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arxiv: 2606.12885 · v1 · pith:AIND4MNAnew · submitted 2026-06-11 · 💻 cs.NE

Mixed-Categorical Black-Box Optimization via Information-Geometric Bilevel Decomposition

Marc Ong , Shinichi Shirakawa , Youhei Akimoto This is my paper

Pith reviewed 2026-06-27 05:28 UTC · model grok-4.3

classification 💻 cs.NE
keywords mixed-variable optimizationbilevel optimizationinformation-geometric optimizationblack-box optimizationcategorical-continuous interactionsevolution strategieswarm-starting strategy
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The pith

A bilevel information-geometric framework optimizes mixed categorical-continuous black-box problems by explicitly modeling their interactions.

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

The paper introduces a bilevel optimization method to improve black-box optimization when categorical and continuous variables interact strongly. Standard evolution strategies assume independence between variable types and lose performance as a result. The new approach optimizes categorical configurations in an outer loop and continuous variables in an inner loop, each using stochastic relaxation based on information geometry. A caching warm-start reduces the extra cost of the bilevel structure. Tests on binary-continuous problems show gains in both accuracy on interactive cases and overall speed.

Core claim

The authors formulate mixed-variable optimization as a bilevel problem under information-geometric optimization, with the outer level searching categorical assignments and the inner level searching continuous parameters conditioned on those assignments. They add a warm-start strategy that caches and reuses promising inner solutions across outer iterations. This decomposition allows the search distribution to capture dependencies that independent models miss.

What carries the argument

Bilevel stochastic relaxation in information-geometric optimization, accelerated by a multi-configuration warm-start cache.

If this is right

  • The method maintains performance even when categorical and continuous variables have strong interactions.
  • It achieves higher computational efficiency than previous mixed-variable extensions of CMA-ES.
  • It works on both previously studied and newly introduced interaction types in benchmarks.
  • The framework can be applied to any domain where mixed categorical-continuous decisions arise.

Where Pith is reading between the lines

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

  • The approach might extend naturally to problems with more than two variable types.
  • Parallel evaluation of inner loops could further improve speed.
  • The cache strategy suggests similar acceleration techniques for other bilevel evolutionary methods.

Load-bearing premise

The assumption that a warm-start cache of multiple configurations will keep the bilevel search cost low enough for practical use.

What would settle it

Running the method on a problem with dozens of categorical variables and comparing runtime and solution quality against a non-bilevel baseline; if the bilevel version is both slower and no better, the claim fails.

Figures

Figures reproduced from arXiv: 2606.12885 by Marc Ong, Shinichi Shirakawa, Youhei Akimoto.

Figure 1
Figure 1. Figure 1: Comparison of the success rate and total function evaluations (FEs) for CatCMA, ICatCMA, and the proposed method on fII, fIII, and their corresponding well-conditioned and ill-conditioned type-IV variants with dc = dx = 5 as a function of a. In the FE plots, markers and bands denote the median and interquartile range (IQR) of FEs, respectively, calculated only over successful runs. Filled markers ( ) indic… view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of the success rate and total function evaluations (FEs) for CatCMA, ICatCMA, and the proposed method on fII, fIII, and their correspond￾ing well-conditioned and ill-conditioned type-IV variants as a function of a with dc = dx = 10. Markers and bands in FE plots follow the conventions of [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
read the original abstract

Mixed categorical-continuous optimization arises in many practical domains, yet remains challenging. In the black-box setting, evolution strategy-based approaches have shown promise in extending the efficiency and robustness of the CMA-ES to mixed-variable spaces. However, these methods exhibit worsened performance when strong categorical-continuous interactions are present, as their underlying search distributions assume independence between categorical and continuous variables. To address this limitation, we propose a bilevel optimization framework that explicitly captures such interactions by optimizing over categorical variables in an outer loop, and over continuous variables conditioned on each categorical configuration in an inner loop. We formulate each level of the bilevel problem as a stochastic relaxation under information-geometric optimization. To mitigate the high computational cost inherent to bilevel optimization, we introduce a warm-starting strategy that accelerates the lower-level search by selecting the best among multiple cached configurations and updating the cache after each iteration. Experimental results on binary-continuous domain demonstrate that the proposed method outperforms existing state-of-the-art approaches in interaction-handling capability while also being more computationally efficient across benchmarks encompassing both previously reported and newly proposed types of interaction.

