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RANDOM

DOI: 10.4230/LIPIcs.APPROX/RANDOM.2026.62

Learning and Testing Convex Functions

Authors: Renato Ferreira Pinto Jr., Cassandra Marcussen, Elchanan Mossel, and Shivam Nadimpalli

Published in: LIPIcs, Volume 392, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)

Abstract

We consider the problems of learning and testing real-valued convex functions over Gaussian space. Despite the extensive study of function convexity across mathematics, statistics, and computer science, its learnability and testability have largely been examined only in discrete or restricted settings - typically with respect to the Hamming distance, which is ill-suited for real-valued functions. In contrast, we study these problems in high dimensions under the standard Gaussian measure, assuming sample access to the function and a mild smoothness condition, namely Lipschitzness. A smoothness assumption is natural and, in fact, necessary even in one dimension: without it, convexity cannot be inferred from finitely many samples. As our main results, we give: - Learning Convex Functions: An agnostic proper learning algorithm for Lipschitz convex functions that achieves error ε using n^O(1/ε²) samples, together with a complementary lower bound of n^poly(1/ε) samples in the correlational statistical query (CSQ) model. - Testing Convex Functions: A tolerant (two-sided) tester for convexity of Lipschitz functions with the same sample complexity (as a corollary of our learning result), and a one-sided tester (which never rejects convex functions) using O(√n/ε)ⁿ samples.

Cite as

Renato Ferreira Pinto Jr., Cassandra Marcussen, Elchanan Mossel, and Shivam Nadimpalli. Learning and Testing Convex Functions. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 62:1-62:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{ferreirapintojr._et_al:LIPIcs.APPROX/RANDOM.2026.62,
  author =	{Ferreira Pinto Jr., Renato and Marcussen, Cassandra and Mossel, Elchanan and Nadimpalli, Shivam},
  title =	{{Learning and Testing Convex Functions}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
  pages =	{62:1--62:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-449-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{392},
  editor =	{Singh, Mohit and Gur, Tom},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.62},
  URN =		{urn:nbn:de:0030-drops-277798},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2026.62},
  annote =	{Keywords: Property testing, learning theory, Gaussian distribution, convex functions}
}

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RANDOM

DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.61

A Fast Coloring Oracle for Average Case Hypergraphs

Authors: Cassandra Marcussen, Edward Pyne, Ronitt Rubinfeld, Asaf Shapira, and Shlomo Tauber

Published in: LIPIcs, Volume 353, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)

Abstract

Hypergraph 2-colorability is one of the classical NP-hard problems. Person and Schacht [SODA'09] designed a deterministic algorithm whose expected running time is polynomial over a uniformly chosen 2-colorable 3-uniform hypergraph. Lee, Molla, and Nagle recently extended this to k-uniform hypergraphs for all k ≥ 3. Both papers relied heavily on the regularity lemma, hence their analysis was involved and their running time hid tower-type constants. Our first result in this paper is a new simple and elementary deterministic 2-coloring algorithm that reproves the theorems of Person-Schacht and Lee-Molla-Nagle while avoiding the use of the regularity lemma. We also show how to turn our new algorithm into a randomized one with average expected running time of only O(n). Our second and main result gives what we consider to be the ultimate evidence of just how easy it is to find a 2-coloring of an average 2-colorable hypergraph. We define a coloring oracle to be an algorithm which, given vertex v, assigns color red/blue to v while inspecting as few edges as possible, so that the answers to any sequence of queries to the oracle are consistent with a single legal 2-coloring of the input. Surprisingly, we show that there is a coloring oracle that, on average, can answer every vertex query in time O(1).

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Cassandra Marcussen, Edward Pyne, Ronitt Rubinfeld, Asaf Shapira, and Shlomo Tauber. A Fast Coloring Oracle for Average Case Hypergraphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 61:1-61:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{marcussen_et_al:LIPIcs.APPROX/RANDOM.2025.61,
  author =	{Marcussen, Cassandra and Pyne, Edward and Rubinfeld, Ronitt and Shapira, Asaf and Tauber, Shlomo},
  title =	{{A Fast Coloring Oracle for Average Case Hypergraphs}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{61:1--61:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-397-3},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{353},
  editor =	{Ene, Alina and Chattopadhyay, Eshan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.61},
  URN =		{urn:nbn:de:0030-drops-244272},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.61},
  annote =	{Keywords: average-case algorithms, local computation algorithms, graph coloring}
}

Document

DOI: 10.4230/LIPIcs.CCC.2025.19

Characterizing the Distinguishability of Product Distributions Through Multicalibration

Authors: Cassandra Marcussen, Aaron Putterman, and Salil Vadhan

Published in: LIPIcs, Volume 339, 40th Computational Complexity Conference (CCC 2025)

Abstract

Given a sequence of samples x_1, … , x_k promised to be drawn from one of two distributions X₀, X₁, a well-studied problem in statistics is to decide which distribution the samples are from. Information theoretically, the maximum advantage in distinguishing the two distributions given k samples is captured by the total variation distance between X₀^{⊗k} and X₁^{⊗k}. However, when we restrict our attention to efficient distinguishers (i.e., small circuits) of these two distributions, exactly characterizing the ability to distinguish X₀^{⊗k} and X₁^{⊗k} is more involved and less understood. In this work, we give a general way to reduce bounds on the computational indistinguishability of X₀ and X₁ to bounds on the information-theoretic indistinguishability of some specific, related variables X̃₀ and X̃₁. As a consequence, we prove a new, tight characterization of the number of samples k needed to efficiently distinguish X₀^{⊗k} and X₁^{⊗k} with constant advantage as k = Θ(d_H^{-2}(X̃₀, X̃₁)), which is the inverse of the squared Hellinger distance d_H between two distributions X̃₀ and X̃₁ that are computationally indistinguishable from X₀ and X₁. Likewise, our framework can be used to re-derive a result of Halevi and Rabin (TCC 2008) and Geier (TCC 2022), proving nearly-tight bounds on how computational indistinguishability scales with the number of samples for arbitrary product distributions. At the heart of our work is the use of the Multicalibration Theorem (Hébert-Johnson, Kim, Reingold, Rothblum 2018) in a way inspired by recent work of Casacuberta, Dwork, and Vadhan (STOC 2024). Multicalibration allows us to relate the computational indistinguishability of X₀, X₁ to the statistical indistinguishability of X̃₀, X̃₁ (for lower bounds on k) and construct explicit circuits to distinguish between X̃₀, X̃₁ and consequently X₀, X₁ (for upper bounds on k).

Cite as

Cassandra Marcussen, Aaron Putterman, and Salil Vadhan. Characterizing the Distinguishability of Product Distributions Through Multicalibration. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 19:1-19:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{marcussen_et_al:LIPIcs.CCC.2025.19,
  author =	{Marcussen, Cassandra and Putterman, Aaron and Vadhan, Salil},
  title =	{{Characterizing the Distinguishability of Product Distributions Through Multicalibration}},
  booktitle =	{40th Computational Complexity Conference (CCC 2025)},
  pages =	{19:1--19:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-379-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{339},
  editor =	{Srinivasan, Srikanth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2025.19},
  URN =		{urn:nbn:de:0030-drops-237130},
  doi =		{10.4230/LIPIcs.CCC.2025.19},
  annote =	{Keywords: Multicalibration, computational distinguishability}
}

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Documents authored by Marcussen, Cassandra

Learning and Testing Convex Functions

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A Fast Coloring Oracle for Average Case Hypergraphs

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Characterizing the Distinguishability of Product Distributions Through Multicalibration

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