The paper revisits the classical Arimoto-Blahut algorithm for computing the capacity of discrete memoryless channels. The key contributions are:
The sequence of approximations generated by the Arimoto-Blahut algorithm converges to an ε-optimal solution, for any constant ε > 0, as long as the current approximation is not ε-optimal.
The rate of convergence to an ε-optimal solution is upper bounded by O(log(m)/ct), for a constant c > 1, where m is the size of the input distribution. This implies at most O(log(log(m)/ε)) iterations to reach an ε-optimal solution.
If the set of optimal solutions has a strictly positive volume, the same convergence bounds apply for achieving an exact optimal solution.
The analysis shows significant improvements over the previously established upper bounds on the convergence rate and complexity of the Arimoto-Blahut algorithm. The key idea is to divide the convergence process into two phases: the initial phase to reach an ε-optimal solution and the subsequent phase to achieve the exact optimal solution. The authors demonstrate that the rate of convergence to an ε-optimal solution is always inverse exponential for any constant ε > 0, regardless of the channel characteristics.
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by Michail Faso... at arxiv.org 09-12-2024
https://arxiv.org/pdf/2407.06013.pdfDeeper Inquiries