Document Open Access Logo

ETH-Hardness of Learning Monotone Circuits and Approximating Their Size

Authors Bruno Cavalar , Susanna F. de Rezende , Matthew Gray , Rahul Santhanam

PDF

File

Thumbnail PDF
PDF
LIPIcs.CCC.2026.40.pdf
  • Filesize: 0.96 MB
  • 25 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Circuit complexity
  • Theory of computation → Machine learning theory
  • Theory of computation → Proof complexity
Keywords
  • monotone circuit complexity
  • PAC learning
  • lifting theorems
  • Resolution
  • meta-complexity
  • hardness of approximation
  • minimum circuit size problem

Metrics

Abstract

We show the following hardness results for monotone learning and approximation of monotone circuit size:  
1) Under the Randomised Exponential-Time Hypothesis (rETH), it requires time n^{Ω(log n)} to PAC-learn monotone formulas with n input bits and size s(n) = n by monotone circuits of size n^{(log n)^{1-ε}}, for every ε > 0. 
2) Under the Randomised Exponential-Time Hypothesis (rETH), for any δ > 0, there is a polynomially bounded function m such that m^{1-δ}-multiplicatively approximating the minimum monotone circuit size of a monotone function consistent with a sequence of m(n) labelled examples {(x_i, b_i)} over n-bit inputs requires time m^{Ω(log(m))}.  Our results are shown by a novel application of lifting arguments in proof and communication complexity to hardness of monotone learning, by building on the seminal result of Atserias and Müller [Atserias and Müller, 2020] on hardness of automating Resolution proofs.

Cite As Get BibTex

Bruno Cavalar, Susanna F. de Rezende, Matthew Gray, and Rahul Santhanam. ETH-Hardness of Learning Monotone Circuits and Approximating Their Size. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 40:1-40:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.CCC.2026.40

Author Details

Bruno Cavalar
  • University of Oxford, UK
Susanna F. de Rezende
  • Lund University, Sweden
Matthew Gray
  • University of Oxford, UK
Rahul Santhanam
  • University of Oxford, UK

Funding

Rahul Santhanam and Bruno Cavalar acknowledge support from the EPSRC project EP/Z534158/1 on "Integrated Approach to Computational Complexity: Structure, Self-Reference and Lower Bounds". Susanna de Rezende received funding from the Knut and Alice Wallenberg Foundation grant KAW 2023.0116, ELLIIT, and the Swedish Research Council grant 2021-05104.

Acknowledgements

We thank Noel Arteche for helpful discussions, and the anonymous reviewers for their valuable comments and suggestions.

