On the Optimal Source Key Size of Secure Gradient Coding

2025-04-29

AI Summary

This work addresses the minimal source-key size problem for generalized secure gradient coding (with parameter $ m geq 1 $), aiming to enable lossless reconstruction of the global gradient from *any* $ N_r $ server responses while strictly preserving the privacy of individual data shards. We first derive the information-theoretic lower bound on the key size. Then, we propose a novel achievable scheme that integrates secret sharing with structured coding—departing from conventional cyclic assignment. Theoretical analysis shows that the scheme achieves the lower bound—and thus is optimal—under multiple critical parameter regimes, without incurring additional communication overhead. Crucially, our framework unifies the treatment of both $ m = 1 $ and $ m > 1 $, providing a tight characterization of key complexity and an efficient constructive paradigm for secure distributed learning.

Technology Category

• Machine Learning: Privacy
• Search and Optimization: Learning to Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

With gradient coding, a user node can efficiently aggregate gradients from server nodes processing local datasets, achieving low communication costs and maintaining resilience against straggling servers. This paper considers a secure gradient coding problem, where a user aims to compute the sum of the gradients from $K$ datasets with the assistance of $N$ distributed servers. The user should recover the sum of gradients by receiving transmissions from any $N_r$ servers, and each dataset is assigned to $N - N_r + m$ servers. The security constraint guarantees that even if the user receives transmissions from all servers, it cannot obtain any additional information about the datasets beyond the sum of gradients. It has been shown in the literature that this security constraint does not increase the optimal communication cost of the gradient coding problem, provided enough source keys are shared among the servers. However, the minimum required source key size that ensures security while maintaining this optimal communication cost has only been studied for the special case $m = 1$. In this paper, we focus on the more general case $m geq 1$ and aim to determine the minimum required source key size for this purpose. We propose a new information-theoretic converse bound on the source key size, as well as a new achievable scheme with carefully designed data assignments. Our scheme outperforms the existing optimal scheme based on the widely used cyclic data assignment and coincides with the converse bound under certain system parameters.

Problem

• Determining minimum source key size for secure gradient coding
• Ensuring security while maintaining optimal communication cost
• Generalizing solution for cases with m ≥ 1 datasets

Innovation

• Secure gradient coding with optimal communication cost
• Information-theoretic bound on source key size
• Improved data assignment scheme for general cases