We study a fitting problem inspired by ontology-mediated querying: given a collection of positive and negative examples of the form $(\mathcal{A},q)$ with $\mathcal{A}$ an ABox and $q$ a Boolean query, we seek an ontology $\mathcal{O}$ that satisfies $\mathcal{A} \cup \mathcal{O} \vDash q$ for all positive examples and $\mathcal{A} \cup \mathcal{O}\not\vDash q$ for all negative examples. We consider the description logics $\mathcal{ALC}$ and $\mathcal{ALCI}$ as ontology languages and a range of query languages that includes atomic queries (AQs), conjunctive queries (CQs), and unions thereof (UCQs). For all of the resulting fitting problems, we provide effective characterizations and determine the computational complexity of deciding whether a fitting ontology exists. This problem turns out to be ${\scriptsize CO}NP$ for AQs and full CQs and $2E{\scriptsize XP}T{\scriptsize IME}$-complete for CQs and UCQs. These results hold for both $\mathcal{ALC}$ and $\mathcal{ALCI}$.

We study gradient flows for loss landscapes of fully connected feed forward neural networks with commonly used continuously differentiable activation functions such as the logistic, hyperbolic tangent, softplus or GELU function. We prove that the gradient flow either converges to a critical point or diverges to infinity while the loss converges to an asymptotic critical value. Moreover, we prove the existence of a threshold $\varepsilon>0$ such that the loss value of any gradient flow initialized at most $\varepsilon$ above the optimal level converges to it. For polynomial target functions and sufficiently big architecture and data set, we prove that the optimal loss value is zero and can only be realized asymptotically. From this setting, we deduce our main result that any gradient flow with sufficiently good initialization diverges to infinity. Our proof heavily relies on the geometry of o-minimal structures. We confirm these theoretical findings with numerical experiments and extend our investigation to real-world scenarios, where we observe an analogous behavior.

In this paper, we advance the understanding of neural network training dynamics by examining the intricate interplay of various factors introduced by weight parameters in the initialization process. Motivated by the foundational work of Luo et al. (J. Mach. Learn. Res., Vol. 22, Iss. 1, No. 71, pp 3327-3373), we explore the gradient descent dynamics of neural networks through the lens of macroscopic limits, where we analyze its behavior as width $m$ tends to infinity. Our study presents a unified approach with refined techniques designed for multi-layer fully connected neural networks, which can be readily extended to other neural network architectures. Our investigation reveals that gradient descent can rapidly drive deep neural networks to zero training loss, irrespective of the specific initialization schemes employed by weight parameters, provided that the initial scale of the output function $\kappa$ surpasses a certain threshold. This regime, characterized as the theta-lazy area, accentuates the predominant influence of the initial scale $\kappa$ over other factors on the training behavior of neural networks. Furthermore, our approach draws inspiration from the Neural Tangent Kernel (NTK) paradigm, and we expand its applicability. While NTK typically assumes that $\lim_{m\to\infty}\frac{\log \kappa}{\log m}=\frac{1}{2}$, and imposes each weight parameters to scale by the factor $\frac{1}{\sqrt{m}}$, in our theta-lazy regime, we discard the factor and relax the conditions to $\lim_{m\to\infty}\frac{\log \kappa}{\log m}>0$. Similar to NTK, the behavior of overparameterized neural networks within the theta-lazy regime trained by gradient descent can be effectively described by a specific kernel. Through rigorous analysis, our investigation illuminates the pivotal role of $\kappa$ in governing the training dynamics of neural networks.

Our emphasis is on the efficiently computable error bounds for the recovery routines. We optimize these bounds with respect to the method parameters to construct the estimators with improved statistical properties. We justify the proposed approach with an oracle inequality which links the properties of the recovery algorithms and the best estimation performance.