arXiv subjects

Timo Kröger

Publications and source records attributed to Timo Kröger.

3 recordsLinked to original sources

arXiv2026

Convergence rates for gradient descent in the training of overparameterized artificial neural networks with piecewise affine activation

In recent years, artificial neural networks have developed into a powerful tool for addressing a multitude of problems for which classical solution approaches reach their limits. However, it is still unclear why gradient descent optimization algorithms with random initialization, such as the well-known batch gradient descent, are able to achieve zero training loss in many situations, even though the objective function is non-convex and non-smooth. One of the most promising approaches to solving this issue in the field of supervised learning is the analysis of gradient descent optimization in the so-called overparameterized regime. In this article, we provide a further contribution to this area of research by considering overparameterized fully connected shallow artificial neural networks with piecewise affine activation, such as the rectified linear unit activation. Specifically, given that the activation function is not affine and the training input data are pairwise distinct, we show that, with high probability, the mean squared error of such a randomly initialized artificial neural network optimized via batch gradient descent converges to zero at a linear convergence rate as long as the width of the artificial neural network is sufficiently large and the learning rate is sufficiently small.

cs.LG↗

arXiv2025

On bounds for norms of reparameterized ReLU artificial neural network parameters: sums of fractional powers of the Lipschitz norm control the network parameter vector

It is an elementary fact in the scientific literature that the Lipschitz norm of the realization function of a feedforward fully-connected rectified linear unit (ReLU) artificial neural network (ANN) can, up to a multiplicative constant, be bounded from above by sums of powers of the norm of the ANN parameter vector. Roughly speaking, in this work we reveal in the case of shallow ANNs that the converse inequality is also true. More formally, we prove that the norm of the equivalence class of ANN parameter vectors with the same realization function is, up to a multiplicative constant, bounded from above by the sum of powers of the Lipschitz norm of the ANN realization function (with the exponents $ 1/2 $ and $ 1 $). Moreover, we prove that this upper bound only holds when employing the Lipschitz norm but does neither hold for Hölder norms nor for Sobolev-Slobodeckij norms. Furthermore, we prove that this upper bound only holds for sums of powers of the Lipschitz norm with the exponents $ 1/2 $ and $ 1 $ but does not hold for the Lipschitz norm alone.

cs.LG↗

arXiv2022

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 $ δ> 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 $ δ$. 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.

math.OC↗