Nearly Tight Rademacher Bounds for Sparsely Activated Neural Networks
Theory paper derives nearly tight Rademacher complexity bounds for sparsely activated one-hidden-layer ReLU networks.
Building on Awasthi et al. (COLT 2024), the authors bound statistical complexity for networks where each input activates at most k of s hidden units. A support-preserving cover and normalized chaining argument remove the explicit dimension factor, with matching lower bounds up to logarithms. They also derive agnostic minimax excess-risk bounds of order min{1, sqrt(s/(km))} for a normalized bounded loss and show bias bounds comparable to WR restore worst-case rates even on domains where sparsity holds globally.
- Upper bound CWR*min{k, sqrt(sk/m)}*log^1.5(2m)+kB/sqrt(m) for width-s networks with k active units
- Lower bounds on i.i.d. marginals match the upper bound up to logarithmic factors
- Zero-bias ball-sparse networks retain complexity O(kWR/sqrt(m)); large biases restore worst-case rate
Full article185 words · extracted from arxiv.org · click to collapse
An input may activate few hidden units even when different inputs collectively use an entire network. We study the statistical complexity of this input-dependent sparsity in the one-hidden-layer ReLU model of Awasthi et al. (COLT 2024). For width $s$, at most $k$ active units per input, and effective weight and bias bounds $W,B$, every size-$m$ sample in the class's fixed radius-$R$ input domain satisfies $\mathcal{R}(S)\le CWR\min\{k,\sqrt{sk/m}\log^{3/2}(2m)\}+kB/\sqrt m$. A support-preserving cover and a single normalized chaining argument remove the previous explicit dimension factor, up to logarithms. Lower bounds on appropriate i.i.d. marginals match up to those logarithms, showing how changing active units across inputs retains a width dependence. The input domain matters: zero-bias networks sparse on the entire ball have at most $2k$ nonzero units and complexity $O(kWR/\sqrt m)$, whereas bias bounds comparable to $WR$ restore the worst-case rate on that same domain in only logarithmic dimension. A spherical-cap construction proves the latter claim without assuming sparsity merely on the sampling support. For a specified normalized bounded loss and biases comparable to $WR$, we also obtain agnostic minimax excess-risk bounds of order $\min\{1,\sqrt{s/(km)}\}$ up to logarithms.
Text extracted automatically; images, tables and formatting may be missing. Original: https://arxiv.org/abs/2609.09130