{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:33:36Z","timestamp":1760146416622,"version":"build-2065373602"},"reference-count":39,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2024,11,5]],"date-time":"2024-11-05T00:00:00Z","timestamp":1730764800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000001","name":"NSF","doi-asserted-by":"publisher","award":["CAREER 1651492","CCF-2100013","CNS-2209951","CNS-1822071","CNS-2317192","DE-SC-ERKJ422","R01-CA261457-01A1"],"award-info":[{"award-number":["CAREER 1651492","CCF-2100013","CNS-2209951","CNS-1822071","CNS-2317192","DE-SC-ERKJ422","R01-CA261457-01A1"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"name":"U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing","award":["CAREER 1651492","CCF-2100013","CNS-2209951","CNS-1822071","CNS-2317192","DE-SC-ERKJ422","R01-CA261457-01A1"],"award-info":[{"award-number":["CAREER 1651492","CCF-2100013","CNS-2209951","CNS-1822071","CNS-2317192","DE-SC-ERKJ422","R01-CA261457-01A1"]}]},{"name":"NIH","award":["CAREER 1651492","CCF-2100013","CNS-2209951","CNS-1822071","CNS-2317192","DE-SC-ERKJ422","R01-CA261457-01A1"],"award-info":[{"award-number":["CAREER 1651492","CCF-2100013","CNS-2209951","CNS-1822071","CNS-2317192","DE-SC-ERKJ422","R01-CA261457-01A1"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Causal graph discovery (CGD) is the process of estimating the underlying probabilistic graphical model that represents the joint distribution of features of a dataset. CGD algorithms are broadly classified into two categories: (i) constraint-based algorithms, where the outcome depends on conditional independence (CI) tests, and (ii) score-based algorithms, where the outcome depends on optimized score function. Because sensitive features of observational data are prone to privacy leakage, differential privacy (DP) has been adopted to ensure user privacy in CGD. Adding the same amount of noise in this sequential-type estimation process affects the predictive performance of algorithms. Initial CI tests in constraint-based algorithms and later iterations of the optimization process of score-based algorithms are crucial; thus, they need to be more accurate and less noisy. Based on this key observation, we present CURATE (CaUsal gRaph AdapTivE privacy), a DP-CGD framework with adaptive privacy budgeting. In contrast to existing DP-CGD algorithms with uniform privacy budgeting across all iterations, CURATE allows for adaptive privacy budgeting by minimizing error probability (constraint-based), maximizing iterations of the optimization problem (score-based) while keeping the cumulative leakage bounded. To validate our framework, we present a comprehensive set of experiments on several datasets and show that CURATE achieves higher utility compared to existing DP-CGD algorithms with less privacy leakage.<\/jats:p>","DOI":"10.3390\/e26110946","type":"journal-article","created":{"date-parts":[[2024,11,5]],"date-time":"2024-11-05T06:30:48Z","timestamp":1730788248000},"page":"946","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["CURATE: Scaling-Up Differentially Private Causal Graph Discovery"],"prefix":"10.3390","volume":"26","author":[{"ORCID":"https:\/\/orcid.org\/0009-0006-7506-8300","authenticated-orcid":false,"given":"Payel","family":"Bhattacharjee","sequence":"first","affiliation":[{"name":"Department of Electrical and Computer Engineering, University of Arizona, Tucson, AZ 85721, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6182-6098","authenticated-orcid":false,"given":"Ravi","family":"Tandon","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, University of Arizona, Tucson, AZ 85721, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,11,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Spirtes, P., Glymour, C., and Scheines, R. (1993). Causation, Prediction, and Search, Springer.","DOI":"10.1007\/978-1-4612-2748-9"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"523","DOI":"10.1126\/science.1105809","article-title":"Causal protein-signaling networks derived from multiparameter single-cell data","volume":"308","author":"Sachs","year":"2005","journal-title":"Science"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"707","DOI":"10.1016\/j.cell.2013.03.030","article-title":"Integrated systems approach identifies genetic nodes and networks in late-onset Alzheimer\u2019s disease","volume":"153","author":"Zhang","year":"2013","journal-title":"Cell"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1141","DOI":"10.1016\/j.tree.2021.08.008","article-title":"Causal assumptions and causal inference in ecological experiments","volume":"36","author":"Kimmel","year":"2021","journal-title":"Trends Ecol. Evol."