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arxiv: 2605.13830 · v1 · submitted 2026-05-13 · 💻 cs.AI · cs.LG

Recognition: no theorem link

Quantifying Sensitivity for Tree Ensembles: A symbolic and compositional approach

Ajinkya Naik, Ashutosh Gupta, Chaitanya Garg, Kuldeep S. Meel, S. Akshay

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Pith reviewed 2026-05-14 17:45 UTC · model grok-4.3

classification 💻 cs.AI cs.LG
keywords decision tree ensemblessensitivity analysisalgebraic decision diagramscompositional computationformal verificationmodel countingcertified bounds
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The pith

Decision tree ensembles can have their sensitivity to small input changes quantified by discretizing the space and counting susceptible regions via algebraic decision diagrams.

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

The paper establishes a quantitative sensitivity measure for decision tree ensembles by discretizing the input space and enumerating regions where small feature changes cause misclassification. It develops an algorithmic technique that encodes the problem as an algebraic decision diagram and decomposes it into independent subproblems solved compositionally. This yields an efficient count of sensitive regions together with certified error and confidence bounds. A sympathetic reader would care because ensembles appear in safety-critical classification tasks where knowing the volume of sensitive regions supports verification and risk analysis. The resulting tool demonstrates clear speedups over direct model counting on benchmarks that vary in number of trees and depth.

Core claim

By representing the discretized sensitivity query as an algebraic decision diagram and splitting the diagram into subproblems that can be solved independently, the count of regions susceptible to misclassification can be obtained efficiently together with explicit error and confidence bounds, outperforming standard model counters on ensembles of increasing size.

What carries the argument

Algebraic decision diagram encoding of the discretized sensitivity function, decomposed into subproblems for compositional evaluation.

If this is right

  • Larger ensembles become feasible to analyze for sensitivity than with direct model counting.
  • The certified bounds allow safe use of the sensitivity number in downstream verification tasks.
  • Compositional decomposition supports parallel execution for further scaling.
  • Sensitivity can be compared across different ensemble sizes and depths under the same certified regime.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same ADD decomposition pattern could be reused for other ensemble properties such as robustness margins.
  • Adaptive choice of discretization granularity based on tree depth might tighten the error bounds further.
  • The method opens a route to sensitivity-guided retraining loops inside automated machine-learning pipelines.

Load-bearing premise

The discretization of the input space together with the algebraic decision diagram encoding captures every sensitivity region without missing or adding errors that would invalidate the certified bounds.

What would settle it

Manually enumerate all sensitive regions on a small decision tree ensemble with known discretization and check whether the computed count lies inside the reported error interval.

Figures

Figures reproduced from arXiv: 2605.13830 by Ajinkya Naik, Ashutosh Gupta, Chaitanya Garg, Kuldeep S. Meel, S. Akshay.

