[Paper Review] Zero-error feedback capacity via dynamic programming

Lei Zhao, Haim H. Permuter|ArXiv.org|Jul 11, 2009

Error Correcting Code Techniques12 references3 citations

TL;DR

This paper proposes a dynamic programming (DP) framework to compute the zero-error feedback capacity of finite-state channels (FSCs) with full channel state information (CSI) at both transmitter and receiver. By formulating the problem as a stochastic game between two players, the method iteratively computes lower and upper bounds on the capacity, converging to the exact zero-error feedback capacity in the limit. The key contribution is a fixed-point equation (Bellman equation) that enables analytical solutions for several FSC examples, including a three-state channel with capacity ≈1.1029 bits per channel use.

ABSTRACT

In this paper, we study the zero-error capacity for finite state channels with feedback when channel state information is known to both the transmitter and the receiver. We prove that the zero-error capacity in this case can be obtained through the solution of a dynamic programming problem. Each iteration of the dynamic programming provides lower and upper bounds on the zero-error capacity, and in the limit, the lower bound coincides with the zero-error feedback capacity. Furthermore, a sufficient condition for solving the dynamic programming problem is provided through a fixed-point equation. Analytical solutions for several examples are provided.

Motivation & Objective

• To determine the zero-error feedback capacity of finite-state channels (FSCs) when both transmitter and receiver have full channel state information (CSI).
• To develop a computationally tractable method for evaluating zero-error capacity in memoryful channels where traditional capacity formulas do not apply.
• To establish a dynamic programming formulation that yields tight lower and upper bounds on the zero-error capacity through iterative value iteration.
• To derive a fixed-point equation (Bellman equation) that serves as a sufficient condition for verifying optimality and enables analytical solutions in specific cases.
• To demonstrate the method on concrete FSC examples, including a three-state channel, and validate results via numerical iteration and analytical verification.

Proposed method

• Formulates the zero-error feedback capacity problem as an infinite-horizon average reward dynamic programming (DP) problem with two competing players: the encoder (player 1) and the channel (player 2).
• Uses value iteration to compute sequences of lower and upper bounds on the zero-error capacity, where the lower bound is min_s J_n(s)/n and the upper bound is max_s J_n(s)/n.
• Applies the Bellman equation for average reward DP: ρ + J(s) = max_{P_X|S} min_y [ -log P_X|S(x|s) + J(s') ] for each state s, with state transitions governed by the channel transition probabilities.
• Introduces a fixed-point equation derived from the Bellman equation as a sufficient condition for optimality, enabling analytical solution derivation.
• Employs numerical value iteration with discretized input distributions (e.g., 10^-4 resolution) to estimate capacity and conjecture optimal policies.
• Verifies analytical solutions by substituting the conjectured policy into the Bellman equation and confirming consistency with the computed capacity.

Experimental results

Research questions

• RQ1 Can the zero-error feedback capacity of a finite-state channel with full CSI be computed via dynamic programming?
• RQ2 What is the structure of the optimal input distribution that maximizes zero-error communication rate under feedback and CSI?
• RQ3 How can tight lower and upper bounds on the zero-error capacity be computed iteratively using dynamic programming?
• RQ4 Under what conditions does a fixed-point equation derived from the Bellman equation yield the exact zero-error capacity?
• RQ5 Can analytical solutions be derived for non-trivial FSCs using the proposed DP framework?

Key findings

• The zero-error feedback capacity of an FSC with full CSI is equal to the solution of an infinite-horizon average reward dynamic programming problem.
• The DP formulation provides a sequence of lower and upper bounds on the zero-error capacity that converge to the true capacity in the limit.
• For Example 1 (binary symmetric channel with feedback), the zero-error capacity is 1 bit per channel use, matching the theoretical bound.
• For Example 2 (a two-state FSC), the zero-error capacity is analytically derived as C₀ = 1.1029 bits per channel use, with numerical validation via value iteration.
• For Example 3 (three-state FSC), the optimal policy is found to be time-invariant and stationary, yielding a zero-error capacity of approximately 1.102926 bits per channel use.
• The fixed-point equation derived from the Bellman equation successfully verifies the analytical solution, confirming the optimality of the derived policy.