{"ID":2866384,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2509.19788","arxiv_id":"2509.19788","title":"Convex Regression with a Penalty","abstract":"A common way to estimate an unknown convex regression function $f_0: Ω\\subset \\mathbb{R}^d \\rightarrow \\mathbb{R}$ from a set of $n$ noisy observations is to fit a convex function that minimizes the sum of squared errors. However, this estimator is known for its tendency to overfit near the boundary of $Ω$, posing significant challenges in real-world applications. In this paper, we introduce a new estimator of $f_0$ that avoids this overfitting by minimizing a penalty on the subgradient while enforcing an upper bound $s_n$ on the sum of squared errors. The key advantage of this method is that $s_n$ can be directly estimated from the data. We establish the uniform almost sure consistency of the proposed estimator and its subgradient over $Ω$ as $n \\rightarrow \\infty$ and derive convergence rates. The effectiveness of our estimator is illustrated through its application to estimating waiting times in a single-server queue.","short_abstract":"A common way to estimate an unknown convex regression function $f_0: Ω\\subset \\mathbb{R}^d \\rightarrow \\mathbb{R}$ from a set of $n$ noisy observations is to fit a convex function that minimizes the sum of squared errors. However, this estimator is known for its tendency to overfit near the boundary of $Ω$, posing sign...","url_abs":"https://arxiv.org/abs/2509.19788","url_pdf":"https://arxiv.org/pdf/2509.19788v1","authors":"[\"Eunji Lim\"]","published":"2025-09-24T06:19:21Z","proceeding":"stat.ML","tasks":"[\"stat.ML\",\"cs.LG\"]","methods":"[]","has_code":false}