-at-risk (CVaR), except that there is an additional probability term being multiplied. The paper would be stronger if it discussed the relationship between the existing CVaR minimization setting and the framework of this paper. No numerical results were done with the CVaR framework.

Author response:
We have also added experimental results for a broader range of quantiles, as well as comparing our approach to CVaR as you have suggested. We notice that the CVaR approach produces decisions that change discretely as the robustness parameter changes (the quantile being targeted), whereas the minimum-violation objective produces continuously changing decisions. This continuity might be explained, as you pointed out, by the additional probability term being multiplied. From the computational experiments we also see that our proposed approach outperforms CVaR (and all approaches we considered) by the following metric: for the same average cost, the cost at the $q^{th}$ quantile is generally lower for our approach. This holds in particular for higher quantiles. Please see the plots in Figure 2 that provides many comparisons.

Reviewer comment (Evaluation):
Could you provide more computational results on a broad set of tolerance levels? For example, 50%, 75%, 90%, 95% quantiles.

Author response:
We have also added experimental results for a broader range of quantiles, as well as comparing our approach to CVaR as you have suggested. We notice that the CVaR approach produces decisions that change discretely as the robustness parameter changes (the quantile being targeted), whereas the minimum-violation objective produces continuously changing decisions. This continuity might be explained, as you pointed out, by the additional probability term being multiplied. From the computational experiments we also see that our proposed approach outperforms CVaR (and all approaches we considered) by the following metric: for the same average cost, the cost at the $q^{th}$ quantile is generally lower for our approach. This holds in particular for higher quantiles. Please see the plots in Figure 2 that provides many comparisons.

Reviewer comment (Writing):
Not fully clear presentation/discussion of their algorithm. Please see my comments in Summary for details.

Author response:
We really appreciate your comments and review. We have done our best to incorporate your suggestions and updated the paper accordingly. We would also like to take the opportunity and answer some of your questions in this response as well.

We agree that it would be very helpful to include a more thorough discussion about how the model is trained, and how the parameters are chosen. In particular, please see the new discussion we added in the revised paper following the theorem about the role of $\epsilon$ on the bound and on learning the  $\hat{p}^{k}({x})$ (the learned probabilities of $w^*({\nu_x}) \in H^\epsilon_k$).

In what follows we try to address some of the questions you raised.  

Indeed, the union of $H_k^\epsilon$ need not cover the entire feasible region. This is part of the bound in Theorem 4 and captured in c which depends on $\mathbb{P}(R_{\nu_y}(w^*(\nu_x) \leq \epsilon)$. This is the probability that the objective of a solution for $\nu_x$ is within $\epsilon$ of the optimal solution with respect to some other ${\nu}_{{y}}$. 

The values of $\hat{p}_k({x})$ learned need not sum to 1. This is because the sets $H^\epsilon_k$ are not disjoint and do not  necessarily cover the entire feasible region. 
There is a lot of freedom in how the $\hat{p}_k({x})$ can be learned. 
In the first experiment on the inventory problem, each $\hat{p}_k({x})$ comes from a separate knn predictor. For the second experiment on the generator scheduling problem, $\hat{p}_k({x})$ are trained jointly and are the output of a neural network with $N$ outputs (one for each $k =1, \dots, N$).

As also pointed out by reviewer 9FbH, the method indeed has similarities to that of (Berstimas and Kallus (2020)). In particular, one can also view the objective function as some new $f_{{\nu}}({w}) = \max \{ R_{{\nu}}({w}) - \phi, 0 \} $ and directly apply their method on this objective. 

To the best of our knowledge, we are the first to propose this new objective function.  More importantly, the significant difference lies in how the weights $\hat{p}_{k}({x})$ are defined. In our case, these are explicitly informed by the optimization problem since $H_k^\epsilon$ are directly dependent on the data and objective function. 

From  a computational viewpoint, indeed the problem increases linearly with the amount of data $N$. However, there are three possible approaches to improve the runtime. One is to simply choose a random subset of the data to use to generate sets $H^\epsilon_k$. The second is, as we run the optimization problem for various out-of-sample points ${x}$, to keep track of the ${x}$ and the corresponding solutions $\hat w_{\epsilon, \gamma}(x)$ found.
 When given a new ${x}'$ we can for example find the nearest neighbor, say ${x}''$ and use $\hat{w}_{\epsilon, \gamma}({x}'')$ as a warm start when solving the optimization problem for ${x}'$. Finally, once calculating all values of $\hat{p}_k({x})$, many of these will have very low value close to zero. One approximation is to simply remove these from the optimization problem. Or alternatively, only keep the largest $k$ values of $\hat{p}_k({x})$ and only optimize with respect to those. We are happy to add more discussion on this in the appendix of the revised paper.