Towards Robust Interpretable Surrogates for Optimization

Marc Goerigk; Michael Hartisch; Sebastian Merten

REVIEW 3 major objections 4 minor 41 references

Towards Robust Interpretable Surrogates for Optimization

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Training a small decision tree that maps observed costs to solutions, under worst-case budgeted perturbations of those observations, gives surrogates that keep most of their nominal quality while performing much better on noisy and…

desk verdict Solid robust-optimization extension of interpretable surrogates, but the out-of-sample global-budget comparison is weakened by a test/training budget mismatch and needs fixing before the headline claim is trusted. read the letter →

arxiv 2412.01264 v1 pith:Y2SZBQAN submitted 2024-12-02 cs.LG math.OC

Marc Goerigk , Michael Hartisch , Sebastian Merten This is my paper · ORCID

classification cs.LGmath.OC MSC 90C2790C1168Q17

keywords data-drivenoptimizationinterpretabilityandexplainabilityinrobustdecisiontreesbudgeteduncertaintysurrogatescenariogenerationmin-max

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper asks whether a decision tree that maps observed cost data to solutions of an optimization problem can be made robust to measurement errors and still stay interpretable. It formalizes the task as choosing a tree that minimizes the worst-case total cost when an adversary perturbs each training observation within a budgeted uncertainty set. The authors design an exact scenario-generation method plus several heuristics and test them on shortest-path instances. Their central finding is that robust trees cost only a small loss in nominal, unperturbed performance, while improving worst-case performance on disturbed observations and also performing better on out-of-sample data. Accepting this claim would make interpretable surrogates usable in settings where input data are known to be unreliable.

What carries the argument

The central object is the robust decision-tree surrogate: a univariate, bounded-depth tree whose internal nodes each test one cost coefficient against a threshold and whose leaves contain feasible solutions of the underlying optimization problem. The argument is carried by the robust interpretable tree (RIT) formulation, whose objective is the worst-case sum over training samples under a budgeted $\ell^1$ perturbation set, either global $U_{\text{glob}}$ (one budget for all perturbations) or local $U_{\text{loc}}$ (a budget per sample). The exact method alternates a flow-based mixed-integer master problem, restricted to split thresholds at midpoints of sorted training values, with an adversary problem that maximizes cost by assigning samples to leaves; the adversary problem is a multiple-choice knapsack. Heuristics split the joint tree-structure and leaf-solution decision by fixing one side and optimizing the other, alternating between the two until convergence.

What would settle it

Take any training instance, solve the robust tree problem with the midpoint grid, then search each chosen split interval on a fine continuous grid; if a single threshold outside the midpoint set lowers the worst-case cost, the method's optimality claim fails. On the empirical side, an out-of-sample dataset containing real measurement noise would be the test: the paper's claim predicts the robust tree achieves lower average cost than the nominally trained tree at small nominal loss.

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Extended reading notes

Core claim

On the paper's own terms, the central claim is that the min-max problem of finding a robust interpretable decision tree, $\min_{T} \max_{\xi \in U} \sum_j c_j^\top T(c_j+\xi_j)$, is tractable enough in practice to be worth solving: exact scenario generation works for small instances, and a heuristic that fixes the tree structure and then optimizes the leaf solutions finds good surrogates quickly. The experiments show that the price of robustness is low—robust trees are within roughly two percent of the nominal tree's training objective—while their worst-case in-sample objective improves by up to about sixteen percent, and on out-of-sample data robust trees beat the nominal tree even when no perturbations are applied. The paper also establishes that with sufficiently large perturbation budgets a single-solution tree is optimal, and that assigning optimal solutions to the leaves of a fixed tree is NP-hard even for a simple selection problem with a unit budget.

Load-bearing premise

The exact method proves optimality only among trees whose split thresholds are midpoints between sorted training values, so the true best robust tree might use a threshold the model never offers.

Editorial extensions

If this is right

A robustly trained tree loses less than about two percent of nominal in-sample performance while improving worst-case in-sample cost by roughly nine to sixteen percent over a nominally trained tree, so robustness does not require sacrificing much everyday quality.
Trees trained for robustness beat the nominally trained tree on out-of-sample test data even when the test data are unperturbed, and the advantage grows with instance size, so robust training appears to act as a regularizer.
Whenever the perturbation budget exceeds $D\cdot N\cdot M$ (global) or $D\cdot M$ (local), a single fixed solution is provably optimal, so large uncertainty budgets make deep trees unnecessary.
Even fixing the tree structure and choosing leaf solutions optimally is NP-hard for the selection problem, so the heuristic decomposition into tree-building and solution-assignment steps is justified.
Worst-case performance under the global and local budgeted uncertainty sets is strongly correlated, so the cheaper local-budget model can serve as a practical proxy for the global model.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The restriction of split thresholds to midpoints of sorted training values may make the reported robust optima conservative; solving the same min-max problem with continuous thresholds could find trees with even smaller nominal loss or better worst-case cost, an improvement the paper does not claim.
The out-of-sample gains in unperturbed settings suggest that robust training is implicitly regularizing the tree, which connects this work to structural-risk arguments without the paper drawing that link.
Because local-budget uncertainty is easier to optimize and strongly correlated with the global set, a practitioner could estimate per-observation budgets from historical measurement residuals and train under $U_{\text{loc}}$ alone, an extension the paper lists as future work.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague —

Worth a look: the authors extend the interpretable-surrogate framework (GH23) by adding robustness to budgeted perturbation of observed cost parameters. The formulation is clean: a min-max problem over observation perturbations under global and local L1 budgets, solved by an iterative scenario-generation method whose adversary is a multiple-choice knapsack, plus three heuristics and two sensible complexity results. Theorem 4.1 (large budget collapses RIT to the single-solution problem) is correct and useful; Theorem 4.2 (NP-hardness of the leaf-solution assignment with Γ=1) looks sound. The code is on GitHub, which is genuinely helpful.

The main empirical claim — robust trees lose little nominal performance and gain on disturbed and out-of-sample data — is supported in-sample, but the out-of-sample global-budget comparison is flawed. Training sets Γglob = NΓloc with N between 3 and 10; the test evaluation then reuses that same absolute Γglob on 1000 test observations. Because Uglob is one global L1 budget over the whole dataset, that makes the test adversary 100–300 times weaker per observation. The natural test analogue is Γglob = TΓloc, or per-observation evaluation. So the Hglob_tree vs. Tnom advantage in the Table 5 test rows mostly reflects a diluted adversary, not robust generalization. I checked Section 5.2.3 and the tables: the paper explicitly says the same absolute budget was used; the stress-test note is right.

Other soft spots, in proportion: no error bars or significance tests anywhere, so some differences in Tables 4–5 could be noise. The exact method solves almost nothing for λ ≥ 0.05 on Uglob (Table 3), so the practical evidence rests on Htree; that is fine for a heuristic paper, but it should be stated that the "optimal" method is basically not solving the real instances. And the threshold grid restriction to midpoints plus a 9-point post-hoc search is a real limitation; I would like the paper to say outright that optimality is only within that grid.

What is genuinely good: the adversary knapsack formulations, the NP-hardness proof, and the local-budget results, which do not suffer from the test-budget mismatch and show robust gains out-of-sample (Table 5, e.g., -6 to -8% for Hloc_tree vs. nominal). That part of the central claim survives.

Recommendation: send it to a serious referee, but expect a major revision. The experimental protocol needs fixing (scale the test budget, add error bars or tests), and the threshold-grid limitation should be discussed head-on.

Referee Report

3 major / 4 minor

Summary. This paper studies decision-tree surrogates for optimization problems when the observed cost parameters are subject to worst-case perturbations. The authors formulate a min-max problem (RIT) over univariate trees, introduce a global budgeted uncertainty set Uglob and a local one Uloc, and propose an exact scenario-generation MIP (master problem (1) plus adversary problem (2)/(3)), a post-processing step, and three heuristics H_tree, H_sol, H_alt, together with a single-solution baseline H1. The main theoretical results are Theorem 4.1 (large-budget regimes make H1 optimal) and Theorem 4.2 (NP-hardness of leaf-solution assignment for a fixed tree). In experiments on synthetic grid shortest-path instances, the paper compares robust trees with a nominal surrogate Tnom and reports that robust models lose little nominal performance and improve worst-case and out-of-sample performance. The code and data are made available.

Significance. The contribution is potentially useful: it extends the interpretable-surrogate framework of Goerigk and Hartisch to a robust setting, and the two uncertainty models plus the exact/heuristic solution methods are clearly specified. Strengths include explicit adversary formulations, a clean large-budget characterization (Theorem 4.1), a correct-looking NP-hardness reduction for LSAP, reproducible code and data, and honest reporting that the exact scenario-generation method solves few instances to optimality for larger budgets. The main caveats are that the exact method is exact only over a discrete threshold grid, and that the global-budget out-of-sample experiment dilutes the test adversary by reusing the training budget on a 1000-point test set. If these are addressed, the empirical claims about robust generalization can be evaluated properly.

major comments (3)

