A framework for inherently interpretable optimization models

Q: What are the constraints for multivariate decision trees?

Multivariate decision trees have constraints to replace (1c-1e) which involve the equation i[n] EQUATION ) q i R i [n], q [Q] (3f). These constraints aim to ensure comprehensibility of the resulting decision tree by allowing at most P non-zero entries in each q, i.e., sparse rules. Additionally, a normalization constraint i[n] | q i | = 1 is used to mitigate numerical issues. Integer coefficients can further reduce numerical issues but may impact computability. When using enumeration or greedy heuristic approaches, more combinations need to be considered, especially when P is a constant. Enumerating all possibilities for binary variables d q i, with at most P active, involves considering all P-dimensional hyperplanes defined through the costs c ji. However, these approaches become computationally expensive with higher values of P.

Q: How can varying splitting criteria improve decision tree quality?

Varying splitting criteria depending on the subtrees in decision trees can improve the quality of solutions obtained. While maintaining comprehensibility, different splits based on the current region reached during tree traversal can lead to higher quality solutions. By adjusting the splitting criteria at each internal node, the decision tree can better represent the scenarios and constraints associated with each leaf. This approach, although slightly decreasing comprehensibility, allows for more variable combinations to be considered in the enumeration method or greedy heuristic. Consequently, more subproblems need to be solved or considered, leading to potentially better solutions in decision tree-based methods.

Question

1. What is the importance of providing easily comprehensible explanations for decisions obtained from optimization processes?

2. What is the main goal of the optimization rule in the given framework?

3. How does the optimization rule allocate solutions?

4. What is the mixed-integer programming formulation for minimizing F p p p (a) when restricting the optimization rule to a univariate, binary decision tree of fixed depth?

Accepted Answer

Providing easily comprehensible explanations for decisions obtained from optimization processes is crucial for motivating actions and fostering trust and positive employee-organization relationships. Complex and difficult-to-understand decision models can result in skepticism and reluctance to adopt or use the system designed to support decision-making. Incorporating workers into the feedback cycle of the modeling process and providing insights into decisions obtained from optimization processes improves the feedback loop. End users often have the need to explain decisions obtained from AI, such as for loan or medical consultants. The European 'right to explanation' aims to ensure that users can ask for an explanation of an algorithmic decision made about them. A framework is explainable if an easily comprehensible, post-hoc justification of the obtained result is available, while a framework is interpretable if an easily comprehensible rule is available that makes it possible to deduce a result from the instance itself. Our approach focuses on creating inherently interpretable models that provide an interpretable optimization rule, mimicking the optimization process and explaining when to use which solution. This approach places our research between K-adaptability and decision rule approaches, emphasizing the importance of comprehensibility in decision-making processes.

Accepted Answer

The main goal of the optimization rule in the given framework is to assign a solution to any cost scenario, where in the case of the observed cost scenarios, the rule assigns a solution x x x X. The decision-dependent random variable C(x x x) takes a value C j (x x x) = c c c j x x x with probability p j for each scenario index j [N ]. The quality of an optimization rule a is evaluated by calculating F p p p (a) = u p p p (C 1 (a(1)), . . . , C N (a(N ))) , where the goal is to find a rule a that minimizes the function F p p p. The rule should be easily comprehensible and contain only a small number of potential solutions, ensuring that it can be applied in several future scenarios.

Accepted Answer

The optimization rule allocates solutions based on the true realization of costs. It uses a univariate, binary tree of fixed depth with the same splits in each level. By fixing the depth of the tree to two, at most four different solutions can be attained at the leaves. The rule selects the best solution for each category defined by the selected splits. For example, if c2 > 5.5 and c3 > 6, solution x2 is chosen. The costs associated with each solution are given in Table 1. However, the rule may not always allocate the optimal solution for every scenario due to its simple structure. It may also result in higher costs compared to a min-sum-min solution, but it can perform better on additional testing data not used during optimization.