Figure 1
Figure 1. Figure 1: Two-tree ensemble represented as decision trees. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Cactus plots comparing execution time across all benchmarks. Tool Solved PAR2 ADD-baseline 153 3428.461 ApproxMC 579 3005.374 Ganak 721 2901.421 XCount 1552 1937.474 [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Relative error of XCount compared to exact counts across all benchmarks. Scalability. We evaluate the scalability of the tool with increasing size of the ensemble. Figure 4a depicts a heat map of experiments over all datasets to illustrate a relation between the number of trees of the ensemble and the max depth of the trees of the ensemble. The higher shades indicated the greater time taken by the instance… view at source ↗
Figure 4
Figure 4. Figure 4: Scalability analysis of XCount wrt ensemble size across all benchmarks [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Cactus plots for ablation ex￾periments on Adult dataset Tool Solved PAR2 Opt 1 (ADD-baseline) 68 3122.31 Opt 1+3 197 2275.374 Opt 1+2 289 1635.355 Subp + CNF 33 3408.22 XCount 345 1304.486 [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Relative sensitivity Cα/C0 for regularized models (α ∈ {1, 5, 10}) against the unregularized baseline. The yellow line marks the median; boxes and whiskers show the IQR and 1.5× IQR range. We used Adult dataset for this study. We trained several models with vary￾ing levels of regularization and compare the number of sensitive regions in each model. We repeat this process by varying the other configurations… view at source ↗
Figure 7
Figure 7. Figure 7: Cactus plots comparing execution time. Lower curves indicate better per [PITH_FULL_IMAGE:figures/full_fig_p030_7.png] view at source ↗
Figure 7
Figure 7. Figure 7: Cactus plots comparing execution time. Lower curves indicate better per [PITH_FULL_IMAGE:figures/full_fig_p031_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Performance of XCount across gap values for models of Adult dataset. Objective Function without Regularization In a typical gradient boosting framework, the goal is to minimize a loss function l that measures the difference between the predicted value yˆi and the actual value yi for each instance i. With￾out regularization, the objective function at iteration t aims solely to minimize this empirical loss [… view at source ↗
Figure 9
Figure 9. Figure 9: The three decision trees in the ensemble. Internal nodes show split con [PITH_FULL_IMAGE:figures/full_fig_p035_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: ADDs for Trees 0, 1, and 2 immediately after pruning with Masks [PITH_FULL_IMAGE:figures/full_fig_p036_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: ADDs for Trees 0, 1, and 2 after pruning and applying constraints. [PITH_FULL_IMAGE:figures/full_fig_p037_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Evolution of the cumulative DiffSum ADD for Subproblem 1. The ADD in 12c shows the final DiffSum for the Subproblem. E.6 Merging Subproblems and Counting After processing all trees, we obtain the symbolic representation of the predic￾tion difference for a subproblem [PITH_FULL_IMAGE:figures/full_fig_p038_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Final DiffSum ADDs for subproblems 0–5. These graphs represent the accumulated ensemble prediction difference for each mask pair before applying the gap threshold [PITH_FULL_IMAGE:figures/full_fig_p039_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Final BDDs for subproblems 1, 3 and 5, where the ADD did not evaluate [PITH_FULL_IMAGE:figures/full_fig_p040_14.png] view at source ↗
read the original abstract

Decision tree ensembles (DTE) are a popular model for a wide range of AI classification tasks, used in multiple safety critical domains, and hence verifying properties on these models has been an active topic of study over the last decade. One such verification question is the problem of sensitivity, which asks, given a DTE, whether a small change in subset of features can lead to misclassification of the input. In this work, our focus is to build a quantitative notion of sensitivity, tailored to DTEs, by discretizing the input space of the model and enumerating the regions which are susceptible to sensitivity. We propose a novel algorithmic technique that can perform this computation efficiently, within a certified error and confidence bound. Our approach is based on encoding the problem as an algebraic decision diagram (ADD), and further splitting it into subproblems that can be solved efficiently and make the computation compositional and scalable. We evaluate the performance of our technique over benchmarks of varying size in terms of number of trees and depth, comparing it against the performance of model counters over the same problem encoding. Experimental results show that our tool XCount achieves significant speedup over other approaches and can scale well with the increasing sizes of the ensembles.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims a novel symbolic method to quantify sensitivity of decision tree ensembles: discretize the input space, encode the resulting sensitivity regions as an algebraic decision diagram (ADD), split the ADD into subproblems that can be solved compositionally, and thereby obtain a sensitivity count together with certified error and confidence bounds. The approach is evaluated on benchmarks of varying tree count and depth, where the tool XCount is reported to achieve substantial speedups over direct model-counting encodings of the same problem.

Significance. If the certified bounds are shown to be sound with respect to the continuous input space, the work supplies a practical, scalable technique for quantitative sensitivity analysis of tree ensembles that are already deployed in safety-critical domains. The compositional ADD encoding and the reported performance gains over off-the-shelf model counters constitute a concrete engineering advance that could be adopted by existing verification tool-chains.

