An Efficient Minimax Optimal Estimator For Multivariate Convex Regression

Gil Kur, Eli Putterman

Research output: Contribution to journalConference articlepeer-review


We study the computational aspects of the task of multivariate convex regression in dimension d ≥ 5. We present the first computationally efficient minimax optimal (up to logarithmic factors) estimators for the tasks of L-Lipschitz and Γ-bounded convex regression under polytopal support. This work is the first to show the existence of efficient minimax optimal estimators for non-Donsker classes whose corresponding Least Squares Estimators are provably minimax suboptimal. The proof of the correctness of these estimators uses a variety of tools from different disciplines, among them empirical process theory, stochastic geometry, and potential theory.

Original languageEnglish
Pages (from-to)1510-1546
Number of pages37
JournalProceedings of Machine Learning Research
StatePublished - 2022
Event35th Conference on Learning Theory, COLT 2022 - London, United Kingdom
Duration: 2 Jul 20225 Jul 2022


  • Minimax Optimality
  • Multivariate Convex Regression
  • Non-Donsker Regime


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