Accepted Answer

The mixed-integer programming formulation for minimizing F p p p (a) when restricting the optimization rule to a univariate, binary decision tree of fixed depth involves the following components: 1. A decision tree with a fixed depth, represented by the number of splits (Q) and the number of leaves (K). 2. Binary variables (d q i) indicating whether an entry i of the cost vector is queried in split q of the decision tree. 3. Indicator variables (a k j) representing whether in scenario j [N ] the traversal of the decision tree reaches leaf k [K]. 4. A cost scenarios vector (c c c j R n) representing the costs associated with each scenario. 5. An optimization rule that selects solutions (x x x k) for the leaves and splits (d q i, b q) at each level of the decision tree. The objective is to minimize F p p p (a) by simultaneously selecting solutions and splits. The formulation includes constraints to ensure comprehensible rules, such as each scenario being mapped to exactly one solution, and each split querying only a single parameter. The model allows for the existence of multiple optimal solutions and can be adapted to different optimization criteria, such as the Laplace criterion or linear models.

Accepted Answer

Restrictions (1c) ensure that each split within the decision tree only considers a single dimension of the cost vector c c c. This feature not only enhances comprehensibility but also provides a computational advantage. Given a constant tree depth Q, there are O(n Q) possibilities to set variables d q i. Once the dimension i to query in split q is fixed, all possible values for b q can be enumerated. By sorting the set {c ji : j [N]} of values in dimension i and considering midpoints between subsequent sorted values, an optimal choice for b q can be determined. This results in O(N) possibilities to set b q. If all d q i and b q variables are fixed, variables a k j become fixed as well. Consequently, the remaining optimization problem involves minimizing the criterion u p p p (k[K] a k 1 c c c 1 x x x k, ..., k[K] a k N c c c N x x x k) with a fixed assignment of solutions to scenarios. These problems are referred to as subproblems. If subproblems can be solved in O(T), the interpretable optimization problem can be solved in O(n Q N Q T). In the case of the Laplace decision criterion, the optimization problem can be further simplified, resulting in K separate nominal problems. Overall, Restrictions (1c) contribute to computational efficiency by reducing the number of possibilities to consider and simplifying the optimization problem.

Accepted Answer

The greedy heuristic optimizes decision tree construction by determining splits one level at a time, rather than considering all possible combinations. It starts by enumerating all possible splits on the first level, solving O(nN) subproblems. Then, the best candidate split is fixed, and the next level is considered, solving O(nN) subproblems again. This process is repeated until the desired depth Q is reached, resulting in a feasible solution in O(QnN T). The greedy heuristic is optimal for Q = 1, and for the Laplace criterion, the number of nominal problems solved is O(nN K). Other heuristic approaches, such as evolutionary algorithms, can also be used to construct decision trees.

Accepted Answer

Meta data can be utilized in optimization rules by expanding the problem domain with dummy variables representing the available meta data. Each dummy variable should be forced to zero to prevent non-cost information from affecting the objective function. For example, in a shortest path setting, meta data such as day of the week can be used to optimize the path. By incorporating meta data, the optimization rules can consider additional factors that impact decision-making, leading to more comprehensive and high-quality rules for future scenarios. Expert knowledge of the underlying problem is useful in obtaining concise optimization rules. Additionally, meta data can be used to preprocess data, such as identifying rush hour scenarios or partitioning time stamp information into clusters, to enhance the optimization process.

Accepted Answer

Uncertainty in constraints can impact the feasibility of solutions by altering the feasible region. In scenarios where changing parameters affect the quality but not feasibility, the feasible set of solutions remains constant. However, if a scenario also affects the feasible region, each scenario induces a different feasibility set. The k-th solution x_k only needs to satisfy the feasibility sets of the scenarios it is assigned to. The feasibility set X_k,j (a_k,j) can be enforced via a linear constraint system A_j x_k <= b_j, where A_j is a matrix, x_k is the solution vector, and b_j is the constraint vector. If a solution is allowed to not obey the feasibility set with probability [0, 1], auxiliary variables g_j can be used to adjust the feasible region. The feasible region is given by A_j x_k <= b_j + (1 - a_k,j + g_j)M_k, where M_k is a vector with sufficiently large entries. The existence of appropriate values for M_k depends on the structure of X. Overall, uncertainty in constraints can lead to different feasibility sets and require adjustments to enforce feasibility.

