The source coding game with a cheating switcher

Motivated by the lossy compression of an active-vision video stream, we consider the problem of finding the rate-distortion function of an arbitrarily varying source (AVS) composed of a finite number of subsources with known distributions. Berger’s paper `The Source Coding Game’, \emph{IEEE Trans. Inform. Theory}, 1971, solves this problem under the condition that the adversary is allowed only strictly causal access to the subsource realizations. We consider the case when the adversary has access to the subsource realizations non-causally. Using the type-covering lemma, this new rate-distortion function is determined to be the maximum of the IID rate-distortion function over a set of source distributions attainable by the adversary. We then extend the results to allow for partial or noisy observations of subsource realizations. We further explore the model by attempting to find the rate-distortion function when the adversary is actually helpful. Finally, a bound is developed on the uniform continuity of the IID rate-distortion function for finite-alphabet sources. The bound is used to give a sufficient number of distributions that need to be sampled to compute the rate-distortion function of an AVS to within a certain accuracy. The bound is also used to give a rate of convergence for the estimate of the rate-distortion function for an unknown IID finite-alphabet source .

💡 Research Summary

The paper revisits the classic “source coding game” introduced by Berger in 1971, but it does so under a much stronger adversarial capability that is motivated by modern active‑vision systems. In the original formulation the switcher (the adversary that chooses which sub‑source supplies each symbol) is strictly causal: it can only use past source symbols when deciding the current switch. Consequently the rate‑distortion function of the arbitrarily varying source (AVS) is obtained by minimizing over all strictly causal strategies. In many practical video‑sensor networks, however, the controller has access to the current frame of each camera or sensor before deciding which stream to forward. This paper therefore assumes a non‑causal (full‑look‑ahead) switcher that knows the entire block of sub‑source realizations in advance.

The authors first formalize the model. There are (K) sub‑sources, each with a known i.i.d. distribution (P_k) on a finite alphabet. The switcher selects, for each time instant (i), a sub‑source index (S_i) based on the full realization (\mathbf{x}^n) of all sub‑sources over a block of length (n). The resulting output sequence (\mathbf{y}^n) therefore follows a distribution (q) that is a convex combination of the (P_k)’s: (q = \sum_{k=1}^K \alpha_k P_k) with (\alpha_k\ge 0) and (\sum_k \alpha_k = 1). The set of all such attainable distributions is denoted (\mathcal{Q}).

Using the type‑covering lemma, the paper shows that for any target distortion (D) and any (q\in\mathcal{Q}) there exists a code of rate (R) that can cover the typical set of sequences generated by (q) provided (R) exceeds the ordinary i.i.d. rate‑distortion function (R_{\text{IID}}(D;q)). Since the switcher can force the source to behave according to any (q\in\mathcal{Q}), the worst‑case (i.e., the adversarial) rate‑distortion function of the AVS is the maximum of these i.i.d. functions: \