major comments (2)
  1. [§3.2] §3.2 (Discretization and ADD construction): the certified error and confidence bounds are stated to hold for the continuous sensitivity measure, yet the manuscript does not supply an explicit argument that every threshold-induced decision boundary in the ensemble is either aligned with or conservatively over-approximated by the chosen discretization grid. Without such a guarantee, a perturbation that crosses a split between two grid cells can be misclassified, rendering the reported bounds unsound for the original continuous problem.
  2. [§4] §4 (Compositional splitting): the complexity analysis of the ADD splitting procedure is only sketched; a precise bound on the number and size of subproblems generated as a function of ensemble depth and number of trees is required to substantiate the scalability claims made in the abstract and §5.
minor comments (2)
  1. [§2] Notation for the sensitivity measure (e.g., the precise definition of the quantitative count versus the indicator) should be introduced once and used consistently; several passages in §2 and §3 reuse the same symbol for both the Boolean and the numeric versions.
  2. [§5] Table 1 and Figure 3 would benefit from explicit column/axis labels that include the units of the reported runtimes and the exact grid resolution used in each experiment.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback. We address the two major comments below and will revise the manuscript accordingly to strengthen the presentation.

read point-by-point responses

  1. Referee: [§3.2] §3.2 (Discretization and ADD construction): the certified error and confidence bounds are stated to hold for the continuous sensitivity measure, yet the manuscript does not supply an explicit argument that every threshold-induced decision boundary in the ensemble is either aligned with or conservatively over-approximated by the chosen discretization grid. Without such a guarantee, a perturbation that crosses a split between two grid cells can be misclassified, rendering the reported bounds unsound for the original continuous problem.

    Authors: We agree that an explicit soundness argument linking the discretization grid to the continuous decision boundaries is required. In the revised version we will insert a new lemma in §3.2 that formally proves the grid either aligns with every split threshold or produces a conservative over-approximation, thereby preserving the certified error and confidence bounds for the original continuous sensitivity measure. The proof will reference the finite set of thresholds extracted from the ensemble and show that any crossing perturbation is captured by the enclosing grid cell. revision: yes

  2. Referee: [§4] §4 (Compositional splitting): the complexity analysis of the ADD splitting procedure is only sketched; a precise bound on the number and size of subproblems generated as a function of ensemble depth and number of trees is required to substantiate the scalability claims made in the abstract and §5.

    Authors: We accept that the current sketch in §4 is insufficient. The revised manuscript will contain a precise complexity statement: the splitting procedure generates at most O(t · d) subproblems, where t is the number of trees and d the maximum depth, with each subproblem of size polynomial in the number of features and the discretization resolution. This bound will be derived from the recurrence relation that governs the compositional decomposition and will be accompanied by a short proof sketch. revision: yes

Circularity Check

0 steps flagged

No circularity in derivation; independent algorithmic encoding

full rationale

The paper defines a quantitative sensitivity measure for decision tree ensembles by discretizing the input space and encoding regions as an algebraic decision diagram (ADD), then solving the resulting subproblems compositionally. This construction is presented as a new algorithmic procedure whose output is compared directly against external model counters on the identical encoding. No step reduces the claimed sensitivity count or certified bounds to a fitted parameter, a self-referential definition, or a load-bearing self-citation; the central claim remains an independent computational method whose soundness is asserted relative to the discretization and ADD representation rather than derived from prior results by the same authors.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that decision tree ensembles admit a faithful symbolic encoding into algebraic decision diagrams and that input discretization preserves the sensitivity property within the stated error bounds.

axioms (1)
  • domain assumption Decision tree ensembles can be faithfully encoded as algebraic decision diagrams for sensitivity analysis
    Invoked in the description of the encoding step that enables splitting into subproblems.

pith-pipeline@v0.9.0 · 5530 in / 1233 out tokens · 36541 ms · 2026-05-14T17:45:33.735570+00:00 · methodology

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Works this paper leans on

47 extracted references · 47 canonical work pages

  1. [1]

    In: The Thirteenth International Conference on Learning Representations,ICLR2025,Singapore,April24-28,2025.OpenReview.net(2025), https://openreview.net/forum?id=h0vC0fm1q7

    Ahmad, A., Tayal, T.V., Gupta, A., Akshay, S.: Sensitivity verification for additive decision tree ensembles. In: The Thirteenth International Conference on Learning Representations,ICLR2025,Singapore,April24-28,2025.OpenReview.net(2025), https://openreview.net/forum?id=h0vC0fm1q7

    work page 2025

  2. [2]

    In: Gurfinkel, A., Ganesh, V

    Athavale, A., Bartocci, E., Christakis, M., Maffei, M., Nickovic, D., Weis- senbacher, G.: Verifying global two-safety properties in neural networks with confidence. In: Gurfinkel, A., Ganesh, V. (eds.) Computer Aided Verification - 36th International Conference, CAV 2024, Montreal, QC, Canada, July 24- 27, 2024, Proceedings, Part II. Lecture Notes in Com...

