Venue: NeurIPS 2022 Conference

Reviews:

final_score: rating: 8: Strong Accept: Technically strong paper, with novel ideas, excellent impact on at least one area, or high-to-excellent impact on multiple areas, with excellent evaluation, resources, and reproducibility, and no unaddressed ethical considerations.
review_content: summary: This paper proposes a method to mitigate some of the computational challenges associated with the decision-focused learning paradigm under stochastic optimization-based decisions with unknown parameters. In particular, the authors construct (a) an energy-based surrogate function for the stochastic optimization-based decisions, (b) a loss function encouraging both good predictive accuracy and good global behavior of the energy-based surrogate, and (c) an efficient training procedure based on a mixture-of-Gaussians proposal. This enables them to avoid expensive implicit KKT differentiation steps that are needed when representing the decision procedure exactly as an optimization problem, as well as better enabling the representation of non-convex decision-making processes. The authors provide extensive experiments on several settings, and show empirically that their method achieves comparable or improved performance over baseline methods while drastically improving on computation time.
strengths_and_weaknesses: Strengths: 
* The proposed method is well-motivated, nicely integrates concepts from energy-based learning into the decision-focused learning literature, and mitigates a number of existing challenges with decision-focused learning paradigms.
* The experimental evaluation is thorough and well-done, demonstrating performance on different kinds of settings with respect to the form of the task loss, and showing insightful ablations. The method strongly outperforms other baselines (and the authors both include error bars *and* transparently leave in one setting where a baseline outperforms their method, both of which I appreciate).
* Everything is very well-explained, making the paper an absolute pleasure to read.

Weaknesses: 
* In the checklist, the authors mention that they describe limitations and potential negative societal impacts of their work, but I cannot find this discussion.

Questions:
* Section 4.1: Why does SO-EBM improve over DFL-QPTH? It would be useful to give some intuition for this, as at first glance it seems strange that a method using a surrogate loss would outperform a method using a more “exact” version of the loss.

Minor points:
* Section 2, Problem Formulation paragraph: The terms $M$ and $a^*(x;\theta)$ should be more formally defined in the text (not just in Figure 1), as the rest of the problem formulation and method hinge on this. In particular, the current statement about learning a model that “takes the features $x$ as input and outputs the optimal decisions $a^*(x;\theta)$” could be misunderstood as a model that maps directly from data to decisions, without the intermediate prediction of the distribution.
* Section 2, Problem Formulation paragraph: As previously defined, $\mathcal{D}$ is the dataset. However, the optimization loss is introduced as “the expected decision cost under the joint distribution.” The wording here should be clarified. The goal is to learn the expected decision cost under the joint test-time distribution, *as approximated by* the dataset at training time, which is what the loss shows.
* Line 92: The left side should be the total derivative, rather than the partial derivative. Similarly, $\partial y/\partial \theta$ should be $\mathrm{d}y/\mathrm{d}\theta$. Similarly, in Equation (6), some of the partial derivatives should be total derivatives.

Typos:
* Line 84: “DFL method” -> “DFL methods”
* Problem Formulation paragraph in Section 2: “optimal decisions minimizes” -> “optimal decisions minimize”
* Line 99: “CVXlayers” -> “cvxpylayers”
* Line 100: “grammer” -> “grammar”
* Line 132: The optimization problem is missing $\mathbf{x}_i$ 
* Line 305: “target node” -> “target nodes”

questions: * Section 4.1: Why does SO-EBM improve over DFL-QPTH? It would be useful to give some intuition for this, as at first glance it seems strange that a method using a surrogate loss would outperform a method using a more “exact” version of the loss.
* What are some of the limitations and potential negative impacts of the method?
limitations: In the checklist, the authors mention that they describe limitations and potential negative societal impacts of their work, but I cannot find this discussion. This discussion should be made more explicit, and should acknowledge that while the applications presented in the paper are social good applications (which I appreciate), the method is general enough that it may also have negative uses.
review_title: null

final_score: rating: 6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
review_content: summary: This paper tackles stochastic optimization problems of the form $\max_{a \in C} f(y, a)$, where $y$ is a stochastic variable and $a$ is an action taken by the learner given an observation $x$. 

