Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the existence of infinitely many realization functions of non-global
  local minima in the training of artificial neural networks with ReLU
  activation   Gradient descent (GD) type optimization schemes are the standard instruments
to train fully connected feedforward artificial neural networks (ANNs) with
rectified linear unit (ReLU) activation and can be considered as temporal
discretizations of solutions of gradient flow (GF) differential equations. It
has recently been proved that the risk of every bounded GF trajectory converges
in the training of ANNs with one hidden layer and ReLU activation to the risk
of a critical point. Taking this into account it is one of the key research
issues in the mathematical convergence analysis of GF trajectories and GD type
optimization schemes, respectively, to study sufficient and necessary
conditions for critical points of the risk function and, thereby, to obtain an
understanding about the appearance of critical points in dependence of the
problem parameters such as the target function. In the first main result of
this work we prove in the training of ANNs with one hidden layer and ReLU
activation that for every $ a, b \in \mathbb{R} $ with $ a < b $ and every
arbitrarily large $ \delta > 0 $ we have that there exists a Lipschitz
continuous target function $ f \colon [a,b] \to \mathbb{R} $ such that for
every number $ H > 1 $ of neurons on the hidden layer we have that the risk
function has uncountably many different realization functions of non-global
local minimum points whose risks are strictly larger than the sum of the risk
of the global minimum points and the arbitrarily large $ \delta $. In the
second main result of this work we show in the training of ANNs with one hidden
layer and ReLU activation in the special situation where there is only one
neuron on the hidden layer and where the target function is continuous and
piecewise polynomial that there exist at most finitely many different
realization functions of critical points.

9
The categories of this paper are: nlin
31
The categories of this paper are: ao-sci
36
The categories of this paper are: cmp-lg
37
The categories of this paper are: supr-con
16
The categories of this paper are: stat
33
The categories of this paper are: plasm-ph
28
The categories of this paper are: alg-geom
11
The categories of this paper are: hep-lat
0
The categories of this paper are: hep-ph
10
The categories of this paper are: q-bio
26
The categories of this paper are: bayes-an
23
The categories of this paper are: solv-int
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The categories of this paper are: hep-ex
30
The categories of this paper are: q-alg
1
The categories of this paper are: cs
18
The categories of this paper are: econ