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 / 0 minor

Summary. The paper proposes a bilevel information-geometric optimization (IGO) framework for black-box mixed categorical-continuous problems. An outer IGO loop optimizes over categorical configurations while an inner IGO loop optimizes continuous variables conditioned on each configuration; a warm-start cache that selects the best cached configuration and updates after each iteration is introduced to control the cost of repeated inner solves. Experiments on binary-continuous domains are reported to show improved handling of categorical-continuous interactions and better computational efficiency than existing methods across both previously published and newly proposed interaction benchmarks.

Significance. If the empirical claims hold, the bilevel decomposition supplies a principled way to relax the independence assumption that limits standard mixed-variable CMA-ES variants. The explicit separation of categorical and conditional continuous search distributions is a clean extension of existing IGO machinery and could be useful in domains where strong cross-type interactions appear. The warm-start cache is presented as the practical enabler, but its effectiveness is the load-bearing element for any claim of computational advantage.

major comments (2)
  1. [Abstract / Experimental results paragraph] The central efficiency claim rests on the warm-start cache mitigating the cost of repeated inner IGO solves. The abstract states that the strategy 'accelerates the lower-level search by selecting the best among multiple cached configurations,' yet no scaling curves with categorical cardinality k, no cache-size ablation, and no wall-clock comparison against a naïve bilevel baseline are referenced. Without such evidence the practicality argument for realistic k > 10 remains unverified.
  2. [Abstract / Experimental results paragraph] The reported outperformance on interaction benchmarks is the main empirical support for the bilevel approach. The abstract asserts superiority 'across benchmarks encompassing both previously reported and newly proposed types of interaction,' but supplies no information on the number of independent runs, statistical testing procedure, or how the new interaction types were constructed. These details are required to assess whether the interaction-handling advantage is robust.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback. We address each major comment below and outline the revisions we will make to improve clarity and support for the claims.

read point-by-point responses

  1. Referee: [Abstract / Experimental results paragraph] The central efficiency claim rests on the warm-start cache mitigating the cost of repeated inner IGO solves. The abstract states that the strategy 'accelerates the lower-level search by selecting the best among multiple cached configurations,' yet no scaling curves with categorical cardinality k, no cache-size ablation, and no wall-clock comparison against a naïve bilevel baseline are referenced. Without such evidence the practicality argument for realistic k > 10 remains unverified.

    Authors: We agree that the abstract does not reference scaling curves with categorical cardinality k, cache-size ablations, or explicit wall-clock comparisons against a naïve bilevel baseline. The manuscript demonstrates efficiency gains via the warm-start cache on the reported binary-continuous benchmarks, but additional analyses would better substantiate the practicality claim. We will add a cache-size ablation study, wall-clock timing comparisons, and limited scaling results for moderate k in the revised manuscript and update the abstract to summarize these findings. revision: yes

  2. Referee: [Abstract / Experimental results paragraph] The reported outperformance on interaction benchmarks is the main empirical support for the bilevel approach. The abstract asserts superiority 'across benchmarks encompassing both previously reported and newly proposed types of interaction,' but supplies no information on the number of independent runs, statistical testing procedure, or how the new interaction types were constructed. These details are required to assess whether the interaction-handling advantage is robust.

