Robust and Resource-Efficient Identification of Two Hidden Layer Neural Networks

We address the structure identification and the uniform approximation of two fully nonlinear layer neural networks of the type f(x)=1^T h(B^T g(A^T x)) on ℝ^d , where g=(g_1,… , g_m_0) , h=(h_1,… , h_m_1) , A=(a_1|… |a_m_0) ∈ℝ^d × m_0 and B=(b_1|… |b_m_1) ∈ℝ^m_0 × m_1 , from a small number of query samples. The solution of the case of two hidden layers presented in this paper is crucial as it can be further generalized to deeper neural networks. We approach the problem by sampling actively finite difference approximations to Hessians of the network. Gathering several approximate Hessians allows reliably to approximate the matrix subspace 𝒲 spanned by symmetric tensors a_1 ⊗ a_1,… ,a_m_0⊗ a_m_0 formed by weights of the first layer together with the entangled symmetric tensors v_1 ⊗ v_1 ,… ,v_m_1⊗ v_m_1 , formed by suitable combinations of the weights of the first and second layer as v_ℓ =A G_0 b_ℓ /‖ A G_0 b_ℓ‖ _2 , ℓ∈ [m_1] , for a diagonal matrix G_0 depending on the activation functions of the first layer. The identification of the 1-rank symmetric tensors within 𝒲 is then performed by the solution of a robust nonlinear program, maximizing the spectral norm of the competitors constrained over the unit Frobenius sphere. We provide guarantees of stable recovery under a posteriori verifiable conditions. Once the 1-rank symmetric tensors {a_i ⊗ a_i, i∈ [m_0]}∪{v_ℓ⊗ v_ℓ , ℓ∈ [m_1] } are computed, we address their correct attribution to the first or second layer ( a_i ’s are attributed to the first layer). The attribution to the layers is currently based on a semi-heuristic reasoning, but it shows clear potential of reliable execution. Having the correct attribution of the a_i,v_ℓ to the respective layers and the consequent de-parametrization of the network, by using a suitably adapted gradient descent iteration, it is possible to estimate, up to intrinsic symmetries, the shifts of the activations functions of the first layer and compute exactly the matrix G_0 . Eventually, from the vectors v_ℓ =A G_0 b_ℓ /‖ A G_0 b_ℓ‖ _2 ’s and a_i ’s one can disentangle the weights b_ℓ ’s, by simple algebraic manipulations. Our method of identification of the weights of the network is fully constructive, with quantifiable sample complexity and therefore contributes to dwindle the black box nature of the network training phase. We corroborate our theoretical results by extensive numerical experiments, which confirm the effectiveness and feasibility of the proposed algorithmic pipeline.