Game of Coding for Vector-Valued Computations

TL;DR

This work extends the game of coding framework from scalar to vector-valued computations in $\mathbb{R}^N$, addressing secure decentralized computation under rational adversaries. It casts the problem as a Stackelberg game where a data collector selects a public acceptance threshold $\eta$ and adversaries choose noise distributions $g(\cdot)$, yielding a two-stage interaction between liveness (acceptance) and accuracy (MSE). The authors derive a scalar intermediary curve $c_{\eta}(\alpha)$ that characterizes the adversary's best responses, show that the entire equilibrium is captured by a 2D search over $\eta$ and $\alpha$, and provide explicit formulas $c_{\eta}(\alpha)=\frac{\tilde{\Psi}_N^{*}(\alpha)}{4\alpha}$ based on the upper concave envelope of geometric kernels tied to lens-like intersections of $N$-balls. They also construct worst-case adversarial noise distributions achieving the bound and illustrate the equilibrium across multiple dimensions, demonstrating that resilience results from the scalar setting extend to high dimensions. Overall, the paper offers a rigorous, dimension-robust foundation for secure, large-scale decentralized computing with no honest-majority assumption and provides practical algorithms to compute the equilibrium in vector-valued tasks.

Abstract

The game of coding is a new framework at the intersection of game theory and coding theory; designed to transcend the fundamental limitations of classical coding theory. While traditional coding theoretic schemes rely on a strict trust assumption, that honest nodes must outnumber adversarial ones to guarantee valid decoding, the game of coding leverages the economic rationality of actors to guarantee correctness and reliable decodability, even in the presence of an adversarial majority. This capability is paramount for emerging permissionless applications, particularly decentralized machine learning (DeML). However, prior investigations into the game of coding have been strictly confined to scalar computations, limiting their applicability to real world tasks where high dimensional data is the norm. In this paper, we bridge this gap by extending the framework to the general $N$-dimensional Euclidean space. We provide a rigorous problem formulation for vector valued computations and fully characterize the equilibrium strategies of the resulting high dimensional game. Our analysis demonstrates that the resilience properties established in the scalar setting are preserved in the vector regime, establishing a theoretical foundation for secure, large scale decentralized computing without honest majority assumptions.

Figures (21)

• Figure 1: System model for the $N$-Dimensional game of coding. The network consists of one honest node and one adversarial node. Each node reports a noisy version of the ground truth $\mathbf{U}$ to the DC. For the honest node, the noise $\mathbf{N}_h$ is uniformly distributed within $\mathcal{B}_N( \Delta)$, while for the adversarial node, the noise $\mathbf{N}_a$ follows an arbitrary distribution $g(\cdot)$ chosen by the adversary. Upon receiving the data, the DC decides whether to accept or reject the inputs based on a consistency threshold $\eta$. If accepted, the DC outputs an estimate of $\mathbf{U}$. In this game, the DC acts as the leader choosing $\eta$, and the adversary acts as the follower choosing $g(\cdot)$.
• Figure 2: The Pareto frontier $c_{\eta}(\alpha)$ and adversarial rationality. Point A (red) is inefficient compared to point B (black), while Point C (gray) lies in the unattainable region beyond the maximum possible error for $\alpha_1$.
• Figure 3: Geometric proof of Lemma \ref{['lemmaJSC']}. Point A represents a suboptimal response where $\beta < c_{\eta}(\alpha)$. By choosing $g'$ rather than $g^*$ to move to Point B on the boundary $\mathcal{C}_{\eta}$ (where the probability of acceptance is at least $\alpha$), the adversary increases their MSE and potentially their probability of acceptance, leading to strictly higher utility.
• Figure 4: Visual representation of the contradiction for $\mathcal{J}_{\eta} \subseteq \mathcal{K}_{\eta}$. Point A is a best response and thus lies on $\mathcal{C}_{\eta}$ by Lemma \ref{['lemmaJSC']}. If A is not in $\mathcal{K}_{\eta}$, there must exist a point B on the same boundary $\mathcal{C}_{\eta}$ that provides strictly higher utility, contradicting the optimality of A.
• Figure 5: The honest noise $\mathbf{N}_h$ is uniformly distributed on the blue ball of radius $\Delta$. Given any adversarial noise $\mathbf{n}_a$ with magnitude $z$, the condition $\|\mathbf{N}_h - \mathbf{n}_a\|_2 \le \eta\Delta$ is satisfied if $\mathbf{N}_h$ falls within the red ball. Due to spherical symmetry, the conditional probability $\Pr(\mathcal{A}_\eta \mid Z=z)$ depends only on the scalar distance $z$.

Theorems & Definitions (14)

• Theorem 1
• Theorem 2
• Remark 1
• Remark 2
• Remark 3
• Lemma 1
• proof
• Lemma 2
• Lemma 3
• Lemma 4