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 O_(k,L)(d^(L²+O(L))δ⁻²ᵉ) 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.