    work page doi:10.1007/978-3-031-65630-9 2024

  3. [3]

    Formal methods in system design10(2), 171–206 (1997)

    Bahar, R.I., Frohm, E.A., Gaona, C.M., Hachtel, G.D., Macii, E., Pardo, A., Somenzi, F.: Algebraic decision diagrams and their applications. Formal methods in system design10(2), 171–206 (1997)

    work page 1997

  4. [4]

    In: 45th IEEE/ACM International Conference on Software En- gineering, ICSE 2023, Melbourne, Australia, May 14-20, 2023

    Biswas, S., Rajan, H.: Fairify: Fairness verification of neural net- works. In: 45th IEEE/ACM International Conference on Software En- gineering, ICSE 2023, Melbourne, Australia, May 14-20, 2023. pp. 1546–1558. IEEE (2023). https://doi.org/10.1109/ICSE48619.2023.00134, https://doi.org/10.1109/ICSE48619.2023.00134

    work page doi:10.1109/icse48619.2023.00134 2023

  5. [5]

    Frontiers in Ar- tificial Intelligence6(2023)

    Blockeel, H., Devos, L., Frénay, B., Nanfack, G., Nijssen, S.: Deci- sion trees: from efficient prediction to responsible ai. Frontiers in Ar- tificial Intelligence6(2023). https://doi.org/10.3389/frai.2023.1124553, https://www.frontiersin.org/journals/artificial-intelligence/articles/10.3389/frai.2023.1124553

    work page doi:10.3389/frai.2023.1124553 2023

  6. [6]

    Machine learning45(1), 5–32 (2001)

    Breiman, L.: Random forests. Machine learning45(1), 5–32 (2001)

    work page 2001

  7. [7]

    IEEE Transactions on Computers100(8), 677–691 (1986)

    Bryant, R.E.: Graph-based algorithms for boolean function manipulation. IEEE Transactions on Computers100(8), 677–691 (1986)

    work page 1986

  8. [8]

    2023 IEEE Conference on Secure and Trustworthy Machine Learning (SaTML) pp

    Calzavara, S., Cazzaro, L., Lucchese, C., Marcuzzi, F.: Explainable global fairness verification of tree-based classifiers. 2023 IEEE Conference on Secure and Trustworthy Machine Learning (SaTML) pp. 1–17 (2022), https://api.semanticscholar.org/CorpusID:252545325

    work page 2023

  9. [9]

    Applied Soft Computing73, 914– 920 (2018)

    Chang, Y.C., Chang, K.H., Wu, G.J.: Application of extreme gra- dient boosting trees in the construction of credit risk assessment models for financial institutions. Applied Soft Computing73, 914– 920 (2018). https://doi.org/https://doi.org/10.1016/j.asoc.2018.09.029, https://www.sciencedirect.com/science/article/pii/S1568494618305465

    work page doi:10.1016/j.asoc.2018.09.029 2018

  10. [10]

    In: Chaudhuri, K., Salakhutdinov, R

    Chen, H., Zhang, H., Boning, D., Hsieh, C.J.: Robust decision trees against adversarial examples. In: Chaudhuri, K., Salakhutdinov, R. (eds.) Proceedings of the 36th International Conference on Machine Learning. Proceedings of Ma- chine Learning Research, vol. 97, pp. 1122–1131. PMLR (09–15 Jun 2019), https://proceedings.mlr.press/v97/chen19m.html

    work page 2019

  11. [11]

    In: Wallach, H.M., Larochelle, H., Beygelzimer, A., d’Alché-Buc, F., Fox, E.B., Garnett, R

    Chen, H., Zhang, H., Si, S., Li, Y., Boning, D.S., Hsieh, C.: Robust- ness verification of tree-based models. In: Wallach, H.M., Larochelle, H., Beygelzimer, A., d’Alché-Buc, F., Fox, E.B., Garnett, R. (eds.) Ad- vances in Neural Information Processing Systems 32: Annual Confer- ence on Neural Information Processing Systems 2019, NeurIPS 2019, December 8-...

