Support journalism that doesn't flinch . Support journalism that doesn't flinch . Elon Musk leaves a meeting with House Republicans in the basement of the US Capitol building on March 5, 2025 in Washington, DC. Get your news from a source that's not owned and controlled by oligarchs. Tennessee teenagers are suing Elon Musk's company xAI over allegations that its artificial intelligence tool Grok undressed photos of them as minors--the latest challenge against the wealthiest living person's chatbot .

Aligning foundation models is essential for their safe and trustworthy deployment.

Further analysis suggests a positive relationship (Spearman's r

Deep learning has achieved remarkable success across a wide range of domains, significantly expanding the frontiers of what is achievable in artificial intelligence. Yet, despite these advances, critical challenges remain -- most notably, ensuring robustness to small input perturbations and generalization to out-of-distribution data. These critical challenges underscore the need to understand the underlying fundamental principles that govern robustness and generalization. Among the theoretical tools available, Lipschitz continuity plays a pivotal role in governing the fundamental properties of neural networks related to robustness and generalization. It quantifies the worst-case sensitivity of network's outputs to small input perturbations. While its importance is widely acknowledged, prior research has predominantly focused on empirical regularization approaches based on Lipschitz constraints, leaving the underlying principles less explored. This thesis seeks to advance a principled understanding of the principles of Lipschitz continuity in neural networks within the paradigm of machine learning, examined from two complementary perspectives: an internal perspective -- focusing on the temporal evolution of Lipschitz continuity in neural networks during training (i.e., training dynamics); and an external perspective -- investigating how Lipschitz continuity modulates the behavior of neural networks with respect to features in the input data, particularly its role in governing frequency signal propagation (i.e., modulation of frequency signal propagation).