Recent work have introduced solutions that involve Deep Neural Networks. One body of work uses DNNs to make a prediction $\hat{y}$ given $x$, and then running an off-the-shelf optimizer for the inferred problem $f(\hat{y}, \cdot)$. This can lead to suboptimal solutions if $\hat{y}$ is not accurate. Another body of work can overcome this in certain scenarios by optimizing $(\hat{y}, a)$ jointly. They do so via end-to-end architectures that use so-called optimization-layers. These layers run a differentiable convex optimizer, and hence the incurred cost $f(\hat{y}, \hat{a} )$ can be optimized by jointly through backpropagation. A limitation of this approach is that the optimization layers must backpropagate through a convex optimiser; hence if it does not have a closed form solution, the computational cost can be steep.

The authors propose a novel architecture for joint optimization of $(\hat{y}, \hat{a})$ that avoids introducing optimization layers by taking an EBM approach. They construct an EBM that takes the expected cost $f(\hat{y}, \hat{a})$ under the model as the energy. To optimize the energy function, the authors operate a two-stage process. First, given a dataset $D = \\{ (x_i, y_i) \\}\_{i=1}^K$, they compute expert actions $a^i$ for each $y^i$ to construct an auxiliary dataset $D' = \\{ (x_i, a_i) \\}\_{i=1}^K$. They then use MLE to optimise the energy model under $D'$. 

However, a main complication of this approach is that it requires stochastic outputs from the DNN. To this end, the authors rely on a Gaussian parameterization and self-normalized importance sampling. This means that the main computational difference between optimization-layer approaches and the author's approach is that they rely on stochastic relaxation instead of exact optimization. 

The authors provide three synthetic experiments to evaluate their architecture: load balancing in power systems, healthcare resource allocation, and network security. In all cases, they show that their proposed method is competitive in terms of final performance while being computationally more efficient than optimization-layer based approaches.   
strengths_and_weaknesses: Strengths
- The paper is generally well written. The main idea, its motivation, mechanics, and intuition for design choices are all clearly spelled out. The experiments are well described with uncertainty estimates.
- The proposed architecture is quite clean and follows naturally from the EBM perspective. 
- The experiments are well executed; clearly described, with thorough analysis of results. 

Weaknesses
- The main contribution stated on page 2 does not seem to be well supported. First, the authors claim that the method is end-to-end, but this is not true since to construct the dataset, an off-the-shelf solver is needed to generate expert actions. Second, it does require solving the optimization problem (though not backpropagating through the solver), and while the EBM objective does not require convexity, imputing expert actions does. Overall, I think the paper can be strengthen significantly by a more nuanced discussion of the relative merits of the proposed method.
- The experiments are somewhat underwhelming in that they are all synthetic. It would have been significantly more interesting to test the proposed architecture on a real world problem with well-developed benchmarks. Moreover, the results themselves suggest that the gains are marginal; in most cases, the two-stage approach of pretraining a predictor $\hat{y}$ is faster and the loss in performance is not statistically significant.
questions: - The ability to generate expert actions is critical for this method to work. The authors suggest that previous method are limited in that they rely on convex solver - but it seems the proposed method suffer from the same limitation in that the solver is required to obtain expert actions?

- The authors opted for an EBM approach. Another equally valid method would have been expert distillation, which would have the benefit of not requiring estimating the energy function. Could the authors comment on why they opted against this simpler approach?
limitations: The authors discuss some limitations of the EBM approach, mainly the risk of overfitting. They provide an ablation to study the severity of the problem. The authors do not discuss other limitations, sensitivity to the choice of sampler and sensitivity to the choice of solver for expert actions.
review_title: null

final_score: rating: 6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
review_content: summary: The authors propose an approach for "decision-focused learning" (DFL).  This is where there is a prediction stage to predict stochastic parameters from some input features, that are then used to solve an optimization problem to determine an optimal action.  For example, an optimization of electricity generator scheduling based on forecasted electricity load / demand.  The old two-stage approach would be to separately fit a forecast model, and then use its predictions in an optimization problem solved by optimization solvers.  The DFL approach is to fit the forecast model in conjunction with the optimization problem, so the forecasts are optimal for the down-stream optimization problems themselves.

To do this for modern models using neural networks for the forecasters would generally require computing and back-propagating the gradient through the optimization loss and problem to the model parameters.  However, the gradient of the optimal action with respect to the forecast / prediction values is not always differentiable and the optimal action is the solution to an optimization problem, so computing gradient updates can be challenging and time consuming.

