Geometric structure of shallow neural networks and constructive ${\mathcal L}^2$ cost minimization

• URL: http://arxiv.org/abs/2309.10370v2
• Date: Sun, 17 Mar 2024 08:09:45 GMT
• Title: Geometric structure of shallow neural networks and constructive ${\mathcal L}^2$ cost minimization
• Authors: Thomas Chen, Patricia Muñoz Ewald,
• Abstract summary: We consider shallow neural networks with one hidden layer, a ReLU activation function, an $mathcal L2$ Schatten class (or Hilbert-Schmidt) cost function. We prove an upper bound on the minimum of the cost function of order $O(delta_P)$ where $delta_P$ measures the signal to noise ratio of training inputs. In the special case $M=Q$, we explicitly determine an exact degenerate local minimum of the cost function, and show that the sharp value differs from the upper bound obtained for $Qleq M$ by a
• Score: 1.189367612437469
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this paper, we approach the problem of cost (loss) minimization in underparametrized shallow neural networks through the explicit construction of upper bounds, without any use of gradient descent. A key focus is on elucidating the geometric structure of approximate and precise minimizers. We consider shallow neural networks with one hidden layer, a ReLU activation function, an ${\mathcal L}^2$ Schatten class (or Hilbert-Schmidt) cost function, input space ${\mathbb R}^M$, output space ${\mathbb R}^Q$ with $Q\leq M$, and training input sample size $N>QM$ that can be arbitrarily large. We prove an upper bound on the minimum of the cost function of order $O(\delta_P)$ where $\delta_P$ measures the signal to noise ratio of training inputs. In the special case $M=Q$, we explicitly determine an exact degenerate local minimum of the cost function, and show that the sharp value differs from the upper bound obtained for $Q\leq M$ by a relative error $O(\delta_P^2)$. The proof of the upper bound yields a constructively trained network; we show that it metrizes a particular $Q$-dimensional subspace in the input space ${\mathbb R}^M$. We comment on the characterization of the global minimum of the cost function in the given context.

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