The Duality Upper Bound for Finite-State Channels with Feedback

TL;DR

The paper addresses computing the feedback capacity of finite-state channels by extending the dual capacity upper bound to directed information and introducing graph-based test distributions on $Q$-graphs. For fixed graph structures, the bound is cast as an MDP with state $z_t=(\beta_t,q_t)$, and simplifies to finite-state MDPs for unifilar FSCs or finite-memory channels, enabling DP/RL methods to yield analytical or near-analytical upper bounds. The authors apply the framework to Noisy Output is the State (NOST) and Noisy Ising (N-Ising) channels, deriving a closed-form capacity for a broad NOST family and novel upper bounds for N-Ising, with RL-assisted exploration producing near-tight bounds. The approach scales to large alphabets and provides a practical toolkit for obtaining bounds on FSCs with memory, combining duality, $Q$-graphs, and DP/RL techniques. Overall, this work offers a tractable methodology to bound FSC feedback capacity and delivers concrete results and bounds for important channel families.

Abstract

This paper investigates the capacity of finite-state channels (FSCs) with feedback. We derive an upper bound on the feedback capacity of FSCs by extending the duality upper bound method from mutual information to the case of directed information. The upper bound is expressed as a multi-letter expression that depends on a test distribution on the sequence of channel outputs. For any FSC, we show that if the test distribution is structured on a $Q$-graph, the upper bound can be formulated as a Markov decision process (MDP) whose state being a belief on the channel state. In the case of FSCs and states that are either unifilar or have a finite memory, the MDP state simplifies to take values in a finite set. Consequently, the MDP consists of a finite number of states, actions, and disturbances. This finite nature of the MDP is of significant importance, as it ensures that dynamic programming algorithms can solve the associated Bellman equation to establish analytical upper bounds, even for channels with large alphabets. We demonstrate the simplicity of computing bounds by establishing the capacity of a broad family of Noisy Output is the State (NOST) channels as a simple closed-form analytical expression. Furthermore, we introduce novel, nearly optimal analytical upper bounds on the capacity of the Noisy Ising channel.

Figures (5)

• Figure 1: Finite-state channel with feedback.
• Figure 2: A $1$st-order Markov $Q$-graph for channel output alphabet $\mathcal{Y}=\{0,1\}$.
• Figure 3: The feedback capacity of the NOST channel as a function of the state parameter $\epsilon$.
• Figure 4: Bounds on the capacity of the N-Ising channel. The upper bound is obtained by a single $Q$-graph of size $4$ (Theorem \ref{['Th: N-Ising_Q4']}). The lower bound is evaluated with two different $Q$-graphs of size $10$ and $12$.
• Figure 5: Histograms of the MDP states visited under an estimated optimal policy learned by RL in the case of the N-Ising channel. The left figure presents the histogram of MDP states when the channel state parameter is $\epsilon=0.1$, while the right figure presents the histogram when the channel state parameter is $\epsilon=0.4$.

Theorems & Definitions (28)

• Theorem 1: Kim08_feedback_directed, Th. 1
• Theorem 2: Dual_capacity, Th. $8.4$
• Theorem 3: Duality UB for Directed Information
• proof : Proof of Theorem \ref{['Th: DB_DI']}
• Remark 1
• Theorem 4: Duality UB for FSCs using $Q$-graphs
• Theorem 5
• Theorem 6: Bellman equation, Arapos93_average_cose_survey
• Theorem 7
• Remark 2