The Map Behind the Flow: Finite-Step Gradient Descent as a Dynamical System
Many phenomena of deep learning are dynamical: they concern not only which minima exist, but how gradient descent reaches, avoids, or selects among them. Edge-of-stability behavior, sharpness oscillations, catapult phases, balancing, and movement toward flatter representations are effects of the training map itself, and are poorly captured by the small-step gradient-flow limit. This paper studies fixed-step gradient descent as a discrete dynamical system in a hierarchy of exactly solvable models retaining basic structures of deep learning: depth, factorization, width, data coupling, activation, and stochasticity. The starting point is the balanced scalar reduction of a deep linear chain, ...
| Authors | Thomas Hofmann |
|---|---|
| Primary category | cs.LG |
| Published | 6 July 2026 |
| Last updated at source | 6 July 2026 |
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