Constrained Learning with Universally Learnable Concept Classes

• URL: http://arxiv.org/abs/2608.08414v1
• Date: Sun, 09 Aug 2026 02:13:57 GMT
• Title: Constrained Learning with Universally Learnable Concept Classes
• Abstract summary: We establish universal PACC learnability of the solutions of the solutions of dual algorithms.<n> Optimality is caught between universality, governed by convexity, and strong Lagrangian duality.
• Score: 2.5998827435861633
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We study constrained statistical learning over infinite-dimensional hypothesis classes in the fully nonconvex setting, and establish universal PACC learnability of the solutions of dual algorithms: Probably Approximately Correct on Constraints, guaranteeing optimality and constraint satisfaction at once. This strengthens near-PACC results, whose feasibility residual no amount of data can remove. Optimality is caught between generalization, governed by Rademacher complexity and favoring small classes, and strong Lagrangian duality, which rests on Lyapunov convexity for vector measures and needs decomposability, a demand pulling the other way. We reconcile the two by posing the population problem over a universal RKHS $\mathcal{H}_K$, dense in a decomposable envelope, and learning over norm balls of growing radius. This yields the Tikhonov complexity $\mathfrak{T}^{\varepsilon}_{n}$, the least RKHS norm reaching an $\varepsilon$-optimal Lagrangian level set; we prove it finite, obtain exact learnability of the optimal value, and make the sample threshold explicit and polynomial in $1/\varepsilon$ under a source condition. Feasibility is harder: absent convexity the Lagrangian may not attain its infimum, and dual information pins down only an averaged constraint-risk vector, not the risks of any returned predictor. We introduce the closure-realization gap $\varepsilon^\star_\infty$, an index of how well $\mathcal{H}_K$ retrieves feasible solutions from dualization; it is a property of the problem, not of a modeling choice. Learnability is exact when $\varepsilon^\star_\infty=0$, in particular under dual differentiability, and near-PACC with residual exactly $\varepsilon^\star_\infty$ otherwise. Finally, no distribution-free threshold exists already in the unconstrained specialization, so universality is the canonical frame for dual algorithms over large hypothesis classes.

Related papers

• Uniform Statistical Convergence of Empirical Sinkhorn Potentials with Exponential and Polynomial Dependence on the Regularization Parameter [3.6552781109515853]
  We study the empirical Sinkhorn estimator of the entropic optimal transport potentials under the uniform loss.<n>For a fixed regularization parameter $varepsilon>0$, we establish a non-asymptotic statistical rate of $n-1/2$.<n>Key requirement is a residual-stability estimate for the population Sinkhorn map.
  arXiv  Detail & Related papers  (2026-08-29T08:59:00Z)
• Provably Adaptive Linear Approximation for the Shapley Value and Beyond [73.0940890296463]
  A fundamental and long-standing challenge is their efficient approximation.<n>We develop a linear-space algorithm that requires $P(|hatboldsymbol-boldsymbol|_2geq)leq $ for all commonly used semi-values.<n>Our algorithm allows explicit minimization of the mean square error for each specific utility function.
  arXiv  Detail & Related papers  (2026-04-09T16:38:14Z)
• Hardness of High-Dimensional Linear Classification [58.29089693778071]
  We establish new exponential in dimension lower bounds for the Maximum Halfspace Discrepancy problem.<n>Both are fundamental problems in computational geometry and machine learning in their exact and approximate forms.
  arXiv  Detail & Related papers  (2026-03-19T15:53:41Z)
• Regularized Online RLHF with Generalized Bilinear Preferences [68.44113000390544]
  We consider the problem of contextual online RLHF with general preferences.<n>We adopt the Generalized Bilinear Preference Model to capture preferences via low-rank, skew-symmetric matrices.<n>We prove that the dual gap of the greedy policy is bounded by the square of the estimation error.
  arXiv  Detail & Related papers  (2026-02-26T15:27:53Z)
• Towards Fundamental Limits for Active Multi-distribution Learning [16.639855803241524]
  We develop new algorithms for active multi-distribution learning and establish improved label complexity upper and lower bounds.<n>We show that the bound in the realizable setting is information-theoretically optimal and that the $knu/varepsilon2$ term in the setting is fundamental for proper learners.
  arXiv  Detail & Related papers  (2025-06-21T06:08:58Z)
• Towards a Sharp Analysis of Offline Policy Learning for $f$-Divergence-Regularized Contextual Bandits [49.96531901205305]
  We analyze $f$-divergence-regularized offline policy learning.<n>For reverse Kullback-Leibler (KL) divergence, we give the first $tildeO(epsilon-1)$ sample complexity under single-policy concentrability.<n>We extend our analysis to dueling bandits, and we believe these results take a significant step toward a comprehensive understanding of $f$-divergence-regularized policy learning.
  arXiv  Detail & Related papers  (2025-02-09T22:14:45Z)
• Learning with Norm Constrained, Over-parameterized, Two-layer Neural Networks [54.177130905659155]
  Recent studies show that a reproducing kernel Hilbert space (RKHS) is not a suitable space to model functions by neural networks. In this paper, we study a suitable function space for over- parameterized two-layer neural networks with bounded norms.
  arXiv  Detail & Related papers  (2024-04-29T15:04:07Z)
• Pseudonorm Approachability and Applications to Regret Minimization [73.54127663296906]
  We convert high-dimensional $ell_infty$-approachability problems to low-dimensional pseudonorm approachability problems. We develop an algorithmic theory of pseudonorm approachability, analogous to previous work on approachability for $ell$ and other norms.
  arXiv  Detail & Related papers  (2023-02-03T03:19:14Z)
• Localization, Convexity, and Star Aggregation [0.0]
  Offset Rademacher complexities have been shown to imply sharp, linear-dependent upper bounds for the square loss. We show that in the statistical setting, the offset bound can be generalized to any loss satisfying certain uniform convexity.
  arXiv  Detail & Related papers  (2021-05-19T00:47:59Z)

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.