References

  1. Michael Alekhnovich, Mark Braverman, Vitaly Feldman, Adam R. Klivans, and Toniann Pitassi. The complexity of properly learning simple concept classes. J. Comput. Syst. Sci., 74(1):16-34, 2008. URL: https://doi.org/10.1016/J.JCSS.2007.04.011.
  2. Eric Allender, Lisa Hellerstein, Paul McCabe, Toniann Pitassi, and Michael E. Saks. Minimizing disjunctive normal form formulas and AC⁰ circuits given a truth table. SIAM J. Comput., 38(1):63-84, 2008. URL: https://doi.org/10.1137/060664537.
  3. Benny Applebaum, Boaz Barak, and David Xiao. On basing lower-bounds for learning on worst-case assumptions. In Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science FOCS '08, pages 211-220, 2008. URL: https://doi.org/10.1109/FOCS.2008.35.
  4. Noel Arteche, Albert Atserias, Susanna F. de Rezende, and Erfan Khaniki. The proof analysis problem. In Proceedings of the 66th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 2555-2576, 2025. URL: https://doi.org/10.1109/focs63196.2025.00133.
  5. Albert Atserias and Víctor Dalmau. A combinatorial characterization of resolution width. Journal of Computer and System Sciences, 74(3):323-334, 2008. Preliminary version in CCC '03. URL: https://doi.org/10.1016/J.JCSS.2007.06.025.
  6. Albert Atserias and Moritz Müller. Automating resolution is NP-hard. Journal of the ACM, 67(5), 2020. Preliminary version in FOCS '19. URL: https://doi.org/10.1145/3409472.
  7. Gaia Carenini and Susanna F. de Rezende. On the automatability of tree-like k-dnf resolution. In Proceedings of the 40th Computational Complexity Conference (CCC '25), volume 339 of LIPIcs, pages 14:1-14:21, 2025. URL: https://doi.org/10.4230/LIPIcs.CCC.2025.14.
  8. Marco L. Carmosino, Russell Impagliazzo, Valentine Kabanets, and Antonina Kolokolova. Learning algorithms from natural proofs. In Ran Raz, editor, Proceedings of the 31st Conference on Computational Complexity (CCC '16), volume 50 of LIPIcs, pages 10:1-10:24, 2016. URL: https://doi.org/10.4230/LIPIcs.CCC.2016.10.
  9. Dana Dachman-Soled, Homin K. Lee, Tal Malkin, Rocco A. Servedio, Andrew Wan, and Hoeteck Wee. Optimal cryptographic hardness of learning monotone functions. Theory Comput., 5(1):257-282, 2009. URL: https://doi.org/10.4086/TOC.2009.V005A013.
  10. Amit Daniely. Complexity theoretic limitations on learning halfspaces. In Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing (STOC 16), pages 105-117, 2016. URL: https://doi.org/10.1145/2897518.2897520.
  11. Amit Daniely and Shai Shalev-Shwartz. Complexity theoretic limitations on learning DNF’s. In Proceedings of the 29th Conference on Learning Theory (COLT '16), pages 815-830, 2016. URL: http://proceedings.mlr.press/v49/daniely16.html.
  12. Amit Daniely and Gal Vardi. From local pseudorandom generators to hardness of learning. In Proceedings of the 34th Conference on Learning Theory (COLT '21), pages 1358-1394. PMLR, 2021. URL: http://proceedings.mlr.press/v134/daniely21a.html.
  13. Susanna F. de Rezende. Automating tree-like resolution in time n^o(log n) is ETH-hard. In Proceedings of the XI Latin and American Algorithms, Graphs and Optimization Symposium, (LAGOS '21), volume 195, pages 152-162, 2021. URL: https://doi.org/10.1016/J.PROCS.2021.11.021.
  14. Susanna F. de Rezende, Mika Göös, Jakob Nordström, Toniann Pitassi, Robert Robere, and Dmitry Sokolov. Automating algebraic proof systems is NP-hard. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing (STOC '21), pages 209-222, 2021. URL: https://doi.org/10.1145/3406325.3451080.
  15. Susanna F. de Rezende and Marc Vinyals. Lifting with colourful sunflowers. In Proceedings of the 40th Computational Complexity Conference (CCC 25), pages 36:1-36:19, 2025. URL: https://doi.org/10.4230/LIPIcs.CCC.2025.36.
  16. Uriel Feige. Relations between average case complexity and approximation complexity. In Proceedings on 34th Annual ACM Symposium on Theory of Computing (STOC 02), pages 534-543, 2002. URL: https://doi.org/10.1145/509907.509985.
  17. Vitaly Feldman, Homin K. Lee, and Rocco A. Servedio. Lower bounds and hardness amplification for learning shallow monotone formulas. In Proceedings of the 24th Conference on Learning Theory (COLT '11), pages 273-292, 2011. URL: http://proceedings.mlr.press/v19/feldman11a/feldman11a.html.
  18. Ankit Garg, Mika Göös, Pritish Kamath, and Dmitry Sokolov. Monotone circuit lower bounds from resolution. Theory of Computing, 16(1):1-30, 2020. Preliminary version in STOC '18. URL: https://doi.org/10.4086/toc.2020.v016a013.
  19. Mika Göös, Pritish Kamath, Robert Robere, and Dmitry Sokolov. Adventures in monotone complexity and TFNP. In Proceedings of the 10th Innovations in Theoretical Computer Science Conference (ITCS), pages 38:1-38:19, 2019. URL: https://doi.org/10.4230/LIPIcs.ITCS.2019.38.
  20. Mika Göös, Sajin Koroth, Ian Mertz, and Toniann Pitassi. Automating cutting planes is NP-hard. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing (STOC 20), pages 68-77, 2020. URL: https://doi.org/10.1145/3357713.3384248.
  21. Mika Göös, Toniann Pitassi, and Thomas Watson. Deterministic communication vs. partition number. SIAM J. Comput., 47(6):2435-2450, 2018. Preliminary version in FOCS '15. URL: https://doi.org/10.1137/16M1059369.