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"878","DOI":"10.1111\/joes.12217","article-title":"Causal inference on education policies: A survey of empirical studies using PISA, TIMSS and PIRLS","volume":"32","author":"Cordero","year":"2018","journal-title":"J. Econ. Surv."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1561\/104.00000036","article-title":"Shock-based causal inference in corporate finance and accounting research","volume":"5","author":"Atanasov","year":"2016","journal-title":"Crit. Financ. Rev."},{"key":"ref_7","unstructured":"Spirtes, P. (2001, January 4\u20137). An Anytime Algorithm for Causal Inference. Proceedings of the International Workshop on Artificial Intelligence and Statistics, PMLR, Key West, FL, USA."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"e1449","DOI":"10.1002\/widm.1449","article-title":"Methods and tools for causal discovery and causal inference","volume":"12","author":"Nogueira","year":"2022","journal-title":"WIREs Data Min. Knowl. Discov."},{"key":"ref_9","unstructured":"Mcdonald, J.H. (2014). Handbook of Biological Statistics, Sparky House Publishing."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"143","DOI":"10.11613\/BM.2013.018","article-title":"The Chi-square test of independence","volume":"23","author":"McHugh","year":"2013","journal-title":"Biochem. Medica"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1093\/biomet\/30.1-2.81","article-title":"A New Measure of Rank Correlation","volume":"30","author":"Kendall","year":"1938","journal-title":"Biometrika"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"441","DOI":"10.2307\/1422689","article-title":"The proof and measurement of association between two things. By C. Spearman, 1904","volume":"100","author":"Spearman","year":"1987","journal-title":"Am. J. Psychol."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1007\/BF00994016","article-title":"Learning Bayesian networks: The combination of knowledge and statistical data","volume":"20","author":"Heckerman","year":"1995","journal-title":"Mach. Learn."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1689","DOI":"10.1214\/14-AOS1217","article-title":"Addendum on the scoring of Gaussian directed acyclic graphical models","volume":"42","author":"Kuipers","year":"2014","journal-title":"Ann. Statist."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1023\/A:1007469629108","article-title":"Efficient approximations for the marginal likelihood of Bayesian networks with hidden variables","volume":"29","author":"Heckerman","year":"1997","journal-title":"Mach. Learn."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Bouckaert, R.R. (1993). Probabilistic network construction using the minimum description length principle. European Conference on Symbolic and Quantitative Approaches to Reasoning and Uncertainty, Springer.","DOI":"10.1007\/BFb0028180"},{"key":"ref_17","unstructured":"Zheng, X., Aragam, B., Ravikumar, P.K., and Xing, E.P. (2018). DAGs with NO TEARS: Continuous Optimization for Structure Learning. Advances in Neural Information Processing Systems, Curran Associates, Inc."},{"key":"ref_18","unstructured":"Murakonda, S.K., Shokri, R., and Theodorakopoulos, G. (2021, January 13\u201315). Quantifying the privacy risks of learning high-dimensional graphical models. Proceedings of the International Conference on Artificial Intelligence and Statistics, PMLR, Virtual."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Vaudenay, S. (2006). Our Data, Ourselves: Privacy Via Distributed Noise Generation. Advances in Cryptology\u2014EUROCRYPT 2006, Springer. Lecture Notes in Computer Science.","DOI":"10.1007\/11761679"},{"key":"ref_20","first-page":"5516","article-title":"Towards practical differentially private causal graph discovery","volume":"Volume 33","author":"Larochelle","year":"2020","journal-title":"Advances in Neural Information Processing Systems"},{"key":"ref_21","unstructured":"Xu, D., Yuan, S., and Wu, X. (2017, January 1\u20134). Differential Privacy Preserving Causal Graph Discovery. Proceedings of the Computer Science and Computer Engineering Faculty Publications and Presentations, Washington, DC, USA."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2324","DOI":"10.1109\/TIFS.2022.3184263","article-title":"NoLeaks: Differentially Private Causal Discovery Under Functional Causal Model","volume":"17","author":"Ma","year":"2022","journal-title":"IEEE Trans. Inf. Forensics Secur."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Halevi, S., and Rabin, T. (2006). Calibrating Noise to Sensitivity in Private Data Analysis. Theory of Cryptography, Springer. Lecture Notes in Computer Science.","DOI":"10.1007\/11681878"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1016\/j.ijar.2022.09.004","article-title":"A survey on causal discovery: Theory and practice","volume":"151","author":"Zanga","year":"2022","journal-title":"Int. J. Approx. Reason."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1561\/0400000042","article-title":"The Algorithmic Foundations of Differential Privacy","volume":"9","author":"Dwork","year":"2013","journal-title":"Found. Trends\u00ae Theor. Comput. Sci."},{"key":"ref_26","unstructured":"Balle, B., Barthe, G., and Gaboardi, M. (2018). Privacy Amplification by Subsampling: Tight Analyses via Couplings and Divergences. Advances in Neural Information Processing Systems, Curran Associates, Inc."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Chickering, D.M. (1996). Learning Bayesian networks is NP-complete. Learning from Data: Artificial Intelligence and Statistics V, Springer.","DOI":"10.1007\/978-1-4612-2404-4_12"},{"key":"ref_28","unstructured":"Dwork, C., and Lei, J. (June, January 31). Differential privacy and robust statistics. Proceedings of the Forty-First Annual ACM Symposium on Theory of Computing, STOC \u201909, New York, NY, USA."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Dwork, C., Rothblum, G.N., and Vadhan, S. (2010, January 23\u201326). Boosting and Differential Privacy. Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science, Las Vegas, NV, USA.","DOI":"10.1109\/FOCS.2010.12"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"4037","DOI":"10.1109\/TIT.2017.2685505","article-title":"The Composition Theorem for Differential Privacy","volume":"63","author":"Kairouz","year":"2017","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_31","unstructured":"Rogers, R.M., Roth, A., Ullman, J., and Vadhan, S. (2016). Privacy Odometers and Filters: Pay-as-you-Go Composition. Advances in Neural Information Processing Systems, Curran Associates, Inc."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Abadi, M., Chu, A., Goodfellow, I., McMahan, H.B., Mironov, I., Talwar, K., and Zhang, L. (2016, January 24\u201328). Deep Learning with Differential Privacy. Proceedings of the 2016 ACM SIGSAC Conference on Computer and Communications Security, Vienna, Austria.","DOI":"10.1145\/2976749.2978318"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Lee, J., and Kifer, D. (2018, January 19\u201323). Concentrated differentially private gradient descent with adaptive per-iteration privacy budget. Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, London, UK.","DOI":"10.1145\/3219819.3220076"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Zhang, X., Ding, J., Wu, M., Wong, S.T., Van Nguyen, H., and Pan, M. (2021, January 5\u20139). Adaptive privacy preserving deep learning algorithms for medical data. Proceedings of the IEEE\/CVF Winter Conference on Applications of Computer Vision, Virtual.","DOI":"10.1109\/WACV48630.2021.00121"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"4422","DOI":"10.1109\/TIFS.2023.3293961","article-title":"Differentially private deep learning with dynamic privacy budget allocation and adaptive optimization","volume":"18","author":"Chen","year":"2023","journal-title":"IEEE Trans. Inf. Forensics Secur."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Korb, K.B., and Nicholson, A.E. (2010). Bayesian Artificial Intelligence, CRC Press.","DOI":"10.1201\/b10391"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Bernardo, J.M., Berger, J.O., Dawid, A.P., Smith, A.F.M., Bernardo, J.M., Berger, J.O., Dawid, A.P., and Smith, A.F.M. (1992). Bayesian Statistics 4: Proceedings of the Fourth Valencia International Meeting: Dedicated to the memory of Morris H. DeGroot, 1931\u20131989: April 15\u201320, 1991, Oxford University Press.","DOI":"10.1093\/oso\/9780198522669.001.0001"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1111\/j.2517-6161.1988.tb01721.x","article-title":"Local Computations with Probabilities on Graphical Structures and Their Application to Expert Systems","volume":"50","author":"Lauritzen","year":"1988","journal-title":"J. R. Stat. Soc. Ser. (Methodol.)"},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Scutari, M., and Denis, J.B. (2014). Bayesian Networks: With Examples in R, Chapman and Hall\/CRC.","DOI":"10.1201\/b17065"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/11\/946\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:26:31Z","timestamp":1760113591000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/11\/946"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,11,5]]},"references-count":39,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2024,11]]}},"alternative-id":["e26110946"],"URL":"https:\/\/doi.org\/10.3390\/e26110946","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2024,11,5]]}}}