Game of Coding for Vector-Valued Computations

TL;DR

This paper extends the game of coding framework to vector-valued computations, analyzing a two-repetition code with rational adversaries to establish foundational equilibrium results for decentralized machine learning applications.

Contribution

It introduces the first analysis of vector-valued coding games, characterizing equilibrium and optimal strategies in a multi-dimensional setting with rational adversaries.

Findings

Fully characterized equilibrium strategies for the two-repetition code.

Extended the game of coding framework from scalar to vector spaces.

Provided foundational results for future multi-dimensional coding game analyses.

Abstract

Traditional coding theory guarantees valid decoding only if a minority of symbols are adversarially manipulated. In contrast, the game of coding framework ensures reliable decoding, even in the presence of an adversarial majority. This formulation is motivated by emerging permissionless applications, particularly decentralized machine learning (DeML), where computation tasks are outsourced to external volunteer nodes that are predominantly rational and reward-seeking. Prior investigations have analyzed the game of coding in the scalar setting. Since the results of most major computations in machine learning are vectors (e.g., computing the gradient of the loss for a machine learning model), we extend the framework in this paper to the general multi-dimensional Euclidean space. As a first, yet fundamental step, in this paper, we study a two-repetition code in which at least one node is controlled by a rational adversary, and we fully characterize the equilibrium and the optimal strategies of the players. Similar to the scalar case, this result serves as a cornerstone for addressing more general scenarios.