Non-approximability of constructive global $\mathcal{L}^2$ minimizers by gradient descent in Deep Learning

We analyze geometric aspects of the gradient descent algorithm in Deep Learning (DL) networks. In particular, we prove that the globally minimizing weights and biases for the $\mathcal{L}^2$ cost obtained constructively in [Chen-Munoz Ewald 2023] for underparametrized ReLU DL networks can generically not be approximated via the gradient descent flow. We therefore conclude that the method introduced in [Chen-Munoz Ewald 2023] is disjoint from the gradient descent method.

Geometric structure of Deep Learning networks and construction of global ${\mathcal L}^2$ minimizers

In this paper, we provide a geometric interpretation of the structure of Deep Learning (DL) networks, characterized by $L$ hidden layers, a ramp activation function, an ${\mathcal L}^2$ Schatten class (or Hilbert-Schmidt) cost function, and input and output spaces ${\mathbb R}^Q$ with equal dimension $Q\geq1$. The hidden layers are defined on spaces ${\mathbb R}^{Q}$, as well. We apply our recent results on shallow neural networks to construct an explicit family of minimizers for the global minimum of the cost function in the case $L\geq Q$, which we show to be degenerate. In the context presented here, the hidden layers of the DL network "curate" the training inputs by recursive application of a truncation map that minimizes the noise to signal ratio of the training inputs. Moreover, we determine a set of $2^Q-1$ distinct degenerate local minima of the cost function.

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

In this paper, we provide a geometric interpretation of the structure of shallow neural networks characterized by one hidden layer, a ramp 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 $N>QM$. 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. We obtain an approximate optimizer using projections adapted to the averages $\overline{x_{0,j}}$ of training input vectors belonging to the same output vector $y_j$, $j=1,\dots,Q$. In the special case $M=Q$, we explicitly determine an exact degenerate local minimum of the cost function; 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 the $Q$-dimensional subspace in the input space ${\mathbb R}^M$ spanned by $\overline{x_{0,j}}$, $j=1,\dots,Q$. We comment on the characterization of the global minimum of the cost function in the given context.