Homomorphically encrypted gradient descent algorithms for quadratic programming

TL;DR

This paper assesses the use of fully homomorphic encryption schemes, especially CKKS, for implementing gradient descent algorithms in quadratic programming, highlighting limitations, trade-offs, and feasibility.

Contribution

It evaluates homomorphic encryption schemes for gradient descent, proposes an implementation, and analyzes their applicability and limitations in quadratic programming.

Findings

CKKS scheme is suitable for encrypted gradient descent

Multiplication depth limits affect iterative algorithms

Feasibility of homomorphic gradient descent demonstrated

Abstract

In this paper, we evaluate the different fully homomorphic encryption schemes, propose an implementation, and numerically analyze the applicability of gradient descent algorithms to solve quadratic programming in a homomorphic encryption setup. The limit on the multiplication depth of homomorphic encryption circuits is a major challenge for iterative procedures such as gradient descent algorithms. Our analysis not only quantifies these limitations on prototype examples, thus serving as a benchmark for future investigations, but also highlights additional trade-offs like the ones pertaining the choice of gradient descent or accelerated gradient descent methods, opening the road for the use of homomorphic encryption techniques in iterative procedures widely used in optimization based control. In addition, we argue that, among the available homomorphic encryption schemes, the one adopted in this work, namely CKKS, is the only suitable scheme for implementing gradient descent algorithms. The choice of the appropriate step size is crucial to the convergence of the procedure. The paper shows firsthand the feasibility of homomorphically encrypted gradient descent algorithms.