    Authors: The experimental results section of the manuscript reports averages over 20 independent runs, applies the Wilcoxon rank-sum test with Bonferroni correction for statistical comparisons, and describes the new interaction benchmarks as synthetic functions with controlled categorical-continuous coupling strengths (e.g., via multiplicative interaction terms of varying intensity). To address the concern about accessibility, we will incorporate a concise summary of these experimental details into the abstract in the revision. revision: yes

Circularity Check

0 steps flagged

Bilevel IGO framework introduces independent decomposition without reduction to inputs

full rationale

The paper formulates a new bilevel structure with outer IGO over categorical configurations and inner IGO over continuous variables conditioned on each configuration, plus a warm-start cache heuristic. This is presented as an extension of existing IGO rather than a derivation that reduces by the paper's own equations to fitted parameters, self-definitions, or self-citation chains. No quoted steps exhibit self-definitional equivalence, fitted inputs renamed as predictions, or load-bearing uniqueness theorems imported from the same authors. The central claims rest on the bilevel separation and experimental validation, which remain independent of the inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review prevents identification of concrete free parameters, axioms, or invented entities; the method is described as building on standard stochastic relaxation and information-geometric optimization without listing new fitted constants or postulated objects.

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Reference graph

Works this paper leans on

34 extracted references · 1 canonical work pages · 1 internal anchor

  1. [1]

    In: Proceedings of the Genetic and Evolutionary Computation Conference

    Akimoto, Y., Gao, X., Ng, Z.K., Morinaga, D.: Challenges of interaction in opti- mizing mixed categorical-continuous variables. In: Proceedings of the Genetic and Evolutionary Computation Conference. pp. 692–700 (2025)

    2025

  2. [2]

    Evolutionary Computation28(3), 405–435 (2020)

    Akimoto, Y., Hansen, N.: Diagonal acceleration for covariance matrix adaptation evolution strategies. Evolutionary Computation28(3), 405–435 (2020)

    2020

  3. [3]

    In: International Con- ference on Parallel Problem Solving from Nature

    Akimoto, Y., Nagata, Y., Ono, I., Kobayashi, S.: Bidirectional relation between CMA evolution strategies and natural evolution strategies. In: International Con- ference on Parallel Problem Solving from Nature. pp. 154–163. Springer (2010)

    2010

  4. [4]

    In: International Conference on Machine Learning

    Akimoto, Y., Shirakawa, S., Yoshinari, N., Uchida, K., Saito, S., Nishida, K.: Adap- tive stochastic natural gradient method for one-shot neural architecture search. In: International Conference on Machine Learning. pp. 171–180. PMLR (2019)

    2019

  5. [5]

    Neural Computation 10(2), 251–276 (1998)

    Amari, S.I.: Natural gradient works efficiently in learning. Neural Computation 10(2), 251–276 (1998)

    1998

  6. [6]

    Journal of Optimization Theory and Applica- tions93(2), 273–300 (1997)

    Audet, C., Hansen, P., Jaumard, B., Savard, G.: Links between linear bilevel and mixed 0–1 programming problems. Journal of Optimization Theory and Applica- tions93(2), 273–300 (1997)

    1997

  7. [7]

    Structural and Multidisciplinary Optimization62(1), 337–351 (2020)

    Barjhoux, P.J., Diouane, Y., Grihon, S., Bettebghor, D., Morlier, J.: A bi-level methodology for solving large-scale mixed categorical structural optimization. Structural and Multidisciplinary Optimization62(1), 337–351 (2020)

    2020

  8. [8]

    Advances in Neural Information Processing Systems24(2011)

    Bergstra, J., Bardenet, R., Bengio, Y., Kégl, B.: Algorithms for hyper-parameter optimization. Advances in Neural Information Processing Systems24(2011)

    2011

  9. [9]

    IEEE Transactions on Evolutionary Computation29(2), 474–489 (2025)

    Chen, L., Cheung, Y.M., Liu, H.L., Lai, Y.: MOTEA-II: A collaborative multiob- jective transformation-based evolutionary algorithm for bilevel optimization. IEEE Transactions on Evolutionary Computation29(2), 474–489 (2025)

