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

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. %when the target function has a genuinely deep compositional structure, obtaining sample-complexity guarantees with controlled dependence on depth remains largely open. A key reason is that most existing methods stop at learning first-layer representations, making it difficult to obtain efficient guarantees for deeper layers. 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 $\delta$-accuracy with sample complexity ${\tilde{O}}_{k,L}\left(d^{L^2+O(L)}{\delta }^{-2e}\right)$ 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.