Hyper-ellipsoidal conditions in XCS: rotation, linear approximation, and solution structure

The learning classifier system XCS is an iterative rule-learning system that evolves rule structures based on gradient-based prediction and rule quality estimates. Besides classification and reinforcement learning tasks, XCS was applied as an effective function approximator. Hereby, XCS learns space partitions to enable a maximally accurate and general function approximation. Recently, the function approximation approach was improved by replacing (1) hyperrectangular conditions with hyper-ellipsoids and (2) iterative linear approximation with the recursive least squares method. This paper combines the two approaches assessing the usefulness of each. The evolutionary process is further improved by changing the mutation operator implementing an angular mutation that rotates ellipsoidal structures explicitly. Both enhancements improve XCS performance in various non-linear functions. We also analyze the evolving ellipsoidal structures confirming that XCS stretches and rotates the evolving ellipsoids according to the shape of the underlying function. The results confirm that improvements in both the evolutionary approach and the gradient approach can result in significantly better performance.