Geometric Regularization from Overparameterization
This work addresses the theoretical understanding of regularization in deep learning for researchers, offering a novel geometric explanation for double descent, though it appears incremental as it builds on existing concepts without empirical validation.
The paper proposes that implicit regularization in overparameterized models arises from the contracting volume of weight distributions in high dimensions, akin to hyperspheres, and uses this geometric perspective to explain the double descent phenomenon by linking it to the intrinsic dimensionality of weight updates during training.
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.