Minimax Theorems for Finite Blocklength Lossy Joint Source-Channel Coding over an AVC

TL;DR

This paper establishes approximate and second order minimax theorems for finite blocklength lossy joint source-channel coding over an arbitrarily varying channel, modeling the problem as a zero-sum game with randomized strategies.

Contribution

It introduces a novel game-theoretic framework for finite blocklength coding over AVCs and proves asymptotic minimax theorems including second order dispersion bounds.

Findings

Minimax and maximin values approach each other asymptotically.

Values approach zero or one depending on the rate relative to a threshold.

Second order bounds characterize the convergence at the threshold.

Abstract

Motivated by applications in the security of cyber-physical systems, we pose the finite blocklength communication problem in the presence of a jammer as a zero-sum game between the encoder-decoder team and the jammer, by allowing the communicating team as well as the jammer only locally randomized strategies. The communicating team's problem is non-convex under locally randomized codes, and hence, in general, a minimax theorem need not hold for this game. However, we show that approximate minimax theorems hold in the sense that the minimax and maximin values of the game approach each other asymptotically. In particular, for rates strictly below a critical threshold, both the minimax and maximin values approach zero, and for rates strictly above it, they both approach unity. We then show a second order minimax theorem, i.e., for rates exactly approaching the threshold with along a specific scaling, the minimax and maximin values approach the same constant value, that is neither zero nor one. Critical to these results is our derivation of finite blocklength bounds on the minimax and maximin values of the game and our derivation of second order dispersion-based bounds.