Most past work tended to either make restrictive assumptions or approximations, or used computational complex and inefficient implicitly differentiable layers.   This work proposes a workaround by modeling the optimal actions / optimization problem solution as a function of the features directly, but ties it to the forecast model as the energy term of the energy model is given by the expected optimization loss of under the forecast model (the forecast gives the distribution over the predicted parameters for the optimization problem).  In this way the energy model can be trained directly on historical examples with optimal solutions previously derived, by minimizing the negative log likelihood (which in turn optimizes the forecast model for the optimization problem) along with a KL-divergence regularizer added to the objective as well.  

The authors compare their proposed approach to past methods on 3 datasets / tasks from different domains, and show it consistently has as good or better performance to other past DFL approaches, while also having as good or better run time. 
strengths_and_weaknesses: Strengths:
-The authors point out key limitations of past work and aim to tackle a significant and important problem.  I.e., most practical applications of ML involve combining predictions and optimization and the DFL approach is arguably the way things should be done for all these applications / ML in general.

-The authors propose a novel solution that is interesting and shows promising results.

-The authors provide experimental results on a variety of different tasks and compare to multiple past DFL methods.  

-The authors pledge to make the code publicly available, which will help a lot with reproducibility and understanding the approach.


Weaknesses:

-One piece of the problem, in particular the optimization part, that seems overlooked and not mentioned is the constraints.  Constraints are not included at all in the model of the end-to-end problem (mapping the features to the action) - they could only be included implicitly since the optimal actions, a, the model is trained on must satisfy the constraints.  It would help to have discussion about why it's not necessary to consider the constraints and what kind of impact it would have.  My speculation is that this can actually have a negative impact, because it puts more burden on the forecast neural network part of the model to account for the constraints (this model then needs to be more complex) - as what naturally fits the data better would have to be abandoned to predict a solution that leads to the best action when dropping the constraints.  I feel the problems chosen had very few and basic constraints which may be why the approach still worked in the experiments, but I would be curious to see how more constraints, and more complex constraints, might impact the performance.  It would also be useful to see how the learned forecast model compares to the separately trained model, and what cases it's making predictions farther off from the true values.

-There is one closely-related work overlooked here that appeared in the AAAI-22 Workshop on AI for Decision Optimization - this pointed out the same limitations and similarly targeted a somewhat similar approach.  However, that approach was different in that it approximated the optimization algorithm itself with a neural net, so the solution could be easily computed and differentiated, as opposed to effectively approximating the entire end-to-end problem as was done here (though here essentially only using the neural net to approximate the forecast piece of the model).  It would be good to mention this alternative approach, and it would be great if the two could be compared, as I'm very curious to see how they compare.  Paper:
"End-to-End Learning via Constraint-Enforcing Approximators for Linear Programs with Applications to Supply Chains"
https://research.ibm.com/haifa/Workshops/AAAI-22-AI4DO/PDF/End-to-End%20Learning%20via%20Constraint-Enforcing%20Approximators%20for%20LinearPrograms%20with%20Applications%20to%20Supply%20Chains.pdf

-The clarity of the presentation and description could be improved.  It's hard to follow the details of the method and the experiments.  For instance, the method description is not easy to follow, especially the KL-divergence regularization and the rationale behind it.  Another example, the experiment procedure is not clearly explained - i.e., for all methods, is the model fit and then just the forecast part used in conjunction with an optimization solver to get the final action / decision?  If so, what solver is used?  How are various hyper parameters set for different methods?  

-Further experimental study would be useful.  In particular it would be good to see the impact on training data size on the performance of the different approaches.  In particular, with enough training data one would not expect the DFL approach to out-perform the two-stage one.  I also wonder if this approach may need more training data points to work well as it directly models the action given the inputs. 
 Additionally run times will also be affected by amount of training data.  It would be good to understand these trade-offs.  Additionally it would be best to see the impact of varying lambda.   Additionally, what is the impact of using different network architectures / sizes?

questions: Please see questions and suggestions raised under weaknesses.  

It would be good to mention and contrast with the other related work mentioned above.

How does the proposed approach handle constraints?  What is the impact of not including constraints?  What is the impact on large numbers of constraints / more complex optimization problems?

How much data does the proposed approach require?  How does the amount of data affect performance?

-Figures 4, 5, and 6 - labels are not lined up with the bars so it's hard to match the methods to the bars
limitations: There doesn't seem to be discussion of what are the limitations of this approach - it could benefit from some thought and discussion around what the limitations are.  Perhaps this would include dealing with constraints, or it could be the case where a large number of solved problems are needed for it to work well, as this should also be investigated ideally.
review_title: null

final_score: rating: 5: Borderline accept: Technically solid paper where reasons to accept outweigh reasons to reject, e.g., limited evaluation. Please use sparingly.
review_content: summary: This paper proposes to address the problem of end-to-end stochastic optimization. That is; finding argmin_a f(y, a) for some cost function f, input variables y, and actions a. The key caveat here is that y is not observed and instead, we observe x, a variable correlated with y (but the relationship between x, y is not provided). 