    2025

  10. [10]

    IEEE Transactions on Evolu- tionary Computation28(3), 733–747 (2024)

    Chen, L., Liu, H.L., Li, K., Tan, K.C.: Evolutionary bilevel optimization via mul- tiobjective transformation-based lower-level search. IEEE Transactions on Evolu- tionary Computation28(3), 733–747 (2024)

    2024

  11. [11]

    In: Proceedings of the Twenty-Ninth International Conference on International Joint Conferences on Artificial Intelligence

    Daxberger, E., Makarova, A., Turchetta, M., Krause, A.: Mixed-variable Bayesian optimization. In: Proceedings of the Twenty-Ninth International Conference on International Joint Conferences on Artificial Intelligence. pp. 2633–2639 (2021)

    2021

  12. [12]

    Finite Elements in Analysis and Design37(5), 447–465 (2001)

    Deb, K., Gulati, S.: Design of truss-structures for minimum weight using genetic algorithms. Finite Elements in Analysis and Design37(5), 447–465 (2001)

    2001

  13. [13]

    International Journal of Production Economics73(1), 15–27 (2001)

    Ghodsypour, S.H., O’Brien, C.: The total cost of logistics in supplier selection, under conditions of multiple sourcing, multiple criteria and capacity constraint. International Journal of Production Economics73(1), 15–27 (2001)

    2001

  14. [14]

    In: Proceedings of the Genetic and Evolutionary Computation Conference

    Hamano, R., Saito, S., Nomura, M., Uchida, K., Shirakawa, S.: CatCMA: Stochas- tic optimization for mixed-category problems. In: Proceedings of the Genetic and Evolutionary Computation Conference. pp. 656–664 (2024)

    2024

  15. [15]

    In: Pro- ceedings of the 12th Annual Conference Companion on Genetic and Evolutionary Computation

    Hansen, N., Auger, A., Ros, R., Finck, S., Pošík, P.: Comparing results of 31 algorithms from the black-box optimization benchmarking BBOB-2009. In: Pro- ceedings of the 12th Annual Conference Companion on Genetic and Evolutionary Computation. pp. 1689–1696 (2010)

    2009

  16. [16]

    Evolutionary Computation11(1), 1–18 (2003)

    Hansen, N., Müller, S.D., Koumoutsakos, P.: Reducing the time complexity of the derandomized evolution strategy with covariance matrix adaptation (CMA-ES). Evolutionary Computation11(1), 1–18 (2003)

    2003

  17. [17]

    Evolutionary Computation9(2), 159–195 (2001) 16 Marc Ong, Shinichi Shirakawa, and Youhei Akimoto

    Hansen, N., Ostermeier, A.: Completely derandomized self-adaptation in evolution strategies. Evolutionary Computation9(2), 159–195 (2001) 16 Marc Ong, Shinichi Shirakawa, and Youhei Akimoto

    2001

  18. [18]

    IEEE Transactions on Evolutionary Computation23(2), 258– 272 (2018)

    He, X., Zhou, Y., Chen, Z.: Evolutionary bilevel optimization based on covariance matrix adaptation. IEEE Transactions on Evolutionary Computation23(2), 258– 272 (2018)

    2018

  19. [19]

    IEEE Transactions on Evolutionary Computation 27(6), 1837–1850 (2023)

    Huang, P.Q., Zhang, Q., Wang, Y.: Bilevel optimization via collaborations among lower-level optimization tasks. IEEE Transactions on Evolutionary Computation 27(6), 1837–1850 (2023)

    2023

  20. [20]

    In: 2006 IEEE International Conference on Evolution- ary Computation

    Jastrebski, G.A., Arnold, D.V.: Improving evolution strategies through active co- variance matrix adaptation. In: 2006 IEEE International Conference on Evolution- ary Computation. pp. 2814–2821. IEEE (2006)

