# SOLE: Hardware-Software Co-design of Softmax and LayerNorm for Efficient Transformer Inference

arXiv: https://arxiv.org/abs/2510.17189 · HTML: https://mlsanity.com/paper/2510.17189/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 39 references that are in the map.
- First posted on arXiv 20 Oct 2025.

## Builds on (most cited first, 28 of 28)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [Swin Transformer: Hierarchical Vision Transformer using Shifted Windows](https://mlsanity.com/paper/2103.14030/) (2021)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [GLUE: A Multi-Task Benchmark and Analysis Platform for Natural Language Understanding](https://mlsanity.com/paper/1804.07461/) (2018)
- [Training data-efficient image transformers & distillation through attention](https://mlsanity.com/paper/2012.12877/) (2020)
- [Zero-Shot Text-to-Image Generation](https://mlsanity.com/paper/2102.12092/) (2021)
- [SQuAD: 100,000+ Questions for Machine Comprehension of Text](https://mlsanity.com/paper/1606.05250/) (2016)
- [Deformable DETR: Deformable Transformers for End-to-End Object Detection](https://mlsanity.com/paper/2010.04159/) (2020)
- [PaLM-E: An Embodied Multimodal Language Model](https://mlsanity.com/paper/2303.03378/) (2023)
- [Rethinking Semantic Segmentation from a Sequence-to-Sequence Perspective with Transformers](https://mlsanity.com/paper/2012.15840/) (2020)
- [Scaling Vision Transformers to 22 Billion Parameters](https://mlsanity.com/paper/2302.05442/) (2023)
- [DoReFa-Net: Training Low Bitwidth Convolutional Neural Networks with Low Bitwidth Gradients](https://mlsanity.com/paper/1606.06160/) (2016)
- [Q-BERT: Hessian Based Ultra Low Precision Quantization of BERT](https://mlsanity.com/paper/1909.05840/) (2019)
- [Post-Training Quantization for Vision Transformer](https://mlsanity.com/paper/2106.14156/) (2021)
- [I-BERT: Integer-only BERT Quantization](https://mlsanity.com/paper/2101.01321/) (2021)
- [SpAtten: Efficient Sparse Attention Architecture with Cascade Token and Head Pruning](https://mlsanity.com/paper/2012.09852/) (2020)
- [Integer Quantization for Deep Learning Inference: Principles and Empirical Evaluation](https://mlsanity.com/paper/2004.09602/) (2020)
- [Online normalizer calculation for softmax](https://mlsanity.com/paper/1805.02867/) (2018)
- [FQ-ViT: Post-Training Quantization for Fully Quantized Vision Transformer](https://mlsanity.com/paper/2111.13824/) (2021)
- [A3: Accelerating Attention Mechanisms in Neural Networks with Approximation](https://mlsanity.com/paper/2002.10941/) (2020)
- [Full Stack Optimization of Transformer Inference: a Survey](https://mlsanity.com/paper/2302.14017/) (2023)
- [Softermax: Hardware/Software Co-Design of an Efficient Softmax for Transformers](https://mlsanity.com/paper/2103.09301/) (2021)
- [Hardware Accelerator for Multi-Head Attention and Position-Wise Feed-Forward in the Transformer](https://mlsanity.com/paper/2009.08605/) (2020)
- [NN-LUT: Neural Approximation of Non-Linear Operations for Efficient Transformer Inference](https://mlsanity.com/paper/2112.02191/) (2021)

## Built on by (most cited first, 5 of 5)

- [QUARK: Quantization-Enabled Circuit Sharing for Transformer Acceleration by Exploiting Common Patterns in Nonlinear Operations](https://mlsanity.com/paper/2511.06767/) (2025)
- [Integrated electro-optic attention nonlinearities for transformers](https://mlsanity.com/paper/2604.09512/) (2026)
- [Hardware-Efficient Softmax and Layer Normalization with Guaranteed Normalization for Edge Devices](https://mlsanity.com/paper/2604.23647/) (2026)
- [Evolving Layer-Specific Scalar Functions for Hardware-Aware Transformer Adaptation](https://mlsanity.com/paper/2605.14047/) (2026)
- [MIVE: A Minimalist Integer Vector Engine for Softmax LayerNorm and RMSNorm Acceleration](https://mlsanity.com/paper/2606.17781/) (2026)

## Abstract (arXiv)

Transformers have shown remarkable performance in both natural language processing (NLP) and computer vision (CV) tasks. However, their real-time inference speed and efficiency are limited due to the inefficiency in Softmax and Layer Normalization (LayerNorm). Previous works based on function approximation suffer from inefficient implementation as they place emphasis on computation while disregarding memory overhead concerns. Moreover, such methods rely on retraining to compensate for approximation error which can be costly and inconvenient. In this paper, we present SOLE, a hardware-software co-design for Softmax and LayerNorm which is composed of E2Softmax and AILayerNorm. E2Softmax utilizes log2 quantization of exponent function and log-based division to approximate Softmax while AILayerNorm adopts low-precision statistic calculation. Compared with state-of-the-art designs, we achieve both low-precision calculation and low bit-width storage on Softmax and LayerNorm. Experiments show that SOLE maintains inference accuracy without retraining while offering orders of magnitude speedup and energy savings over GPU, achieving 3.04x, 3.86x energy-efficiency improvements and 2.82x, 3.32x area-efficiency improvements over prior state-of-the-art custom hardware for Softmax and LayerNorm, respectively.