[Section 5.2.3, Table 5] The out-of-sample evaluation for the global budget reuses the training budget Gammaglob = N Gammaloc (with N = 3 to 10 training samples) on a test set of T = 1000 observations. Since Uglob is a single L1 budget across the whole set, the per-observation adversary power on the test set is smaller by a factor of order T/N, roughly 100 to 300, than on the training set. The reported advantages of H_glob_tree over Tnom on the test data therefore likely reflect a much weaker test adversary rather than robust generalization. Please re-evaluate with Gammaglob_test = T Gammaloc, or with a per-observation evaluation using Gammaloc, and report error bars or significance tests over the 20 instances.
[Section 2.2, Problem (1)] The exact method restricts split thresholds to the finite midpoint set Theta(i), while Problem (RIT) as defined in Section 2.1 allows any threshold in the interval [min_j c_{j,i}, max_j c_{j,i}]. The robust training objective is not constant on such intervals because the adversary costs rho_j^k depend linearly on the distance from c_{j,i} to the threshold; hence the midpoint restriction is not without loss of optimality. The post-processing over the nine-point grid Pi = {0.1, ..., 0.9} only explores a finite subset of each interval and does not restore global optimality. The paper should either define the solved problem as restricted to Theta(i) and adjust the 'optimal' claims accordingly, or justify that an optimal threshold can always be found on the grid.
[Section 5.2.2, Table 3 and Figures 6-7] The comparison between SG and the heuristics is difficult to interpret because SG is not solved to optimality for most instances; Table 3 reports only counts of instances with a gap of at most 0.001, not the actual observed gaps. Since the plotted SG objective values are therefore upper bounds on the true robust optimum for these instances, the claim that H_sol performs noticeably worse than other robust methods is not fully established. Please report average optimality gaps, or mark the points in Figures 6 and 7 where the one-hour time limit was reached.

minor comments (4)

[Section 4.2, Theorem 4.2] The reduction starts from a partition problem with the additional constraint p = n/2, but the standard Partition problem does not constrain cardinality. Please state the cardinality-constrained variant explicitly and justify its NP-completeness, or give a short reduction from the unconstrained variant.
[Section 5.1, Tables 4 and 5] No error bars or significance tests are reported for any of the numerical comparisons beyond averaging over 20 instances. Adding standard errors, especially for the out-of-sample rows in Tables 4 and 5, would make the relative rankings much more informative.
[Section 2.2, Constraint (1h)] The big-M values M^j are defined as max_{x in X} c_j^T x or any upper bound; this assumes bounded X or a known finite upper bound. For the shortest-path experiments this is fine, but the text should state the boundedness assumption explicitly.
[Section 4.1, Theorem 4.1] The proof's equality/inequality chain would benefit from one sentence explaining why, under the stated budgets, the adversary can simultaneously send every sample to any leaf, rather than only each sample individually.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: robust-tree formulation and experiments are self-contained, with only a benchmarking confound in the global-budget test evaluation.

full rationale

The paper's central derivation is not circular. Problem (RIT) in Section 2.1 defines a new min-max objective, and the exact scenario-generation method, the adversary MIPs (Models (2)-(3)), and the heuristics are all built from this definition; no fitted constant is later relabeled as a prediction. The nominal baseline Tnom is re-implemented using Model (1) rather than taken on faith from the authors' earlier framework, and comparisons also include the trivial single-solution heuristic H1, so the robust claim is tested against independent or intentionally weak alternatives. Optimality is explicitly scoped: thresholds are restricted to the midpoint grid Θ(i) in Section 2.2, with a post-processing search over a fixed 9-point discretization, so the exact method is optimal only within that stated class. The self-citations to [GH23], [GH24], and [GHMS24] are contextual background (framework, textbook, feature-based extension) and do not carry any load-bearing theorem; Theorems 4.1 and 4.2 are new proofs with self-contained constructions. The most notable weakness is in Section 5.2.3, where Γglob=NΓloc is trained on N samples and then reused as an absolute budget on T=1000 test observations, diluting the per-observation global-budget threat; the authors themselves acknowledge this: 'This can be explained by the use of the same absolute values for Γglob in the training and test environment, which reduces its relative impact in the latter case.' That is an evaluation limitation, not a circular reduction: test data are not used to fit the model, and the out-of-sample objective is not equal to the training objective by construction. No circular step can be exhibited.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central results do not rely on fitted constants or invented entities. The main assumptions are the midpoint threshold restriction, the observation-only perturbation model, MIP representability of the underlying problem, and the use of balanced partition NP-completeness in the reduction. The user-chosen budget λ and numerical constants are experiment settings, not fitted determinants of the main conclusions.

free parameters (3)