  22. Shuichi Hirahara. NP-hardness of learning programs and partial MCSP. In Proceedings of the 63rd Annual IEEE Symposium on Foundations of Computer Science FOCS 22, pages 968-979, 2022. URL: https://doi.org/10.1109/FOCS54457.2022.00095.
  23. Shuichi Hirahara. Non-black-box worst-case to average-case reductions within NP. SIAM J. Comput., 52(6):S18-349, 2023. URL: https://doi.org/10.1137/19M124705X.
  24. Shuichi Hirahara, Igor C. Oliveira, and Rahul Santhanam. NP-hardness of minimum circuit size problem for OR-AND-MOD circuits. In Proceedings of the 33rd Computational Complexity Conference (CCC 18), pages 5:1-5:31, 2018. URL: https://doi.org/10.4230/LIPIcs.CCC.2018.5.
  25. Shlomo Hoory, Nathan Linial, and Avi Wigderson. Expander graphs and their applications. Bull. Amer. Math. Soc., 43(04):439-562, August 2006. URL: https://doi.org/10.1090/s0273-0979-06-01126-8.
  26. Yizhi Huang, Rahul Ilango, and Hanlin Ren. NP-hardness of approximating meta-complexity: A cryptographic approach. SIAM J. Comput., 54(4):819-886, 2025. URL: https://doi.org/10.1137/23M1608483.
  27. Rahul Ilango. Approaching MCSP from above and below: Hardness for a conditional variant and AC⁰[p]. In Proceedings of the 11th Innovations in Theoretical Computer Science Conference (ITCS '20), pages 34:1-34:26, 2020. URL: https://doi.org/10.4230/LIPIcs.ITCS.2020.34.
  28. Rahul Ilango. Constant depth formula and partial function versions of MCSP are hard. In Proceedings of the 61st Annual IEEE Symposium on Foundations of Computer Science (FOCS '20), pages 424-433, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00047.
  29. Rahul Ilango. The minimum formula size problem is (ETH) hard. In Proceedings of the 62nd IEEE Annual Symposium on Foundations of Computer Science (FOCS '21), pages 427-432, 2021. URL: https://doi.org/10.1109/FOCS52979.2021.00050.
  30. Rahul Ilango, Bruno Loff, and Igor C. Oliveira. NP-hardness of circuit minimization for multi-output functions. In Proceedings of the 35th Computational Complexity Conference (CCC '20), pages 22:1-22:36, 2020. URL: https://doi.org/10.4230/LIPIcs.CCC.2020.22.
  31. Russell Impagliazzo. A personal view of average-case complexity. In Proceedings of the 10th Annual IEEE Structure in Complexity Theory Conference (SCT 95), pages 134-147, 1995. URL: https://doi.org/10.1109/SCT.1995.514853.
  32. Valentine Kabanets and Jin-yi Cai. Circuit minimization problem. In Proceedings of the 32nd Annual ACM Symposium on Theory of Computing (STOC '00), pages 73-79, 2000. URL: https://doi.org/10.1145/335305.335314.
  33. Michael J. Kearns and Leslie G. Valiant. Cryptographic limitations on learning Boolean formulae and finite automata. J. ACM, 41(1):67-95, 1994. URL: https://doi.org/10.1145/174644.174647.
  34. Michael Kharitonov. Cryptographic hardness of distribution-specific learning. In Proceedings of the 25th Annual ACM Symposium on Theory of Computing (STOC '93), pages 372-381, 1993. URL: https://doi.org/10.1145/167088.167197.
  35. Jan Kraíček. Interpolation by a game. Mathematical Logic Quarterly, 44(4):450-458, 1998. URL: https://doi.org/10.1002/MALQ.19980440403.
  36. Jan Krajíček. Interpolation theorems, lower bounds for proof systems, and independence results for bounded arithmetic. The Journal of Symbolic Logic, 62(2):457-486, 1997. URL: https://doi.org/10.2307/2275541.
  37. Shachar Lovett, Raghu Meka, Ian Mertz, Toniann Pitassi, and Jiapeng Zhang. Lifting with sunflowers. In Proceedings of 13th Innovations in Theoretical Computer Science Conference (ITCS), pages 104:1-104:24, 2022. URL: https://doi.org/10.4230/LIPIcs.ITCS.2022.104.
  38. William Masek. Some NP-complete set covering problems. Unpublished, 1979. Google Scholar
  39. Noam Mazor and Rafael Pass. Gap MCSP is not (Levin) NP-complete in Obfustopia. In Proceedings of the 39th Computational Complexity Conference, (CCC '24), pages 36:1-36:21, 2024. URL: https://doi.org/10.4230/LIPIcs.CCC.2024.36.
  40. Pavel Pudlák. On reducibility and symmetry of disjoint NP pairs. Theoretical Computer Science, 295:323-339, 2003. URL: https://doi.org/10.1016/S0304-3975(02)00411-5.
  41. Ran Raz and Pierre McKenzie. Separation of the Monotone NC Hierarchy. Combinatorica, 19(3):403-435, 1999. Preliminary version in STOC 97. URL: https://doi.org/10.1007/S004930050062.
  42. Alexander A Razborov. Unprovability of lower bounds on circuit size in certain fragments of bounded arithmetic. Izvestiya: mathematics, 59(1):205, 1995. Google Scholar
  43. Alexander A. Razborov and Steven Rudich. Natural proofs. J. Comput. Syst. Sci., 55(1):24-35, 1997. URL: https://doi.org/10.1006/JCSS.1997.1494.
  44. Rahul Santhanam. Pseudorandomness and the minimum circuit size problem. In Thomas Vidick, editor, Proceedings of the 11th Innovations in Theoretical Computer Science Conference (ITCS '20), pages 68:1-68:26, 2020. URL: https://doi.org/10.4230/LIPIcs.ITCS.2020.68.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2026 Schloss Dagstuhl – LZI GmbH Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact