# Beyond Overparameterization: Provable Learning of Input-Convex Multi-Layer Polynomial Networks with Active Queries

arXiv: https://arxiv.org/abs/2610.04999 · HTML: https://mlsanity.com/paper/2610.04999/

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- First posted on arXiv 4 Oct 2026.

## Abstract (arXiv)

The theoretical understanding of multi-layer neural networks is largely confined to overparameterized settings, which obscure parameter identifiability and incur high sample complexity. Neural tangent kernel (NTK) provides a general theory for wide networks, but does not offer efficient sample-complexity guarantees. Recent feature-learning results go beyond kernel methods for single-neuron, multi-index, and hierarchical targets. However, the analysis is often restricted to shallow or specific architectures and to the overparameterized regime. We break this paradigm to achieve parameter-level recovery of deep target networks, albeit by using active data queries. Specifically, we study L-layer polynomial networks with even degree-k monomial activations and nonnegative higher-layer weights. This structure makes the target network input-convex, while the optimization landscape remains highly nonconvex with respect to the parameters. Leveraging input convexity and active queries, we propose ASPIRE (Active SamPling for Iterative Recovery via Eigendirections), a layerwise sampling-based diagonalization algorithm that recovers all network parameters to δ-accuracy with sample complexity  widetilde Ok,Lleft(dL2+O(L)δ-2eright)  in polynomial time. To our knowledge, this is the first parameter-recovery guarantee for deep target networks whose exponent grows only polynomially with depth, as well as the first justification for the effectiveness of using high-quality data in neural network training, with a remarkably exponential separation.