Efficient Softmax Reformulation for Homomorphic Encryption via Moment Generating Function

• URL: http://arxiv.org/abs/2602.01621v2
• Date: Mon, 09 Feb 2026 06:55:35 GMT
• Title: Efficient Softmax Reformulation for Homomorphic Encryption via Moment Generating Function
• Authors: Hanjun Park, Byeong-Seo Min, Jiheon Woo, Min-Wook Jeong, Jongho Shin, Yongwoo Lee, Young-Sik Kim, Yongjune Kim,
• Abstract summary: MGF-softmax is a novel softmax reformulation based on the moment generating function.<n>It provides an efficient and accurate approximation of softmax in encrypted inference.
• Score: 6.124433622051914
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Homomorphic encryption (HE) is a prominent framework for privacy-preserving machine learning, enabling inference directly on encrypted data. However, evaluating softmax, a core component of transformer architectures, remains particularly challenging in HE due to its multivariate structure, the large dynamic range induced by exponential functions, and the need for accurate division during normalization. In this paper, we propose MGF-softmax, a novel softmax reformulation based on the moment generating function (MGF) that replaces the softmax denominator with its moment-based counterpart. This reformulation substantially reduces multiplicative depth while preserving key properties of softmax and asymptotically converging to the exact softmax as the number of input tokens increases. Extensive experiments on Vision Transformers and large language models show that MGF-softmax provides an efficient and accurate approximation of softmax in encrypted inference. In particular, it achieves inference accuracy close to that of high-depth exact methods, while requiring substantially lower computational cost through reduced multiplicative depth.

Related papers

• Unpacking Softmax: How Temperature Drives Representation Collapse, Compression, and Generalization [15.458541841436967]
  We study the pivotal role of the softmax function in shaping the model's representation.<n>We introduce the concept of rank deficit bias - a phenomenon in which softmax-based deep networks find solutions of rank much lower than the number of classes.<n>We demonstrate how to exploit the softmax dynamics to learn compressed representations or to enhance their performance on out-of-distribution data.
  arXiv  Detail & Related papers  (2025-06-02T11:38:10Z)
• Self-Adjust Softmax [62.267367768385434]
  The softmax function is crucial in Transformer attention, which normalizes each row of the attention scores with summation to one.<n>We propose Self-Adjust Softmax (SA-Softmax) to address this issue by modifying $softmax(x)$ to $x cdot softmax(x)$ and its normalized variant $frac(x - min(x_min,0))max(0,x_max)-min(x_min,0) cdot softmax(x)$.
  arXiv  Detail & Related papers  (2025-02-25T15:07:40Z)
• Scalable-Softmax Is Superior for Attention [0.0]
  Transformer-based language models rely on Softmax to compute attention scores.<n>SSMax replaces Softmax in scenarios where the input vector size varies.<n>Models using SSMax not only achieve faster loss reduction during pretraining but also significantly improve performance in long contexts.
  arXiv  Detail & Related papers  (2025-01-31T18:55:35Z)
• r-softmax: Generalized Softmax with Controllable Sparsity Rate [11.39524236962986]
  We propose r-softmax, a modification of the softmax, outputting sparse probability distribution with controllable sparsity rate. We show on several multi-label datasets that r-softmax outperforms other sparse alternatives to softmax and is highly competitive with the original softmax.
  arXiv  Detail & Related papers  (2023-04-11T14:28:29Z)
• Convex Bounds on the Softmax Function with Applications to Robustness Verification [69.09991317119679]
  The softmax function is a ubiquitous component at the output of neural networks and increasingly in intermediate layers as well. This paper provides convex lower bounds and concave upper bounds on the softmax function, which are compatible with convex optimization formulations for characterizing neural networks and other ML models.
  arXiv  Detail & Related papers  (2023-03-03T05:07:02Z)
• Spectral Aware Softmax for Visible-Infrared Person Re-Identification [123.69049942659285]
  Visible-infrared person re-identification (VI-ReID) aims to match specific pedestrian images from different modalities. Existing methods still follow the softmax loss training paradigm, which is widely used in single-modality classification tasks. We propose the spectral-aware softmax (SA-Softmax) loss, which can fully explore the embedding space with the modality information.
  arXiv  Detail & Related papers  (2023-02-03T02:57:18Z)
• Softmax-free Linear Transformers [90.83157268265654]
  Vision transformers (ViTs) have pushed the state-of-the-art for visual perception tasks. Existing methods are either theoretically flawed or empirically ineffective for visual recognition. We propose a family of Softmax-Free Transformers (SOFT)
  arXiv  Detail & Related papers  (2022-07-05T03:08:27Z)
• Sparse-softmax: A Simpler and Faster Alternative Softmax Transformation [2.3813678058429626]
  The softmax function is widely used in artificial neural networks for the multiclass classification problems. In this paper, we provide an empirical study on a simple and concise softmax variant, namely sparse-softmax, to alleviate the problem that occurred in traditional softmax in terms of high-dimensional classification problems.
  arXiv  Detail & Related papers  (2021-12-23T09:53:38Z)
• SOFT: Softmax-free Transformer with Linear Complexity [112.9754491864247]
  Vision transformers (ViTs) have pushed the state-of-the-art for various visual recognition tasks by patch-wise image tokenization followed by self-attention. Various attempts on approximating the self-attention with linear complexity have been made in Natural Language Processing. We identify that their limitations are rooted in keeping the softmax self-attention during approximations. For the first time, a softmax-free transformer or SOFT is proposed.
  arXiv  Detail & Related papers  (2021-10-22T17:57:29Z)
• Optimal Approximation -- Smoothness Tradeoffs for Soft-Max Functions [73.33961743410876]
  A soft-max function has two main efficiency measures: approximation and smoothness. We identify the optimal approximation-smoothness tradeoffs for different measures of approximation and smoothness. This leads to novel soft-max functions, each of which is optimal for a different application.
  arXiv  Detail & Related papers  (2020-10-22T05:19:58Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.