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

Jinqi Tang, Qian Chen, Shihong Ding, Cong Fang

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 \textbf{ASPIRE} (\textbf{A}ctive \textbf{S}am\textbf{P}ling for \textbf{I}terative \textbf{R}ecovery via \textbf{E}igendirections), a layerwise sampling-based diagonalization algorithm that…