Trainable at Depth, Solvable at Small Size: What the Loss Surface of a Polynomial Activation Dictates for Designing, Initializing and Training Networks
cs.LG, cs.AI
Submitted: 2026-10-14
Updated: 2026-10-14
License: http://creativecommons.org/licenses/by-nc-sa/4.0/
The gist: What makes a deep network train, and what shape is its loss surface? We answer both with one construction at two scales: a polynomial activation f (z) = z + λz 3, clamped and projected onto the unit
Abstract
What makes a deep network train, and what shape is its loss surface? We answer both with one construction at two scales: a polynomial activation f (z) = z + λz 3, clamped and projected onto the unit sphere, plus an unnormalized identity path. As a deep stack it trains to one hundred layers on MNIST with no normalization machinery; as a small model its square loss is a polynomial system we solve exactly. The small scale explains the large one: it prescribes how to design, initialize and train the stack.
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