Uniform Convergence Bounds for Halfspaces Beyond VC Dimension
A recent paper in theoretical computer science, available on arXiv (2605.06004), examines the detailed uniform convergence characteristics of halfspaces, extending beyond the traditional worst-case VC bounds. For inhomogeneous halfspaces in ℝ^d where d ≥ 2, the authors demonstrate that typical first-order VC bounds are nearly tight: consistent hypotheses can lead to a population error of Θ(d ln(n/d)/n), while in the agnostic scenario, the deviation is proportional to √(τ ln(1/τ)) with true error τ. Conversely, homogeneous halfspaces in ℝ^2 show significantly different patterns. In the realizable case, any hypothesis consistent with the sample has an error of O(1/n). In the agnostic case, they establish a log-free deviation bound on each dyadic risk band using a critical-wedge localization technique. When combined across bands, only a ln ln n overhead is added, and they provide a matching lower bound to prove that this overhead is necessary. Collectively, these findings offer a nuanced and nearly complete understanding of uniform convergence for halfspaces.
Key facts
- Paper studies fine-grained uniform convergence of halfspaces beyond worst-case VC bounds.
- For inhomogeneous halfspaces in ℝ^d (d ≥ 2), VC bounds are essentially tight.
- Consistent hypotheses can incur population error Θ(d ln(n/d)/n).
- Agnostic deviation scales as √(τ ln(1/τ)) at true error τ.
- Homogeneous halfspaces in ℝ^2 show different behavior.
- Realizable case: error O(1/n) for consistent hypotheses.
- Agnostic case: bandwise log-free deviation bound via critical-wedge localization.
- Union over bands incurs ln ln n overhead, shown to be unavoidable.
- Matching lower bound established for the overhead.
- Results provide a nearly complete picture of uniform convergence for halfspaces.
Entities
Institutions
- arXiv