Hyperbolic Aware Minimization: Implicit Bias for Sparsity

TL;DR

This paper introduces Hyperbolic Aware Minimization (HAM), a new optimization technique that leverages hyperbolic geometry to promote sparsity and improve generalization in deep models, especially when combined with sparsification methods.

Contribution

HAM is a novel optimization method that alternates standard steps with hyperbolic mirror steps, overcoming the small-inverse-metric issue and enhancing model sparsity and performance.

Findings

HAM improves performance in underdetermined linear regression.

HAM enhances sparsity when combined with sparsification methods.

Experiments show HAM's effectiveness on standard vision benchmarks.

Abstract

Understanding the implicit bias of optimization algorithms is key to explaining and improving the generalization of deep models. The hyperbolic implicit bias induced by pointwise overparameterization promotes sparsity, but also yields a small inverse Riemannian metric near zero, slowing down parameter movement and impeding meaningful parameter sign flips. To overcome this obstacle, we propose Hyperbolic Aware Minimization (HAM), which alternates a standard optimizer step with a lightweight hyperbolic mirror step. The mirror step incurs less compute and memory than pointwise overparameterization, reproduces its beneficial hyperbolic geometry for feature learning, and mitigates the small-inverse-metric bottleneck. Our characterization of the implicit bias in the context of underdetermined linear regression provides insights into the mechanism how HAM consistently increases performance --even in the case of dense training, as we demonstrate in experiments with standard vision benchmarks. HAM is especially effective in combination with different sparsification methods, advancing the state of the art.