Online Convex Optimization of a Multi-task Fuzzy Rule-based Evolving System

This paper extends the recently conceived learning mechanism called EVeP (Extreme Value evolving Predictor), an evolving fuzzy-rule-based predictor characterized by innovative procedures to define the antecedent and consequent parts of the existing fuzzy rules. In EVeP, information granules are recursively updated and associated with Weibull distributions, a generalization of Gaussian distributions which incorporates more robust statistics to establish the region of influence of each fuzzy rule. Shared information from all the rules, in a multi-task formulation, is adopted to set the consequent parameters in EVeP. Given that the multi-task formulation is solved using batch learning and gradient descent, the computational cost per iteration tends to be high, being a concern in practical applications. Therefore, here the multi-task framework at the consequent part of the rules was revised to incorporate online convex optimization, given rise to EVeP_OCO. Now, antecedent and consequent parts of the rules are updated in a fully recursive way, with a clear reduction in the computational burden per iteration, particularly when the worst case scenarios are considered: the cost per iteration depends on the current number of rules to be updated. The case studies are composed of a variety of benchmark time series prediction problems. They demonstrate the significant gain in terms of computational cost per iteration, with an admissible reduction in performance by replacing a batch multi-task learning procedure by an online counterpart.