On the Value of Look-Ahead in Competitive Online Convex Optimization

Although using look-ahead information is known to improve the competitive ratios of online convex optimization (OCO) problems with switching costs, the competitive ratios obtained from existing results often depend on the cost coefficients of the problem, and can potentially be large. In this paper, we propose new online algorithms that can utilize look-ahead to achieve much lower competitive ratios for OCO problems with switching costs and hard constraints. For the perfect look-ahead case where the algorithm is provided with the exact inputs in a future look-ahead window of size K, we design an Averaging Regularized Moving Horizon Control (ARMHC) algorithm that can achieve a competitive ratio of K+1/K. To the best of our knowledge, ARMHC is the first to attain a low competitive ratio that is independent of either the coefficients of the switching costs and service costs, or the upper and lower bounds of the inputs. Then, for the case when the future look-ahead has errors, we develop a Weighting Regularized Moving Horizon Control (WRMHC) algorithm that carefully weights the decisions inside the look-ahead window based on the accuracy of the look-ahead information. As a result, WRMHC also achieves a low competitive ratio that is independent of the cost coefficients, even with uncertain hard constraints. Finally, our analysis extends online primal-dual analysis to the case with look-ahead by introducing a novel "re-stitching" idea, which is of independent interest.