    work page 2019

  12. [12]

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

    Chen, T., Guestrin, C.: XGBoost: A scalable tree boosting system. In: Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. pp. 785–794. KDD ’16, ACM, New York, NY, USA (2016). https://doi.org/10.1145/2939672.2939785, http://doi.acm.org/10.1145/2939672.2939785

    work page doi:10.1145/2939672.2939785 2016

  13. [13]

    In: Proceedings of the Thirty-Third International Joint Conference on Artifi- cial Intelligence, IJCAI 2024, Jeju, South Korea, August 3-9, 2024

    de Colnet, A., Szeider, S., Zhang, T.: Compilation and fast model counting beyond CNF. In: Proceedings of the Thirty-Third International Joint Conference on Artifi- cial Intelligence, IJCAI 2024, Jeju, South Korea, August 3-9, 2024. pp. 3315–3323. ijcai.org (2024), https://www.ijcai.org/proceedings/2024/367

    work page 2024

  14. [14]

    https://xgboost.readthedocs.io, accessed: 2026

    Developers, X.: Xgboost documentation. https://xgboost.readthedocs.io, accessed: 2026

    work page 2026

  15. [15]

    Proceedings of the AAAI Conference on Artificial Intelligence 38(19), 21019–21028 (Mar 2024)

    Devos, L., Cascioli, L., Davis, J.: Robustness verification of multi-class tree ensembles. Proceedings of the AAAI Conference on Artificial Intelligence 38(19), 21019–21028 (Mar 2024). https://doi.org/10.1609/aaai.v38i19.30093, https://ojs.aaai.org/index.php/AAAI/article/view/30093

    work page doi:10.1609/aaai.v38i19.30093 2024

  16. [16]

    In: Meila, M., Zhang, T

    Devos, L., Meert, W., Davis, J.: Versatile verification of tree ensembles. In: Meila, M., Zhang, T. (eds.) Proceedings of the 38th International Conference on Machine Learning. Proceedings of Machine Learning Research, vol. 139, pp. 2654–2664. PMLR (18–24 Jul 2021), https://proceedings.mlr.press/v139/devos21a.html

    work page 2021

  17. [17]

    In: Ogata, K., Méry, D., Sun, M., Liu, S

    Downing, M., Eiers, W., DeLong, E., Lodha, A., Burns, B.O., Kadron, I.B., Bultan, T.: Quantitative symbolic robustness verification for quantized neural networks. In: Ogata, K., Méry, D., Sun, M., Liu, S. (eds.) Formal Methods and Software Engineering - 25th International Conference on Formal Engineer- ing Methods, ICFEM 2024, Hiroshima, Japan, December 2...

    work page doi:10.1007/978-981-96-0617-7 2024

  18. [18]

    Dudek, J.M., Phan, V., Vardi, M.Y.: ADDMC: weighted model count- ing with algebraic decision diagrams. In: The Thirty-Fourth AAAI Con- ference on Artificial Intelligence, AAAI 2020, The Thirty-Second Inno- vative Applications of Artificial Intelligence Conference, IAAI 2020, The Tenth AAAI Symposium on Educational Advances in Artificial Intelli- gence, EA...

    work page doi:10.1609/aaai.v34i02.5505 2020

  19. [19]

    In: Proceedings of the 3rd innovations in theoretical computer science con- ference

    Dwork, C., Hardt, M., Pitassi, T., Reingold, O., Zemel, R.: Fairness through aware- ness. In: Proceedings of the 3rd innovations in theoretical computer science con- ference. pp. 214–226 (2012)

    work page 2012

  20. [20]

    In: Proceedings of the 2017 11th Joint Meeting on Foundations of Soft- ware Engineering

    Galhotra, S., Brun, Y., Meliou, A.: Fairness testing: testing software for discrim- ination. In: Proceedings of the 2017 11th Joint Meeting on Foundations of Soft- ware Engineering. p. 498–510. ESEC/FSE 2017, Association for Computing Ma- chinery, New York, NY, USA (2017). https://doi.org/10.1145/3106237.3106277, https://doi.org/10.1145/3106237.3106277

    work page doi:10.1145/3106237.3106277 2017

  21. [21]

