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Game of Coding With an Unknown Adversary

Hanzaleh Akbarinodehi, Parsa Moradi, Mohammad Ali Maddah-Ali

TL;DR

This work tackles learning near-optimal decision rules in a game of coding with an unknown adversary under incomplete information. It shows that the equilibrium link between acceptance probability and estimation error is invariant to utility functions, enabling a decentralized collector to converge to near-optimal strategies without knowing the adversary’s payoff. By framing the interaction as a Stackelberg game and leveraging a discretization plus concentration bounds, the authors prove polynomial-sample algorithms that achieve prescribed accuracy with probabilistic guarantees, and they further improve efficiency via dynamic candidate elimination. The results extend beyond the two-node baseline to larger networks and offer practical convergence guarantees for decentralized coding systems with rational, potentially malicious participants.

Abstract

Motivated by emerging decentralized applications, the \emph{game of coding} framework has been recently introduced to address scenarios where the adversary's control over coded symbols surpasses the fundamental limits of traditional coding theory. Still, the reward mechanism available in decentralized systems, motivates the adversary to act rationally. While the decoder, as the data collector (DC), has an acceptance and rejection mechanism, followed by an estimation module, the adversary aims to maximize its utility, as an increasing function of (1) the chance of acceptance (to increase the reward), and (2) estimation error. On the other hand, the decoder also adjusts its acceptance rule to maximize its own utility, as (1) an increasing function of the chance of acceptance (to keep the system functional), (2) decreasing function of the estimation error. Prior works within this framework rely on the assumption that the game is complete, that is, both the DC and the adversary are fully aware of each other's utility functions. However, in practice, the decoder is often unaware of the utility of the adversary. To address this limitation, we develop an algorithm enabling the DC to commit to a strategy that achieves within the vicinity of the equilibrium, without knowledge of the adversary's utility function. Our approach builds on an observation that at the equilibrium, the relationship between the probability of acceptance and the mean squared error (MSE) follows a predetermined curve independent of the specific utility functions of the players. By exploiting this invariant relationship, the DC can iteratively refine its strategy based on observable parameters, converging to a near-optimal solution. We provide theoretical guarantees on sample complexity and accuracy of the proposed scheme.

Game of Coding With an Unknown Adversary

TL;DR

This work tackles learning near-optimal decision rules in a game of coding with an unknown adversary under incomplete information. It shows that the equilibrium link between acceptance probability and estimation error is invariant to utility functions, enabling a decentralized collector to converge to near-optimal strategies without knowing the adversary’s payoff. By framing the interaction as a Stackelberg game and leveraging a discretization plus concentration bounds, the authors prove polynomial-sample algorithms that achieve prescribed accuracy with probabilistic guarantees, and they further improve efficiency via dynamic candidate elimination. The results extend beyond the two-node baseline to larger networks and offer practical convergence guarantees for decentralized coding systems with rational, potentially malicious participants.

Abstract

Motivated by emerging decentralized applications, the \emph{game of coding} framework has been recently introduced to address scenarios where the adversary's control over coded symbols surpasses the fundamental limits of traditional coding theory. Still, the reward mechanism available in decentralized systems, motivates the adversary to act rationally. While the decoder, as the data collector (DC), has an acceptance and rejection mechanism, followed by an estimation module, the adversary aims to maximize its utility, as an increasing function of (1) the chance of acceptance (to increase the reward), and (2) estimation error. On the other hand, the decoder also adjusts its acceptance rule to maximize its own utility, as (1) an increasing function of the chance of acceptance (to keep the system functional), (2) decreasing function of the estimation error. Prior works within this framework rely on the assumption that the game is complete, that is, both the DC and the adversary are fully aware of each other's utility functions. However, in practice, the decoder is often unaware of the utility of the adversary. To address this limitation, we develop an algorithm enabling the DC to commit to a strategy that achieves within the vicinity of the equilibrium, without knowledge of the adversary's utility function. Our approach builds on an observation that at the equilibrium, the relationship between the probability of acceptance and the mean squared error (MSE) follows a predetermined curve independent of the specific utility functions of the players. By exploiting this invariant relationship, the DC can iteratively refine its strategy based on observable parameters, converging to a near-optimal solution. We provide theoretical guarantees on sample complexity and accuracy of the proposed scheme.

Paper Structure

This paper contains 5 sections, 6 theorems, 18 equations, 1 figure, 3 algorithms.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Preliminaries
  4. Main Results
  5. Proof of Main Results

Key Result

Theorem 1

Assume that the function $\mathsf{U}(.)$, defined in def:u_function, is $(L,d)$-piecewise Lipschitz function. Additionally, for any $\eta \in [a,b]$ and $0 \leq \alpha \leq 1$, the function $\mathsf{Q}_\mathsf{DC}(\alpha, c_\eta(\alpha))$ is $\ell$-Lipschitz with respect to $\alpha$. For any $\delta

Figures (1)

  • Figure 1: Game of Coding involves with one honest and one adversarial node providing noisy versions of $\mathbf{u}$ to the DC, which decides whether to accept the inputs and estimate $\mathbf{u}$. While the honest node's noise distribution is known, the adversary selects its noise to optimize its utility function in response to the DC's acceptance strategy. The DC also strategically commits to an acceptance rule, optimizing its utility function. This paper addresses practical scenarios where the DC is unaware of the adversary’s utility function.

Theorems & Definitions (13)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Remark 1
  • Theorem 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 3 more