Nonstationary online convex optimization with multiple predictions

Abstract

This work focuses on dynamic regret for non-stationary online convex optimization with full information. State-of-the-art analysis shows that Implicit Online Mirror Descent (IOMD) combined with Hedge achieves an O˜(min⁡{VT,(1+PT)T}) dynamic regret, where VT denotes the temporal variability of the loss functions and PT measures the path length reflecting the non-stationarity of the comparator sequence, and Optimistic IOMD (OptIOMD) enjoys the dynamic regret of O(min⁡{VT′,(1+P)T}), where VT′ denotes the cumulative distance from the loss functions to an arbitrary predictor sequence, and P is an upper bound of the path-length PT. In order to further suppress dynamic regret, we propose an algorithm named Hedge-OptIOMD, which achieves an O˜(min⁡{minj∈1:n⁡VTj,(1+PT)T}) dynamic regret via multiple predictors, where VTj represents the cumulative distance from the loss functions to the j-th predictor sequence. We also verify the advantages of Hedge-OptIOMD through numerical experiments.