A Novel Family of Adaptive Filtering Algorithms Based on the Logarithmic Cost

We introduce a novel family of adaptive filtering algorithms based on a relative logarithmic cost inspired by the “competitive methods” from the online learning literature. The competitive or regret based approaches stabilize or improve the convergence performance of adaptive algorithms through relative cost functions. The new family elegantly and gradually adjusts the conventional cost functions in its optimization based on the error amount. We introduce important members of this family of algorithms such as the least mean logarithmic square (LMLS) and least logarithmic absolute difference (LLAD) algorithms. However, our approach and analysis are generic such that they cover other well-known cost functions as described in the paper. The LMLS algorithm achieves comparable convergence performance with the least mean fourth (LMF) algorithm and enhances the stability performance significantly. The LLAD and least mean square (LMS) algorithms demonstrate similar convergence performance in impulse-free noise environments while the LLAD algorithm is robust against impulsive interferences and outperforms the sign algorithm (SA). We analyze the transient, steady-state and tracking performance of the introduced algorithms and demonstrate the match of the theoretical analyses and simulation results. We show the enhanced stability performance of the LMLS algorithm and analyze the robustness of the LLAD algorithm against impulsive interferences. Finally, we demonstrate the performance of our algorithms in different scenarios through numerical examples.