Geometric Regularization from Overparameterization

TL;DR

This paper proposes a geometric regularization conjecture explaining overparameterization effects like double descent through volume contraction in high-dimensional weight distributions, linking geometric properties to model capacity phenomena.

Contribution

It introduces a new geometric regularization conjecture and connects volume contraction in high-dimensional spaces to the double descent phenomenon in neural networks.

Findings

Volume contraction explains implicit regularization.

Data complexity influences double descent thresholds.

Geometric forms suggest universality in model capacity behavior.

Abstract

The volume of the distribution of weight sets associated with a loss value may be the source of implicit regularization from overparameterization due to the phenomenon of contracting volume with increasing dimensions for geometric figures demonstrated by hyperspheres. We introduce the geometric regularization conjecture and extract to an explanation for the double descent phenomenon by considering a similar property resulting from shrinking intrinsic dimensionality of the distribution of potential weight set updates available along training path, where if that distribution retracts across a volume verses dimensionality curve peak when approaching the global minima we could expect geometric regularization to re-emerge. We illustrate how data fidelity representational complexity may influence model capacity double descent interpolation thresholds. The existence of epoch and model capacity double descent curves originating from different geometric forms may imply universality of closed n-manifolds having dimensionally adjusted n-sphere volumetric correspondence.