    Computers in Biology and Medicine128, 104089 (2021)

    Ghiasi, M.M., Zendehboudi, S.: Application of decision tree- based ensemble learning in the classification of breast can- cer. Computers in Biology and Medicine128, 104089 (2021). https://doi.org/https://doi.org/10.1016/j.compbiomed.2020.104089, https://www.sciencedirect.com/science/article/pii/S0010482520304200

    work page doi:10.1016/j.compbiomed.2020.104089 2021

  22. [22]

    Ghosh, B., Basu, D., Meel, K.S.: Algorithmic fairness verification with graphi- cal models. In: Thirty-Sixth AAAI Conference on Artificial Intelligence, AAAI Quantifying Sensitivity for Tree Ensembles 23 2022, Thirty-Fourth Conference on Innovative Applications of Artificial Intelli- gence, IAAI 2022, The Twelveth Symposium on Educational Advances in Ar- ...

    work page doi:10.1609/aaai.v36i9.21187 2022

  23. [23]

    In: Antonie, L., Pei, J., Yu, X., Chierichetti, F., Lauw, H.W., Sun, Y., Parthasarathy, S

    Gupta, A., Henzinger, T.A., Kueffner, K., Mallik, K., Pape, D.: Mon- itoring robustness and individual fairness. In: Antonie, L., Pei, J., Yu, X., Chierichetti, F., Lauw, H.W., Sun, Y., Parthasarathy, S. (eds.) Pro- ceedings of the 31st ACM SIGKDD Conference on Knowledge Discov- ery and Data Mining, V.2, KDD 2025, Toronto ON, Canada, August 3- 7, 2025. pp...

    work page doi:10.1145/3711896.3737054 2025

  24. [24]

    Ignatiev, A., Izza, Y., Stuckey, P.J., Marques-Silva, J.: Using maxsat for efficient explanations of tree ensembles. In: Thirty-Sixth AAAI Conference on Artificial In- telligence, AAAI 2022, Thirty-Fourth Conference on Innovative Applications of Ar- tificial Intelligence, IAAI 2022, The Twelveth Symposium on Educational Advances in Artificial Intelligence...

    work page doi:10.1609/aaai.v36i4.20292 2022

  25. [25]

    In: Proceedings of the Thirty-Fourth International Joint Conference on Artificial In- telligence, IJCAI 2025, Montreal, Canada, August 16-22, 2025

    Izza, Y., Ignatiev, A., Rubin, S., Marques-Silva, J., Stuckey, P.J.: Most general explanations of tree ensembles. In: Proceedings of the Thirty-Fourth International Joint Conference on Artificial In- telligence, IJCAI 2025, Montreal, Canada, August 16-22, 2025. pp. 5463–5471. ijcai.org (2025). https://doi.org/10.24963/IJCAI.2025/608, https://doi.org/10.24...

    work page doi:10.24963/ijcai.2025/608 2025

  26. [26]

    In: Adams, R.P., Gogate, V

    John, P.G., Vijaykeerthy, D., Saha, D.: Verifying individual fairness in machine learning models. In: Adams, R.P., Gogate, V. (eds.) Proceedings of the Thirty- Sixth Conference on Uncertainty in Artificial Intelligence, UAI 2020, virtual online, August 3-6, 2020. Proceedings of Machine Learning Research, vol. 124, pp. 749–

    work page 2020

  27. [27]

    AUAI Press (2020), http://proceedings.mlr.press/v124/george-john20a.html

    work page 2020

  28. [28]

    In: Balcan, M., Weinberger, K.Q

    Kantchelian,A.,Tygar,J.D.,Joseph,A.D.:Evasionandhardeningoftreeensemble classifiers. In: Balcan, M., Weinberger, K.Q. (eds.) Proceedings of the 33nd Inter- national Conference on Machine Learning, ICML 2016, New York City, NY, USA, June 19-24, 2016. JMLR Workshop and Conference Proceedings, vol. 48, pp. 2387–

    work page 2016

  29. [29]

    JMLR.org (2016), http://proceedings.mlr.press/v48/kantchelian16.html

    work page 2016

  30. [30]

    Advances in neural information processing systems30, 3146–3154 (2017)

