Retracted on December 2, 2020 : On the Value of Look-Ahead in Competitive Online Convex Optimization

Abstract

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. CCS Concepts: • Theory of computation → Online algorithms; • Mathematics of computing → Convex optimization; • Networks → Network algorithms.

Online Optimization with Predictions and Non-convex Losses

“…In the case when costs are convex, Deterministic SFHC provides a competitive ratio of max 1 + η+η 2 2λ , η 2 without access to predictions and a competitive ratio of 1+O(1/w) in the case of predictions (Theorem 3). Thus, SFHC unifies two distinct lines of inquiry in the literature: how to design algorithms take advantage of predictions when they are available [21,22,35,37,46] and how to design algorithms that work when predictions are not available [12,23,27,28,38]. SFHC is the first algorithm to provide a constant-competitive guarantee in both settings.…”

“…The design of SFHC is inspired by the design of Averaging Fixed Horizon Control (AFHC) [37], which has served as the basis for many algorithms in this space, e.g., [22,46]. Like AFHC, Deterministic SFHC works by averaging the choices of w different subroutines.…”

“…In many applications the online learner has some information about future costs, and making use of these predictions of future costs is crucial. This has prompted a great deal of work involving the design of algorithms that leverage predictions, e.g., [10,21,22,24,36,46]. Most of this work considers models where the online learner has a prediction window of length w, i.e.…”

“…However, while it is possible to provide dimensionfree, constant competitive algorithms in settings where the learner has no information about future hitting cost functions, in most applications where SOCO is used it is possible to make accurate predictions of future costs. Such predictions are extremely valuable for the online learner and, as a result, a growing literature has considered situations where the learner has access to predictions of future costs, e.g., [9,21,22,24,35,37,46]. Most typically, this stream of work considers that the learner has access to perfect predictions of the next w costs, but in some cases it is possible to extend such results to noisy predictions as well, e.g., [21,22].…”

“…The first such result to appear was [37], which provides an algorithm that has a competitive ratio of 1 + O(1/w) when hitting costs are the operating cost for servers and movement costs are incurred by toggling into and out of a power-saving mode between timeslots. Since then, other results have followed, e.g., [9,21,22,35,46], and when more stringent requirements on the cost functions are considered it is possible to design algorithms whose competitive ratio converges to 1 exponentially quickly in w [35].…”