At the heart of deep learning’s transformative impact lies the concept of scale--encompassing both data and computational resources, as well as their interaction with neural network architectures.

Scale, however, presents critical challenges, such as increased instability during training and prohibitively expensive model-specific tuning. Given the substantial resources required to train such models, formulating high-confidence scaling hypotheses backed by rigorous theoretical research has become paramount. The first part of the tutorial will provide an overview of significant advances in the theory of scaling in deep learning, covering its historical foundations, recent breakthroughs, and practical implications for training large-scale models.

To bridge theory and practice, the tutorial explores another key mathematical ingredient of scaling: the numerical solution algorithms commonly employed in deep learning, spanning domains from vision to language models. We unify these algorithms under a common master template, making their foundational principles transparent. In doing so, we reveal the interplay between adaptation to smoothness structures via online learning and the exploitation of optimization geometry through non-Euclidean norms.

Our exposition moves beyond simply building larger models--it emphasizes strategic scaling, offering insights that promise to advance the field while economizing on resources.