The DOI matches its Crossref record; 22 of the 23 references it cites were found in a registry or an index.
Softermax: Hardware/Software Co-Design of an Efficient Softmax for Transformers
Open accessAbstract
Transformers have transformed the field of natural language processing. Their superior performance is largely attributed to the use of stacked “self-attention” layers, each of which consists of matrix multiplies as well as softmax operations. As a result, unlike other neural networks, the softmax operation accounts for a significant fraction of the total run-time of Transformers. To address this, we propose Softermax, a hardware-friendly softmax design. Softermax consists of base replacement, low-precision softmax computations, and an online normalization calculation. We show Softermax results in 2.35x the energy efficiency at 0.90x the size of a comparable baseline, with negligible impact on network accuracy.
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![Fig. 4 (a) The Unnormed Softmax Unit determines the local max, performs the power of 2 calculation using the current max, and accumulates the denominator. (b) The Normalization Unit performs the renormalization of the numerator, as well as the final division of the numerator by the accumulated sum. (c) In an example accelerator [17], the Unnormed Softmax can be integrated into the post-processing vector unit on a per PE basis, while the Normalization Unit can be shared across multiple PEs and integrated between the PEs and the Global Buffer](/paper-figures/46f03c2ffabd423f83b1ca0c67587f1a/softermax-hardware-software-co-design-of-an-efficient-softmax-for-transformers-figure-4.png)
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“Non-linear operations like Softmax, LayerNorm, and GELU, though less prevalent, present computational challenges on conventional hardware due to their complexity, potentially restricting the enhancement of end-to-end transformer inference [31].”
From Model Quantization and Hardware Acceleration for Vision Transformers: A Comprehensive Survey (2024) · cites this paper as [31]
Experimental — accuracy not yet measuredThe source backs part of the claim, not all of it — or backs it for a different population, condition or comparison.
In the citing paper
“A large portion of the inference time in ViT models is consumed by floating-point non-linear activations, normalizations, and the associated quantization and dequantization operations [31].”
From Model Quantization and Hardware Acceleration for Vision Transformers: A Comprehensive Survey (2024) · cites this paper as [31]
Experimental — accuracy not yet measuredThe source backs part of the claim, not all of it — or backs it for a different population, condition or comparison.
In the citing paper
“In addition, various approximation methods that hold floating-point arithmetic are presented [36, 50]; while they lower certain computational costs, they cannot meet the demands of integer arithmetic.”
From I-ViT: Integer-only Quantization for Efficient Vision Transformer Inference (2023) · cites this paper as [36]
Experimental — accuracy not yet measuredThe source backs part of the claim, not all of it — or backs it for a different population, condition or comparison.
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“Due to the non-linearity, Softmax cannot follow the dyadic arithmetic pipeline discussed above, and the exponential arithmetic in Eq. 11 is typically unsupported by integer-only logic units [36].”
From I-ViT: Integer-only Quantization for Efficient Vision Transformer Inference (2023) · cites this paper as [36]
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Citations
Cites
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In-Datacenter Performance Analysis of a Tensor Processing Unit
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Megatron-LM: Training Multi-Billion Parameter Language Models Using Model Parallelism
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 3: Accelerating attention mechanisms in neural networks with approximation
Ham, T. J., Jung, S. J., Kim, S., Oh, Y. H., Park, Y., Song, Y., Park, J.-H., Lee, S., Park, K., Lee, J. W., et al. (2020) · 2020 IEEE International Symposium on High Performance Computer Architecture (HPCA) · doi:10.1109/hpca47549.2020.00035 · Cited by 242
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Self-citationEfficient Precision-Adjustable Architecture for Softmax Function in Deep Learning
Zhu, D., Lu, S., Wang, M., Lin, J., & Wang, Z. (2020) · IEEE Transactions on Circuits and Systems II: Express Briefs · doi:10.1109/tcsii.2020.3002564 · Cited by 115
Design and Implementation of an Approximate Softmax Layer for Deep Neural Networks
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Fully Quantized Transformer for Machine Translation
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Online normalizer calculation for softmax
Milakov, M., & Gimelshein, N. (2018) · arXiv · doi:10.48550/arxiv.1805.02867 · Cited by 27 · cited 2×
“Prior work [18] has proposed an “online normalizer” calculation, where the max is calculated continuously along with the normalization value ( i.e , the summation for the denominator).”
BackgroundOptimizing Transformers with Approximate Computing for Faster, Smaller and more Accurate NLP ModelsPreprint
Nagarajan, A., Sen, S., Stevens, J. R., & Raghunathan, A. (2021) · arXiv (Cornell University) · Cited by 4
Self-citation
Research works printed but not linked: 2
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Cited by
Sorted by citations
Yenduri, G., Ramalingam, M., Selvi, G. C., Supriya, Y., Srivastava, G., Maddikunta, P. K. R., Raj, G. D., Jhaveri, R. H., Prabadevi, B., Wang, W., et al. (2024) · IEEE Access · doi:10.1109/access.2024.3389497 · Cited by 916
I-ViT: Integer-only Quantization for Efficient Vision Transformer Inference
Li, Z., & Gu, Q. (2023) · 2023 IEEE/CVF International Conference on Computer Vision (ICCV) · doi:10.1109/iccv51070.2023.01565 · Cited by 141
“11 is typically unsupported by integer-only logic units [36].”
BackgroundMethodX-Former: In-Memory Acceleration of Transformers
Sridharan, S., Stevens, J. R., Roy, K., & Raghunathan, A. (2023) · IEEE Transactions on Very Large Scale Integration (VLSI) Systems · doi:10.1109/tvlsi.2023.3282046 · Cited by 74
Self-citationA 95.6-TOPS/W Deep Learning Inference Accelerator With Per-Vector Scaled 4-bit Quantization in 5 nm
Keller, B., Venkatesan, R., Dai, S. X., Tell, S. G., Zimmer, B., Sakr, C., Dally, W. J., Gray, C. T., & Khailany, B. (2023) · IEEE Journal of Solid-State Circuits · doi:10.1109/jssc.2023.3234893 · Cited by 66
Self-citationSaha, S., & Xu, L. (2025) · Neurocomputing · doi:10.1016/j.neucom.2025.130417 · Cited by 85
Metadata
- Authors
- Stevens, J. R., Venkatesan, R., Dai, S., Khailany, B., & Raghunathan, A.
- Affiliations
- Purdue University West Lafayette (US)
- Nvidia (United Kingdom) (GB)
- Venue
- 2021 58th ACM/IEEE Design Automation Conference (DAC)
- Publisher
- IEEE
- Published
- 2021-12-05
- Pages
- 469-474
- ISSN
- 0738-100X (ISSN-L)
- OpenAlex
- W3137649401
- Cited by
- 148
- References
- 23
- Citations by year
- 2026: 22 · 2025: 68 · 2024: 35 · 2023: 16 · 2022: 5 · 2021: 2
- FWCI
- 4.74
- Type
- conference paper
- Language
- en
- Open access
- green
- arxiv.org
- Licence
- Locations
- ACM/IEEE Design Automation Conference (DAC) (publishedVersion)
- arXiv (Cornell University) (submittedVersion, cc-by-nc-nd)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Computer Science › Computer Vision and Pattern Recognition › Advanced Neural Network Applications
- Physical Sciences › Computer Science › Hardware and Architecture › Parallel Computing and Optimization Techniques
- Physical Sciences › Engineering › Electrical and Electronic Engineering › Low-power high-performance VLSI design
- Keywords
- softmax, transformer models, hardware-software co-design, self-attention, low-precision arithmetic, energy efficiency, neural network acceleration
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