Accepted Answer

Multivariate decision trees have constraints to replace (1c-1e) which involve the equation i[n] EQUATION ) q i R i [n], q [Q] (3f). These constraints aim to ensure comprehensibility of the resulting decision tree by allowing at most P non-zero entries in each q, i.e., sparse rules. Additionally, a normalization constraint i[n] | q i | = 1 is used to mitigate numerical issues. Integer coefficients can further reduce numerical issues but may impact computability. When using enumeration or greedy heuristic approaches, more combinations need to be considered, especially when P is a constant. Enumerating all possibilities for binary variables d q i, with at most P active, involves considering all P-dimensional hyperplanes defined through the costs c ji. However, these approaches become computationally expensive with higher values of P.

Accepted Answer

Varying splitting criteria depending on the subtrees in decision trees can improve the quality of solutions obtained. While maintaining comprehensibility, different splits based on the current region reached during tree traversal can lead to higher quality solutions. By adjusting the splitting criteria at each internal node, the decision tree can better represent the scenarios and constraints associated with each leaf. This approach, although slightly decreasing comprehensibility, allows for more variable combinations to be considered in the enumeration method or greedy heuristic. Consequently, more subproblems need to be solved or considered, leading to potentially better solutions in decision tree-based methods.

Accepted Answer

First-stage costs in multi-stage robust optimization refer to the costs associated with implementing a solution in a given scenario. These costs are independent of the scenario realization and are necessary for preparatory actions in various fields, such as disaster management. In the proposed model, the costs of implementing solution k in scenario j are represented as c c c j x x x k. However, the model also considers the costs of preparatory actions, represented by the variable vector y y y R p, which are associated with all potential future solutions x x x k. These preparatory actions ensure that the performed preparations enable the realization of solution x x x k. By adding the cost term c c c y y y to the objective function and modeling the connection between preparation and implementation in X, the original model can be adapted to account for first-stage costs.

Accepted Answer

The section presents three experiments to compare the performance of the greedy heuristic with an optimal solution for the interpretable optimization problem. The experiments aim to evaluate the efficiency and effectiveness of the greedy heuristic in solving the problem. The code is implemented in C++ using Cplex version 12.8 to solve mixed-integer programs. The experiments focus on the sum of objective values as the decision criterion. The results of these experiments can provide insights into the trade-offs between using a greedy heuristic and an optimal solution in interpretable optimization problems.

Accepted Answer

In Figure 2 (a), G2 and IP2 have identical performance, indicating that both methods place a single decision optimally. However, on test instances, there may be differences in performance values between G2 and IP2, as optimal solutions are not necessarily unique. Overall, the greedy heuristic provides a reasonable approximation of the optimal solution, with a relatively small loss in performance compared to the improvements in median computation times shown in Figure 3. The greedy method offers a good trade-off between small computation times and larger performance values, making it a preferred method for subsequent experiments.

Accepted Answer

In Experiment 2, G4 achieves nearly the same performance as MSM4 on training instances, with a performance difference of about one to two percentage points. This comparison is shown in Figure 4 (a). Interestingly, G2 even outperforms MSM2, which is only possible due to the time limit applied to the latter method. The results indicate that optimization rules only lead to a small decrease in performance compared to general decision rules, especially when sufficient training scenarios are available. Additionally, the linear solution time of the greedy method (G4) means that the approach scales well in problem size.

