Remove the logarithmic loss in the Rademacher complexity bound

Prove the logarithm-free Rademacher complexity bound conjectured for one-hidden-layer ReLU networks with at most k active units per input, width s, effective weight bound W, bias bound B, and input radius R, thereby removing the remaining logarithmic factors from the established dimension-free rate.

Background

The paper studies the Rademacher complexity of one-hidden-layer ReLU networks whose activation pattern may vary with the input but contains at most k active units on every point of a fixed input domain. Earlier work conjectured a rate proportional to (WR+B)√(sk/m), without the explicit √n dependence appearing in its bound.

The paper removes the explicit input-dimension factor and obtains a nearly matching upper bound, but incurs logarithmic losses, principally through the affine covering estimate and multiscale chaining. The authors explicitly state that the logarithm-free version remains unresolved.

References

The logarithm-free form of their conjecture remains unresolved here.

— Nearly Tight Rademacher Bounds for Sparsely Activated Neural Networks  (2609.09130 - Li et al., 8 Sep 2026) in Section 1, paragraph 'Comparison with the original bound'; Section 6, paragraph 'The logarithmic gap remains'

The sufficient value $B=(\sqrt3/2)WR$ is not shown to be a critical threshold.

— Nearly Tight Rademacher Bounds for Sparsely Activated Neural Networks  (2609.09130 - Li et al., 8 Sep 2026) in Section 6, paragraph 'The domain results leave a more geometric question'