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 recovers all network parameters to $δ$-accuracy with sample complexity $ \widetilde O_{k,L}\left(d^{L^2+O(L)}δ^{-2e}\right) $ in polynomial time. To our knowledge, this is the \emph{first} parameter-recovery guarantee for deep target networks whose exponent grows only polynomially with depth, as well as the \emph{first} justification for the effectiveness of using high-quality data in neural network training, with a remarkably \emph{exponential} separation.

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