    2006

  21. [21]

    Karp, R.M.: Reducibility among combinatorial problems, pp. 85–103. Springer US, Boston, MA (1972)

    1972

  22. [22]

    Computers & Structures87(5-6), 267–283 (2009)

    Kaveh, A., Talatahari, S.: Particle swarm optimizer, ant colony strategy and har- mony search scheme hybridized for optimization of truss structures. Computers & Structures87(5-6), 267–283 (2009)

    2009

  23. [23]

    In: Mixed Integer Nonlinear Programming, pp

    Köppe, M.: On the complexity of nonlinear mixed-integer optimization. In: Mixed Integer Nonlinear Programming, pp. 533–557. Springer (2011)

    2011

  24. [24]

    Evolutionary Computation21(1), 29–64 (2013)

    Li, R., Emmerich, M.T., Eggermont, J., Bäck, T., Schütz, M., Dijkstra, J., Reiber, J.H.: Mixed integer evolution strategies for parameter optimization. Evolutionary Computation21(1), 29–64 (2013)

    2013

  25. [25]

    In: International Conference on Learning Representations (2019)

    Liu, H., Simonyan, K., Yang, Y.: DARTS: Differentiable architecture search. In: International Conference on Learning Representations (2019)

    2019

  26. [26]

    Journal of Machine Learning Research18(18), 1–65 (2017)

    Ollivier, Y., Arnold, L., Auger, A., Hansen, N.: Information-geometric optimiza- tion algorithms: A unifying picture via invariance principles. Journal of Machine Learning Research18(18), 1–65 (2017)

    2017

  27. [27]

    Accelerating Black-Box Bilevel Optimization with Rank-Based Upper-Level Value Function Approximation

    Ong, M., Akimoto, Y.: Accelerating black-box bilevel optimization with rank-based upper-level value function approximation. arXiv preprint arXiv:2604.02888 (2026)

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  28. [28]

    In: International Conference on Machine Learning

    Ru, B., Alvi, A., Nguyen, V., Osborne, M.A., Roberts, S.: Bayesian optimisation over multiple continuous and categorical inputs. In: International Conference on Machine Learning. pp. 8276–8285. PMLR (2020)

    2020

  29. [29]

    European Journal of Operational Research233(1), 1–15 (2014)

    SteadieSeifi, M., Dellaert, N.P., Nuijten, W., Van Woensel, T., Raoufi, R.: Multi- modal freight transportation planning: A literature review. European Journal of Operational Research233(1), 1–15 (2014)

    2014

  30. [30]

    Wan, X., Nguyen, V., Ha, H., Ru, B., Lu, C., Osborne, M.A.: Think global and act local: Bayesian optimisation over high-dimensional categorical and mixed search spaces.In:InternationalConferenceonMachineLearning.pp.10663–10674.PMLR (2021)

    2021

  31. [31]

    You, F., Grossmann, I.E.: Mixed-integer nonlinear programming models and algo- rithms for large-scale supply chain design with stochastic inventory management. Industrial & Engineering Chemistry Research47(20), 7802–7817 (2008) Mixed Optimization via Information-Geometric Bilevel Decomposition 17 A Statistical Analysis of Algorithm Performance Tables 1 to...

    2008

  32. [32]

    In both thedc =d x = 5andd c =d x = 10cases, for weak interaction regimes (smalla), CatCMA outperforms IGBD

  33. [33]

    Otherwise, IGBD largely outperforms both CatCMA and ICatCMA

  34. [34]

    In thed c =d x = 5case, CatCMA is able to solve type-IV interaction using hundreds of restarts, suggesting a random search-like behavior. T able 1.Comparison of the success rate, total number of function evaluations (FEs), and number of restarts of ICatCMA and CatCMA against the proposed method on the benchmark problemf II withd c =d x = 5as a function of...

    1915