发表机构
MIT Laboratory for Information and Decision Systems (LIDS); Harvard John A. Paulson School of Engineering and Applied Sciences, Harvard University; Technical University of Munich; MIT Computer Science and Artificial Intelligence Laboratory (CSAIL)（麻省理工学院信息与决策系统实验室; 哈佛大学约翰·A·保尔森工程与应用科学学院; 慕尼黑工业大学; 麻省理工学院计算机科学与人工智能实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出首个统一适用于有限与无限群的多项式时间精确不变学习算法，并解决未知对称性发现，证明可恢复对称性且匹配最优样本复杂度。
AI 中文摘要
具有群不变性的学习是许多科学和几何学习问题的核心，但其计算基础仍然知之甚少。即使在经典的监督回归设置中，也不清楚是否能够高效地计算一个对给定群作用精确不变的回归函数。最近的研究表明，当底层群是有限且已知时，可以在多项式时间内强制实现精确不变性，但留下了无限群和未知对称性的情况未解决。在本文中，我们解决了这两个挑战。首先，我们提出了第一个适用于有限群和无限群统一的多项式时间算法，用于具有精确群不变性的学习。其运行时间在数据维度和样本大小上是多项式的，并且与群无关，同时实现了强泛化保证。这为几何机器学习中不变和等变方法的经验成功提供了计算解释，并部分回答了文献中最近的一个开放问题。其次，我们研究了对称性发现设置中的学习，其中不变群是未知的。聚焦于有限群的子群格，我们证明了精确对称性可以从数据中识别并用于多项式时间的学习。对于有限维特征空间上的回归，我们的算法可证明地恢复底层对称性，匹配已知对称性设置中的极小极大最优样本复杂度，并且运行时间在数据维度和样本大小上是多项式的。我们的分析依赖于随机凯莱图和扩展器理论中的工具，这些工具可能具有独立的意义。
英文摘要
Learning with group invariances is central to many scientific and geometric learning problems, yet its computational foundations remain poorly understood. Even for classical supervised regression settings, it has been unclear whether one can efficiently compute a regression function that is exactly invariant to a given group action. Recent work showed that exact invariance can be enforced in polynomial time when the underlying group is finite and known, but left open the cases of infinite groups and unknown symmetries. In this paper, we resolve both challenges. First, we present the first polynomial-time algorithm for learning with exact group invariances that applies uniformly to finite and infinite groups. The runtime is polynomial in the data dimension and sample size, and independent of the group, while achieving strong generalization guarantees. This provides a computational explanation for the empirical success of invariant and equivariant methods in geometric machine learning and partially answers a recent open question in the literature. Second, we study learning in the symmetry discovery setting, where the invariance group is unknown. Focusing on the subgroup lattice of a finite group, we show that exact symmetries can be identified from data and exploited for learning in polynomial time. For regression over finite-dimensional feature spaces, our algorithm provably recovers the underlying symmetry, matches the minimax-optimal sample complexity of the known-symmetry setting, and runs in time polynomial in the data dimension and sample size. Our analysis relies on tools from random Cayley graphs and expander theory, which may be of independent interest.
Comments29 pages. Published at COLT 2026
Journal refConference on Learning Theory (COLT) 2026
