Research map / Learning theory and algorithms / Kernels and regression theory
Tight Generalization Bounds for Large-Margin Halfspaces
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Proves a generalization bound for large-margin halfspaces that matches the known lower bound across the full tradeoff in margin, margin loss, confidence and sample size, via a refined random-discretization analysis with Rademacher contraction.
Problem
Generalization bounds for large-margin halfspaces were not tight across the tradeoff between margin, fraction of training points with that margin (margin loss), failure probability and sample size; existing upper bounds had a logarithmic gap to the known lower bound.
Approach
Refines the random Johnson-Lindenstrauss-plus-randomized-rounding discretization so that the probability over the rounding becomes a Lipschitz function of the margin alone, enabling Rademacher contraction with a Lipschitz constant that decays exponentially in k, and uses infinitely many grids to remove the residual exp(-k) term.
Key results
- Proves a generalization bound for large-margin halfspaces that matches the known lower bound up to constants across margin, margin loss, failure probability and sample size, closing the logarithmic gap left by prior work.
- The bound holds for all margins n^{-1/2} <= gamma <= 1 and all w simultaneously with probability 1-delta.
- Improves the state-of-the-art first-order-type bound by removing the ln(n) factor inside the square root, replacing it with ln(e/L_S^gamma(w)).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use the proposed first-order generalization bound instead of the prior state-of-the-art bound | Large-margin halfspaces · Generalization from a training sample | generalization error bound | Helped |
| tighten the bound across the margin-loss tradeoff | Large-margin halfspaces · Generalization from a training sample | tightness to lower bound | Helped |
Limitations
- Improvement is small in magnitude (a logarithmic factor), as the authors acknowledge
- Analysis restricted to halfspaces and requires reduction to unit-norm inputs with constant-scale margins
Experimental setup
Baselines: Grønlund et al. 2019 bound Kakade et al. 2008 bound Bartlett-Mendelson Rademacher bound
Builds on: Kakade-Sridharan-Tewari margin bounds Johnson-Lindenstrauss transform Rademacher contraction inequality
Closest papers by meaning
- A Near-optimal Algorithm for Learning Margin Halfspaces with Massart Noise (NeurIPS 2024 · similarity 0.88)
- Reliable Learning of Halfspaces under Gaussian Marginals (NeurIPS 2024 · similarity 0.87)
- Replicable Learning of Large-Margin Halfspaces (ICML 2024 · similarity 0.87)
- Fantastic Generalization Measures are Nowhere to be Found (ICLR 2024 · similarity 0.86)
- Learning Noisy Halfspaces with a Margin: Massart is No Harder than Random (NeurIPS 2024 · similarity 0.85)
- Active Learning of General Halfspaces: Label Queries vs Membership Queries (NeurIPS 2024 · similarity 0.85)
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