Algorithmic Principles For Multiclass Learning Are Hard To Come By: Limits of Regularization and Proper Learning

• URL: http://arxiv.org/abs/2608.26516v1
• Date: Thu, 27 Aug 2026 01:39:28 GMT
• Title: Algorithmic Principles For Multiclass Learning Are Hard To Come By: Limits of Regularization and Proper Learning
• Abstract summary: Two of the most fundamental questions in statistical learning theory are which prediction problems are learnable, and how should they be learned?<n>We show that proper learning can require training error and characterize this phenomenon precisely.<n>We exhibit a properly learnable class that cannot be learned by any Structural Risk Minimization learner, and a learnable class that cannot be learned by any local regularizer.
• Score: 17.483318390877916
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: Two of the most fundamental questions in statistical learning theory are the following: which prediction problems are learnable, and how should they be learned? For the former, elegant answers often take the form of combinatorial dimensions. The latter question, however, has proved considerably more elusive: all known general-purpose multiclass learners rely on intricate orientations of exponentially large one-inclusion structures, and familiar algorithmic principles such as proper learning and regularization remain poorly understood. Motivated by prior work, we ask whether learning reduces to proper learning---possibly over a larger hypothesis class---and whether proper or improper multiclass learning can ultimately be captured by suitable regularizers. Our primary results answer both questions negatively, resolving three open problems from prior work. First, we exhibit a learnable multiclass problem that cannot be embedded in any properly learnable class, meaning learning cannot be reduced to proper learning by enlarging the hypothesis class. Second, we demonstrate that proper learning can require training error and characterize this phenomenon precisely: every properly learnable class admits a proper learner making $o(m)$ errors on samples of size $m$, but every prescribed sublinear scale $a_m=o(m)$ is necessary for some properly learnable problem. Third, regularization is not a general learner: we exhibit a properly learnable class that cannot be learned by any Structural Risk Minimization (SRM) learner, and a learnable class that cannot be learned by any local regularizer. We complement these impossibility results with a positive theory that gives two sufficient conditions for SRM learnability and characterizes SRM representability through integrability of revealed preferences.

Related papers

• Online Set Learning from Precision and Recall Feedback [60.00180898830079]
  We consider the problem of learning an unknown subset $N_texttarget$ of a domain in an online setting.<n>This simple online set learning problem abstracts a variety of learning scenarios with precision- and recall-type feedback.<n>We show that a hypothesis class is learnable in this setting if and only if it has finite Vapnik-Chervonenkis dimension.
  arXiv  Detail & Related papers  (2026-05-10T14:28:05Z)
• Characterizing Online and Private Learnability under Distributional Constraints via Generalized Smoothness [63.833913892018536]
  We study sequential decision making under distributional adversaries that can adaptively choose data-generating distributions from a fixed family $U$.<n>We provide a near complete characterization of families $U$ that admit learnability in terms of a notion known as generalized smoothness.<n>We show that the generalized smoothness also characterizes private learnability under distributional constraints.
  arXiv  Detail & Related papers  (2026-02-24T06:15:59Z)
• Computable universal online learning [1.7961752949636303]
  We show that universal online learning does not imply computable universal online learning.<n>We also consider a variant of proper universal online learning and show exactly when it is possible.
  arXiv  Detail & Related papers  (2025-10-21T07:21:32Z)
• Proper Learnability and the Role of Unlabeled Data [10.168670899305232]
  We show that there are problems whose proper learnability is logically undecidable, i.e., independent of the ZFC axioms.<n>We then show all impossibility results which obstruct any characterization of proper learnability in the realizable PAC model.
  arXiv  Detail & Related papers  (2025-02-14T18:41:53Z)
• Ticketed Learning-Unlearning Schemes [57.89421552780526]
  We propose a new ticketed model for learning--unlearning. We provide space-efficient ticketed learning--unlearning schemes for a broad family of concept classes.
  arXiv  Detail & Related papers  (2023-06-27T18:54:40Z)
• Adversarially Robust Learning: A Generic Minimax Optimal Learner and Characterization [39.51923275855131]
  We present a minimax optimal learner for the problem of learning predictors robust to adversarial examples at test-time. In particular, we show, in a strong negative sense, the suboptimality of the robust learner proposed by Montasser, Hanneke, and Srebro.
  arXiv  Detail & Related papers  (2022-09-15T15:32:42Z)
• Realizable Learning is All You Need [21.34668631009594]
  equivalence of realizable and agnostic learnability is a fundamental phenomenon in learning theory. We give the first model-independent framework explaining the equivalence of realizable and agnostic learnability.
  arXiv  Detail & Related papers  (2021-11-08T19:00:00Z)
• Exploring Bayesian Deep Learning for Urgent Instructor Intervention Need in MOOC Forums [58.221459787471254]
  Massive Open Online Courses (MOOCs) have become a popular choice for e-learning thanks to their great flexibility. Due to large numbers of learners and their diverse backgrounds, it is taxing to offer real-time support. With the large volume of posts and high workloads for MOOC instructors, it is unlikely that the instructors can identify all learners requiring intervention. This paper explores for the first time Bayesian deep learning on learner-based text posts with two methods: Monte Carlo Dropout and Variational Inference.
  arXiv  Detail & Related papers  (2021-04-26T15:12:13Z)
• Constrained Learning with Non-Convex Losses [119.8736858597118]
  Though learning has become a core technology of modern information processing, there is now ample evidence that it can lead to biased, unsafe, and prejudiced solutions.
  arXiv  Detail & Related papers  (2021-03-08T23:10:33Z)
• Probably Approximately Correct Constrained Learning [135.48447120228658]
  We develop a generalization theory based on the probably approximately correct (PAC) learning framework. We show that imposing a learner does not make a learning problem harder in the sense that any PAC learnable class is also a constrained learner. We analyze the properties of this solution and use it to illustrate how constrained learning can address problems in fair and robust classification.
  arXiv  Detail & Related papers  (2020-06-09T19:59:29Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.