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Life sciences · Preprint
arXiv · September 8, 2026
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This is a preprint presenting theoretical Rademacher complexity bounds for one-hidden-layer ReLU networks with input-dependent sparsity. The work provides mathematical proofs relating statistical complexity to network width, sparsity, sample size, and weight/bias bounds, with matching lower bounds in several regimes. No empirical validation, clinical application, or real-world outcome measurement is provided.
Preprint.
Rademacher complexity upper bound: $\mathcal{R}(S)\le CWR\min\{k,\sqrt{sk/m}\log^{3/2}(2m)\}+kB/\sqrt m$ for sample size m, width s, active units k, weight bound W, bias bound B, and radius R. For zero-bias networks sparse on entire ball: complexity is $O(kWR/\sqrt m)$ with at most $2k$ nonzero units. Agnostic minimax excess-risk bounds of order $\min\{1,\sqrt{s/(km)}\}$ up to logarithms obtained for normalized bounded loss and biases comparable to $WR$.
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This is a theoretical analysis of statistical complexity bounds for sparse neural networks with no empirical validation, experimental data, or clinical application.
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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.
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