We prove existence of global minima in the loss landscape for the approximation of continuous target functions using shallow feedforward artificial neural networks with ReLU activation. This property is one of the fundamental artifacts separating ReLU from other commonly used activation functions. We propose a kind of closure of the search space so that in the extended space minimizers exist. In a second step, we show under mild assumptions that the newly added functions in the extension perform worse than appropriate representable ReLU networks. This then implies that the optimal response in the extended target space is indeed the response of a ReLU network.

Many mathematical convergence results for gradient descent (GD) based algorithms employ the assumption that the GD process is (almost surely) bounded and, also in concrete numerical simulations, divergence of the GD process may slow down, or even completely rule out, convergence of the error function. In practical relevant learning problems, it thus seems to be advisable to design the ANN architectures in a way so that GD optimization processes remain bounded. The property of the boundedness of GD processes for a given learning problem seems, however, to be closely related to the existence of minimizers in the optimization landscape and, in particular, GD trajectories may escape to infinity if the infimum of the error function (objective function) is not attained in the optimization landscape. This naturally raises the question of the existence of minimizers in the optimization landscape and, in the situation of shallow residual ANNs with multi-dimensional input layers and multi-dimensional hidden layers with the ReLU activation, the main result of this work answers this question affirmatively for a general class of loss functions and all continuous target functions. In our proof of this statement, we propose a kind of closure of the search space, where the limits are called generalized responses, and, thereafter, we provide sufficient criteria for the loss function and the underlying probability distribution which ensure that all additional artificial generalized responses are suboptimal which finally allows us to conclude the existence of minimizers in the optimization landscape.