HORST: Composing Optimizer Geometries for Sparse Transformer Training

TL;DR

HORST is a novel modular optimizer that combines adaptive optimizer stability with an $L_1$ bias to effectively train sparse transformers, outperforming standard methods across vision and language tasks.

Contribution

We introduce HORST, a compositional optimizer using non-commutative operators and hyperbolic mirror maps to enable stable, sparse transformer training.

Findings

HORST outperforms AdamW across all sparsity levels.

Significant improvements at higher sparsity levels.

Effective for both vision and language transformer tasks.

Abstract

Sparsifying transformers remains a fundamental challenge, as standard optimizers fail to simultaneously encourage sparsity and maintain training stability. Effective adaptive optimizers exhibit an implicit $L_{\infty}$ bias favoring stability, yet, sparsity requires an $L_1$ bias. To integrate sparsity, we propose a composition of optimizer steps, which we cast as non-commutative operators to analyze and combine their optimization geometry in a principled way. This yields HORST (Hyperbolic Operator for Robust Sparse Training), a modular optimizer that inherits stability from adaptive methods while inducing $L_1$ sparsity bias through a hyperbolic mirror map. Our experiments demonstrate its utility for sparse training of transformers on both vision and language tasks. HORST consistently and significantly outperforms AdamW baselines across all sparsity levels, with large gains at higher sparsity.