Uncertainty budget scaling λ
In Section 5.1, λ ∈ [0,1] is chosen by the user to scale Γloc and Γglob in experiments; it is an experimental hyperparameter, not fitted to data.
Threshold refinement grid Π = 0.1, 0.2, ..., 0.9
Post-processing discretization in Section 2.2; chosen by hand, affects final tree quality.
Tie-breaking constant ϵ = 0.001
Used for strict inequalities in adversary perturbation costs (Section 5.1); a numerical convention.

assumptions (4)

domain assumption Split thresholds are restricted to midpoints of sorted training values for each item; the optimal tree is assumed to be findable within this discrete grid with post-hoc refinement.
Section 2.2 defines Θ(i) as midpoints between sorted entries. This restricts the hypothesis space; post-processing over a 9-point grid partially mitigates the restriction but does not remove it.
domain assumption The true cost vector of an instance is fixed; only its observation can be perturbed by an adversary within a budgeted L1 uncertainty set.
Section 2.1 states that observed values may deviate from the true underlying values, which do not change. This observation-only perturbation model is the basis of the robust formulation.
standard math The underlying optimization problem min_{x∈X} c^T x is representable as a mixed-integer linear program.
Section 2.1 assumes MIP representability, which is needed for the master problem and for the experiments on shortest path.
standard math Balanced partition with an exact cardinality of n/2 is NP-complete, used as the source problem in the reduction for Theorem 4.2.
Theorem 4.2 reduces from this partition variant; the NP-completeness of the cardinality-constrained partition is standard.

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Cite this review

Pith. "Pith review of Towards Robust Interpretable Surrogates for Optimization." pith.science (2026). https://pith.science/paper/Y2SZBQAN

@misc{pith2026241201264,
  author       = {Pith},
  title        = {Pith review of: Towards Robust Interpretable Surrogates for Optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y2SZBQAN}},
  note         = {Machine review of arXiv:2412.01264}
}

read the original abstract

An important factor in the practical implementation of optimization models is the acceptance by the intended users. This is influenced among other factors by the interpretability of the solution process. Decision rules that meet this requirement can be generated using the framework for inherently interpretable optimization models. In practice, there is often uncertainty about the parameters of an optimization problem. An established way to deal with this challenge is the concept of robust optimization. The goal of our work is to combine both concepts: to create decision trees as surrogates for the optimization process that are more robust to perturbations and still inherently interpretable. For this purpose we present suitable models based on different variants to model uncertainty, and solution methods. Furthermore, the applicability of heuristic methods to perform this task is evaluated. Both approaches are compared with the existing framework for inherently interpretable optimization models.

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Min--max and min--max regret versions of combinatorial optimization problems: A survey

Hassene Aissi, Cristina Bazgan, and Daniel Vanderpooten. Min--max and min--max regret versions of combinatorial optimization problems: A survey. European Journal of Operational Research , 197(2):427--438, 2009

work page 2009

[2] [2]

Optimal policy trees

Maxime Amram, Jack Dunn, and Ying Daisy Zhuo. Optimal policy trees. Machine Learning , 111(7):2741--2768, 2022

work page 2022

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A framework for data-driven explainability in mathematical optimization

Kevin-Martin Aigner, Marc Goerigk, Michael Hartisch, Frauke Liers, and Arthur Miehlich. A framework for data-driven explainability in mathematical optimization. Proceedings of the AAAI Conference on Artificial Intelligence , 38(19):20912--20920, Mar. 2024

work page 2024

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Sina Aghaei, Andr \'e s G \'o mez, and Phebe Vayanos. Strong optimal classification trees. Operations Research , 2024

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Explainable predict-and-optimize

Rafael Blanquero, Emilio Carrizosa, and Nuria G \'o mez-Vargas. Explainable predict-and-optimize. Preprint available online , 2023

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Decision trees: from efficient prediction to responsible AI

Hendrik Blockeel, Laurens Devos, Beno \^ t Fr \'e nay, G \'e raldin Nanfack, and Siegfried Nijssen. Decision trees: from efficient prediction to responsible AI . Frontiers in Artificial Intelligence , 6:1124553, 2023

work page 2023

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Interpretable operations research for high-stakes decisions: Designing the Greek COVID -19 testing system

Hamsa Bastani, Kimon Drakopoulos, Vishal Gupta, Jon Vlachogiannis, Christos Hadjichristodoulou, Pagona Lagiou, Gkikas Magiorkinis, Dimitrios Paraskevis, and Sotirios Tsiodras. Interpretable operations research for high-stakes decisions: Designing the Greek COVID -19 testing system. INFORMS Journal on Applied Analytics , 52(5):398--411, 2022

work page 2022

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Improving Stability in Decision Tree Models

Dimitris Bertsimas and Vassilis Digalakis Jr. Improving stability in decision tree models. arXiv preprint arXiv:2305.17299 , 2023

work page Pith review arXiv 2023