Symmetry-Compatible Principle for Optimizer Design: Embeddings, LM Heads, SwiGLU MLPs, and MoE Routers

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optimizer symmetry equivariance neural-networks training deep-learning pre-training

Summary

Researchers introduce symmetry-compatible optimizers that respect the equivariance structures of neural network parameters, improving training stability and performance over traditional methods like Adam. The approach is validated on various language model architectures including Qwen3-0.6B, Gemma 3 1B, and OLMoE-1B-7B.

A striking geometric disparity has long persisted in the practice of deep learning. While modern neural network architectures naturally exhibit rich symmetry and equivariance properties, popular optimizers such as Adam and its variants operate inherently coordinate-wise, rendering them unable to respect the equivariance structures of the parameter space. We address this disparity by introducing a symmetry-compatible principle for optimizer design: the gradient update rule should be equivariant under the symmetry group acting on the corresponding weight block. Following this principle, we first provide a unified perspective on bi-orthogonally equivariant updates for general matrix layers, as employed by stochastic spectral descent, Muon, Scion, and polar gradient methods. More importantly, by moving from orthogonal groups to permutation and shared-shift symmetries, we derive symmetry-compatible optimizers for parameter blocks whose symmetries differ from those of general matrix layers: embedding and LM head matrices, SwiGLU MLP projections, and MoE router matrices. These constructions include one-sided spectral, row-norm, hybrid row-norm/spectral, row-aware, column-aware, centered row-norm, and left-spectral updates. They yield an end-to-end layerwise optimizer stack in which each major matrix-valued parameter class is assigned an update whose equivariance matches its symmetry group. We corroborate this principle through pre-training experiments on dense and sparse MoE language models, including Qwen3-0.6B-style, Gemma 3 1B-style, OLMoE-1B-7B-style, and downsized gpt-oss architectures. Across these experiments, symmetry-compatible updates consistently improve final validation loss, and in several cases training stability, over corresponding AdamW updates.
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Paper page - Symmetry-Compatible Principle for Optimizer Design: Embeddings, LM Heads, SwiGLU MLPs, and MoE Routers

Source: https://huggingface.co/papers/2605.18106

Abstract

Researchers developed symmetry-compatible optimizers that respect the equivariance structures of neural network parameters, improving training stability and performance over traditional coordinate-wise methods like Adam.

A striking geometric disparity has long persisted in the practice of deep learning. While modern neural network architectures naturally exhibit rich symmetry andequivarianceproperties, popular optimizers such as Adam and its variants operate inherently coordinate-wise, rendering them unable to respect theequivariancestructures of theparameter space. We address this disparity by introducing a symmetry-compatible principle for optimizer design: the gradient update rule should be equivariant under the symmetry group acting on the correspondingweight block. Following this principle, we first provide a unified perspective onbi-orthogonally equivariant updatesfor general matrix layers, as employed bystochastic spectral descent,Muon,Scion, andpolar gradient methods. More importantly, by moving from orthogonal groups to permutation and shared-shift symmetries, we derivesymmetry-compatible optimizersfor parameter blocks whose symmetries differ from those of general matrix layers: embedding andLM head matrices,SwiGLU MLP projections, andMoE router matrices. These constructions include one-sided spectral, row-norm, hybrid row-norm/spectral, row-aware, column-aware, centered row-norm, andleft-spectral updates. They yield an end-to-end layerwise optimizer stack in which each major matrix-valued parameter class is assigned an update whoseequivariancematches its symmetry group. We corroborate this principle throughpre-trainingexperiments on dense andsparse MoElanguage models, includingQwen3-0.6B-style,Gemma 3 1B-style,OLMoE-1B-7B-style, and downsizedgpt-ossarchitectures. Across these experiments, symmetry-compatible updates consistently improve final validation loss, and in several cases training stability, over corresponding AdamW updates.

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