Standard approaches to this problem learn a model of p(y|x), then use an off-the-shelf stochastic optimizer to find argmin_a E_{p(y|x)}[f(y, a)]. This approach of course can accrue errors due to the p(y|x) model not being perfect leading to subpar results. 

This work proposes to learn the p(y|x) model, not to predict y, but such that the induced optimal action a performs best. To do this, they parametrize an energy-based model by taking the expected value of the cost function over the p(y|x) model, i.e. E(a; x) = E_{p(y|x)}[f(y, a)] and train this EBM to maximize likelihood of obtaining the ground truth optimal actions a, given the input x. 

The model is trained using contrastive divergence but an importance sampling estimator is used to deal with the partition function. 

Given that we also have knowledge of the true y for each x in the dataset, we also have access to a noiseless version of the loss function, not available at test time. This information is utilized by adding a regularizer to encourage low KL divergence. Between the noisy model and the noise-free model. 

The authors present results on 3 stochastic optimization problems and show that on average their approach either outperforms the baselines or performs comparable to the strongest baselines while having a considerably lower training cost. 
strengths_and_weaknesses: Strengths:

This is an interesting application of the theory of energy-based models. It is always nice to see work embracing the methods that have been developed to work with and train unnormalized models, rather than replacing them with normalized surrogates. The proposed method is simple and enables end-to-end optimization of the target lost function rather than a multi-stage approach. Unfortunately, I am not very familiar with this problem-space and the baselines, so I cannot speak much to the quality of the experimental evaluation. 

Weaknesses:

I am somewhat concerned by the procedure used to train the models and I feel like the authors could provide more evidence to verify that the method is working as expected. Partition function estimation and training in energy-based models is notoriously difficult and, in my experience, importance sampling estimators only tend to work in *very* small scale settings. In larger scale settings, MCMC-based training such as PCD or CD tend to scale much better. I would expect to see at least 1 MCMC-based training procedure as a baseline. But, given that all of the experiments are in relatively low-dimensional problems, we are in a regime where alternative EBM training procedures may also such such as score-matching, noise contrastive estimation, or conditional noise contrastive estimation. I believe the work would be improved if some investigation was done into the method chosen to train the model.

Having said that, I would love to see a larger-scale application -- something maybe with a few hundred or thousand variables. EBM training has been successfully scaled to very high-dimensional data so I expect the method should work. Again, I am not very familiar with this space so I am not sure if any such benchmark problems exist, but if they do, that would be nice to include. 

Besides this, I found the experimental details to be very lacking. There is no mention of the number of importance samples used in training and the scales of the importance weighting distribution. The neural networks used in the model are not described very well and I had to re-read a few times to understand that p(y|x) = N(y; neural_net1(x), neural_net2(x)). Is that right? The types of neural networks should be explained.

There is also no discussion of how long the models are trained, how they are optimized, and how model selection is performed. This is very difficult for EBMs, and therefore people typically use proxies for model fit instead of test-set likelihood. For that reason, I feel like it would be important for the authors to comment on this in the paper. 
questions: 
After reading this paper, I am left with a few questions. First, did you explore alternative methods for EBM training such as contrastive divergence, PCD, noise contrastive estimation, or score matching? As I mentioned above, at the scale of the presented experiments, I think it is likely that a number of these techniques could be successful. 

Next, given that your training procedure relies on self-normalized importance sampling, did you perform any verification that the SNIS estimator is accurate? One way to do this would be to show the effective sample size of the importance samples. This can also be used to help tune many parts of the sampler such as the proposal distribution.
limitations: The work discusses some of the limitations of EBM training and evaluation but does not spend much time talking about the specific drawbacks of the proposed method. I think the work would benefit from a limitations section.
review_title: null

Meta-review: metareview: In agreement with all the reviewers, I recommend acceptance. In the final version, the authors should take into account the reviewers’ recommendations and update the paper accordingly. Also, regarding the suggestion of Reviewer kQia, I recommend that the author include a high-dimensional synthetic experiment to analyze how their method behaves in that scenario.
confidence: Certain
Decision: Accept