Re-adapting the Regularization of Weights for Non-stationary Regression

The goal of a learner in standard online learning is to have the cumulative loss not much larger compared with the best-performing prediction-function from some fixed class. Numerous algorithms were shown to have this gap arbitrarily close to zero compared with the best function that is chosen off-line. Nevertheless, many real-world applications (such as adaptive filtering) are non-stationary in nature and the best prediction function may not be fixed but drift over time. We introduce a new algorithm for regression that uses per-feature-learning rate and provide a regret bound with respect to the best sequence of functions with drift. We show that as long as the cumulative drift is sub-linear in the length of the sequence our algorithm suffers a regret that is sub-linear as well. We also sketch an algorithm that achieves the best of the two worlds: in the stationary settings has log(T) regret, while in the non-stationary settings has sub-linear regret. Simulations demonstrate the usefulness of our algorithm compared with other state-of-the-art approaches.