    Ke, G., Meng, Q., Finley, T., Wang, T., Chen, W., Ma, W., Ye, Q., Liu, T.Y.: Lightgbm: A highly efficient gradient boosting decision tree. Advances in neural information processing systems30, 3146–3154 (2017)

    work page 2017

  31. [31]

    In: 47th IEEE/ACM International Conference on Software Engineering, ICSE 2025, Ottawa, ON, Canada, April 26 - May 6,

    Kim, B.H., Wang, J., Wang, C.: Fairquant: Certifying and quantifying fair- ness of deep neural networks. In: 47th IEEE/ACM International Conference on Software Engineering, ICSE 2025, Ottawa, ON, Canada, April 26 - May 6,

    work page 2025

  32. [32]

    pp. 527–539. IEEE (2025). https://doi.org/10.1109/ICSE55347.2025.00016, https://doi.org/10.1109/ICSE55347.2025.00016

    work page doi:10.1109/icse55347.2025.00016 2025

  33. [33]

    In: Chakraborty, S., Jiang, J.R

    Lagniez, J., Marquis, P., Biere, A.: Dynamic blocked clause elimina- tion for projected model counting. In: Chakraborty, S., Jiang, J.R. (eds.) 27th International Conference on Theory and Applications of Satisfiability Testing, SAT 2024, Pune, India, August 21-24, 2024. LIPIcs, vol. 305, pp. 21:1–21:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2...

    work page doi:10.4230/lipics.sat.2024.21 2024

  34. [34]

    IOP Conference Series: Materials Science and Engineering 1022(1), 012042 (jan 2021)

    Madaan, M., Kumar, A., Keshri, C., Jain, R., Nagrath, P.: Loan de- fault prediction using decision trees and random forest: A compara- tive study. IOP Conference Series: Materials Science and Engineering 1022(1), 012042 (jan 2021). https://doi.org/10.1088/1757-899X/1022/1/012042, https://dx.doi.org/10.1088/1757-899X/1022/1/012042

    work page doi:10.1088/1757-899x/1022/1/012042 2021

  35. [35]

    In: Libkin, L., Pichler, R., Guagliardo, P

    Meel, K.S., Vinodchandran, N.V., Chakraborty, S.: Estimating the size of union of sets in streaming models. In: Libkin, L., Pichler, R., Guagliardo, P. (eds.) PODS’21: Proceedings of the 40th ACM SIGMOD-SIGACT-SIGAI Sym- posium on Principles of Database Systems, Virtual Event, China, June 20- 25, 2021. pp. 126–137. ACM (2021). https://doi.org/10.1145/3452...

    work page doi:10.1145/3452021.3458333 2021

  36. [36]

    and Jordan, Michael I

    Ng, A.Y.: Feature selection, l1 vs. l2 regularization, and rotational in- variance. In: Proceedings of the Twenty-First International Conference on Machine Learning. p. 78. ICML ’04, Association for Computing Machin- ery, New York, NY, USA (2004). https://doi.org/10.1145/1015330.1015435, https://doi.org/10.1145/1015330.1015435

    work page doi:10.1145/1015330.1015435 2004

  37. [37]

    Environmental Modelling and Software174, 105971 (2024)

    Niazkar, M., Menapace, A., Brentan, B., Piraei, R., Jimenez, D., Dhawan, P., Righetti, M.: Applications of xgboost in wa- ter resources engineering: A systematic literature review (dec 2018–may 2023). Environmental Modelling and Software174, 105971 (2024). https://doi.org/https://doi.org/10.1016/j.envsoft.2024.105971, https://www.sciencedirect.com/science...

    work page doi:10.1016/j.envsoft.2024.105971 2018

  38. [38]

    Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learninginpython.Journalofmachinelearningresearch12(Oct),2825–2830(2011)

    work page 2011

  39. [39]

    Proceedings of the AAAI Conference on Artificial Intelli- gence34(04), 5478–5486 (Apr 2020)

    Ranzato, F., Zanella, M.: Abstract interpretation of decision tree ensem- ble classifiers. Proceedings of the AAAI Conference on Artificial Intelli- gence34(04), 5478–5486 (Apr 2020). https://doi.org/10.1609/aaai.v34i04.5998, https://ojs.aaai.org/index.php/AAAI/article/view/5998

    work page doi:10.1609/aaai.v34i04.5998 2020

  40. [40]

