Explore the complex interplay of competing mechanisms that govern neural network training dynamics in this conference talk from the 2025 Mathematical and Scientific Foundations of Deep Learning Annual Meeting. Delve into the theoretical foundations underlying how different optimization processes interact and compete during the training of deep learning models. Examine the mathematical frameworks that explain why certain training behaviors emerge and how multiple mechanisms can simultaneously influence learning trajectories. Gain insights into the fundamental principles that determine which mechanisms dominate under different conditions and how this competition affects model performance and convergence. Understand the implications of these competing dynamics for designing more effective training algorithms and predicting training outcomes in deep learning systems.