CGF-Softmax: A Cumulant-Based Softmax Reformulation for Efficient Inference under Homomorphic Encryption

TL;DR

This paper introduces CGF-softmax, a reformulation of softmax using cumulant generating functions, enabling efficient and accurate encrypted inference with lower computational depth in homomorphic encryption.

Contribution

It presents a novel softmax reformulation that reduces multiplicative depth and computational cost in homomorphic encryption-based inference.

Findings

Achieves inference accuracy close to high-depth exact methods.

Reduces multiplicative depth significantly compared to traditional softmax.

Demonstrates efficiency on Vision Transformers and large language models.

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 costly division operation. In this paper, we propose CGF-softmax, which reformulates the softmax denominator through the cumulant generating function (CGF). By eliminating both homomorphic division and explicit maximum subtraction, this reformulation substantially reduces multiplicative depth while preserving key properties of softmax. Extensive experiments on Vision Transformers and large language models show that CGF-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.