    In: Proceedings of International Joint Conference on Artificial Intelligence (IJCAI) (2019)

    Sharma, S., Roy, S., Soos, M., Meel, K.S.: Ganak: A scalable probabilistic ex- act model counter. In: Proceedings of International Joint Conference on Artificial Intelligence (IJCAI) (2019)

    work page 2019

  41. [41]

    Procedia Computer Science 215, 652–661 (2022)

    Shrestha, S.M., Shakya, A.: A customer churn prediction model using xg- boost for the telecommunication industry in nepal. Procedia Computer Science 215, 652–661 (2022). https://doi.org/https://doi.org/10.1016/j.procs.2022.12.067, https://www.sciencedirect.com/science/article/pii/S187705092202138X, 4th In- ternational Conference on Innovative Data Communi...

    work page doi:10.1016/j.procs.2022.12.067 2022

  42. [42]

    In: Proceedings of the Thirty-Second International Joint Conference on Artificial Intelli- gence, IJCAI 2023, 19th-25th August 2023, Macao, SAR, China

    Soos, M., Aggarwal, D., Chakraborty, S., Meel, K.S., Obremski, M.: Engineering an efficient approximate dnf-counter. In: Proceedings of the Thirty-Second International Joint Conference on Artificial Intelli- gence, IJCAI 2023, 19th-25th August 2023, Macao, SAR, China. pp. 2031–2038. ijcai.org (2023). https://doi.org/10.24963/IJCAI.2023/226, https://doi.or...

    work page doi:10.24963/ijcai.2023/226 2023

  43. [43]

    Usage of Not

    Soos, M., Gocht, S., Meel, K.S.: Tinted, detached, and lazy CNF-XOR solving and its applications to counting and sampling. In: Lahiri, S.K., Wang, C. (eds.) Computer Aided Verification - 32nd International Conference, CAV 2020, Los An- geles, CA, USA, July 21-24, 2020, Proceedings, Part I. Lecture Notes in Computer Science, vol. 12224, pp. 463–484. Spring...

    work page doi:10.1007/978-3- 2020

  44. [44]

    In: Proceedings of the International Conference on Computer Aided Verification (CAV) (2025)

    Soos, M., Meel, K.S.: Engineering an efficient probabilistic exact model counter. In: Proceedings of the International Conference on Computer Aided Verification (CAV) (2025)

    work page 2025

  45. [45]

    In: The Fourteenth International Conference on Learning Representations, ICLR 2026, Rio de Janeiro, April 23-27, 2026

    Varshney, N., Gupta, A., Ahmad, A., Tayal, T.V., Akshay, S.: Data- aware and scalable sensitivity analysis for decision tree ensembles. In: The Fourteenth International Conference on Learning Representations, ICLR 2026, Rio de Janeiro, April 23-27, 2026. OpenReview.net (2026), https://openreview.net/pdf?id=q8KqAvdfZK

    work page 2026

  46. [46]

    In: Proceedings of the 37th International Conference on Machine Learning, ICML 2020, 13-18 July 2020, Virtual Event

    Wang, Y., Zhang, H., Chen, H., Boning, D.S., Hsieh, C.: On lp-norm robustness of ensemble decision stumps and trees. In: Proceedings of the 37th International Conference on Machine Learning, ICML 2020, 13-18 July 2020, Virtual Event. Pro- ceedings of Machine Learning Research, vol. 119, pp. 10104–10114. PMLR (2020), http://proceedings.mlr.press/v119/wang20aa.html

    work page 2020

  47. [47]

    after processing the firstisubproblems, the algorithm has sampling probabilitypj

    Yang, J., Meel, K.S.: Rounding meets approximate model counting. In: Enea, C., Lal, A. (eds.) Computer Aided Verification - 35th International Con- ference, CAV 2023, Paris, France, July 17-22, 2023, Proceedings, Part II. Lecture Notes in Computer Science, vol. 13965, pp. 132–162. Springer (2023). https://doi.org/10.1007/978-3-031-37703-7\_7, https://doi....

    work page doi:10.1007/978-3-031-37703-7 2023