Degree-Constrained Interval Optimization for Minimax Polynomial Approximation in Homomorphic Encryption

2026-07-08

AI Summary

This work addresses the challenge of approximating non-polynomial activation functions in homomorphic encryption, where fixed-degree polynomial approximations are required and the choice of approximation interval critically affects both approximation error and robustness to large inputs. The authors formulate interval selection as a distribution-aware optimization problem and introduce an analytically tractable surrogate objective that jointly captures the minimax error within the interval and the clipping error outside it. They establish a theoretical connection between this objective and feasible homomorphic constructions. By integrating domain extension functions (DEFs), domain extension polynomials (DEPs), the Remez algorithm, and error bound analysis, the proposed method optimizes the approximation interval based on the pre-activation input distribution for a given polynomial degree, significantly reducing overall mean squared error and enhancing numerical stability in homomorphic inference.

Abstract

Homomorphic encryption (HE) enables privacy-preserving inference under arithmetic constraints that restrict encrypted evaluation to additions and multiplications. As a result, non-polynomial activation functions must be replaced by polynomial approximations. Among polynomial approximation methods, minimax approximation, typically computed by the Remez algorithm, is a standard approach because it minimizes the maximum approximation error over a given design interval. For minimax polynomial design, the approximation interval is a critical hyperparameter: a wider interval improves robustness to large-magnitude inputs while increasing the minimax approximation error under a fixed degree budget. In this paper, we formulate this trade-off as a distribution-aware interval optimization problem, where the approximation interval is chosen to minimize the mean-squared error (MSE) with respect to the pre-activation distribution of interest. To effectively control outside-interval inputs, we combine minimax polynomials with domain extension functions (DEFs) and their HE-realizable polynomial counterparts, domain extension polynomials (DEPs), which approximate a clipping operation outside the design interval and thereby suppress uncontrolled polynomial extrapolation. We first derive an analytically tractable DEF-based proxy objective that captures the trade-off between within-interval minimax approximation error and outside-interval clipping error. We then connect this idealized objective to HE-realizable DEP constructions through an implementation-error decomposition with an accompanying upper bound.

Problem

• homomorphic encryption
• minimax polynomial approximation
• degree-constrained optimization
• approximation interval
• domain extension

Innovation

• degree-constrained optimization
• minimax polynomial approximation
• homomorphic encryption
• domain extension polynomials
• interval optimization