Accepted Answer

The dataset used in Experiment 3 is real-world data collected in (Chassein, Dokka, & Goerigk, 2019) with 538 nodes and 1308 edges representing the road network of Chicago. It contains 4363 scenarios, split into training and testing halves. A total of 100 random source-sink pairs are generated, with nominal paths ensuring less than 50 edges. Two artificial edges are added for meta data, with the first edge using days of the week and the second edge using time in seconds since midnight. The greedy heuristic is used to find optimization rules using all 1310 edges, resulting in median computation times of 1209.5 and 3450.5 seconds for the first setting, and 126.4 and 363.1 seconds for the second setting. The first splitting decisions are based on the day of the week and time of the day, with edges along Lake Michigan's shore being frequently split. The data shows traffic speed volatility and snapshots at specific times, leading to small improvements from meta data rules.

Accepted Answer

The comparison between the greedy heuristic and mixed-integer programming formulation focuses on finding optimization rules in the form of a decision tree. Recent research on decision trees for classification models shows promising results, but computation times become prohibitive for larger problems. The greedy heuristic scales well and is a viable choice for larger optimization problems, with solution quality close to exact solutions. The heuristic is preferred for remaining experiments. The second question examines the loss when using optimization rules instead of arbitrary rules, showing little difference on training data but significant differences on test data. With more training data, optimization rules perform similarly to general rules. Finally, the performance on real-world data shows improvements towards the lower bound, with high-density edges providing reliable predictions. The results are valid within the scope of the setup used, but the methods were not specifically designed for the shortest path problem.

Accepted Answer

Interpretability in optimization models is crucial for practical application and stakeholder acceptance. The black-box nature of optimization solvers poses a major barrier, as solutions are only useful if they can be applied in practice and accepted by stakeholders. Interpretability and explainability, central concepts in artificial intelligence, have received less attention in operations research and management science. In the provided section, an inherently interpretable optimization model is proposed to derive solutions with optimization rules, making them easily comprehensible and applicable. Decision tree-based rules are focused on, as they are easily understandable and can be applied. The model can be extended to include various features and can incorporate nominal solution algorithms to improve applicability. Experiments show that using interpretable optimization rules instead of complex, incomprehensible rules results in a minimal loss of only a few percentage points. The paper emphasizes the need for further research in interpretable optimization methods and testing them with decision makers to determine acceptance. The section also considers different approaches, such as producing a single solution with a simple and communicable structure, as demonstrated by the example of a car driver seeking directions.

A framework for inherently interpretable optimization models

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Most frequently asked questions

1. What is the importance of providing easily comprehensible explanations for decisions obtained from optimization processes?

2. What is the main goal of the optimization rule in the given framework?

3. How does the optimization rule allocate solutions?

4. What is the mixed-integer programming formulation for minimizing F p p p (a) when restricting the optimization rule to a univariate, binary decision tree of fixed depth?

5. What is the computational benefit of Restrictions (1c) in the proposed model?

6. How does the greedy heuristic optimize decision tree construction?

7. How can meta data be utilized in optimization rules?

8. How does uncertainty in constraints affect the feasibility of solutions?

9. What are the constraints for multivariate decision trees?

10. How can varying splitting criteria improve decision tree quality?

11. What are first-stage costs in multi-stage robust optimization?

12. How does greedy heuristic compare to optimal solution?

13. How do G2 and IP2 compare in performance?

14. How does G4 compare to MSM4 on training instances?

15. What is the dataset used in Experiment 3?

16. How do greedy heuristic and mixed-integer programming compare?

17. What is the importance of interpretability in optimization models?

Citations

Explainable AI for Operational Research: A Defining Framework, Methods, Applications, and a Research Agenda

Explainable interactive evolutionary multiobjective optimization

Fair and Efficient Resource Allocation via Vehicle-Edge Cooperation in 5G-V2X Networks

A Framework for Data-Driven Explainability in Mathematical Optimization

OptiChat: Bridging Optimization Models and Practitioners with Large Language Models

References

A unified approach to interpreting model predictions

Stop explaining black box machine learning models for high stakes decisions and use interpretable models instead

Explainable Artificial Intelligence (XAI): Concepts, Taxonomies, Opportunities and Challenges toward Responsible AI.

Explanation in artificial intelligence: Insights from the social sciences

Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining

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