Learning-Augmented Weighted Paging

Abstract

We consider a natural semi-online model for weighted paging, where at any time the algorithm is given predictions, possibly with errors, about the next arrival of each page. The model is inspired by Belady's classic optimal offline algorithm for unweighted paging, and extends the recently studied model for learning-augmented paging [45,50,52] to the weighted setting.For the case of perfect predictions, we provide an -competitive deterministic and an O(log )-competitive randomized algorithm, where is the number of distinct weight classes. Both these bounds are tight, and imply an O(log W)-and O(log log W)-competitive ratio, respectively, when the page weights lie between 1 and W. Previously, it was not known how to use these predictions in the weighted setting and only bounds of k and O(log k) were known, where k is the cache size. Our results also generalize to the interleaved paging setting and to the case of imperfect predictions, with the competitive ratios degrading smoothly from O( ) and O(log ) to O(k) and O(log k), respectively, as the prediction error increases.Our results are based on several insights on structural properties of Belady's algorithm and the sequence of page arrival predictions, and novel potential functions that incorporate these predictions. For the case of unweighted paging, the results imply a very simple potential function based proof of the optimality of Belady's algorithm, which may be of independent interest.

Discrete-Convex-Analysis-Based Framework for Warm-Starting Algorithms with Predictions

“…Algorithms with predictions. Many recent studies [36,41,4,34,43,2,5] improved competitive ratios of online algorithms with predictions. Dinitz et al [17] proposed to warm-start algorithms with predictions.…”

“…Since such instances arising in the same domain often have some tendencies, using predictions made from past instances to improve algorithms' performance is a natural and promising idea. A recent line of work [36,41,4,34,43,2,5] successfully combined online algorithms with predictions and showed that those algorithms perform provably better than known worst-case bounds if predictions are good while enjoying worst-case guarantees even if predictions are poor. See [37] for a survey.…”

Shortest Paths without a Map, but with an Entropic Regularizer

“…The online version has applications to the design and analysis of hybrid algorithms. In particular, the disjoint paths case has applications in derandomizing online algorithms [20], in the design of divide-and-conquer online algorithms [19,2], and in the design of advice and learning augmented online algorithms [25,1,5]. In this context, Kao et al [23] resolve exactly the randomized competitive ratio of width 2 layered graph traversal: it is roughly 4.59112, precisely the solution for in the equation ln( − 1) = −1 ; see also [13].…”

“…Width layered graph traversal includes as a special case metrical task systems in -point metric spaces. 5 There is a tight bound of 2 − 1 on the deterministic competitive ratio of metrical task systems in any -point metric [9], and the randomized competitive ratio lies between an Ω(log /log log ) lower bound (Bartal et al [6,7]) and an (log 2 ) upper bound (Bubeck et al [10]). Thus, width layered graph traversal is strictly a more general problem than -point metrical task systems.…”

Permutation Predictions for Non-Clairvoyant Scheduling

“…Further related work There has been signi cant interest in the recent framework of learning-augmented online algorithms. Many problems have been considered, e.g., caching [49,55,4], further scheduling [36,8,7,56,10,44,45,50], rent-or-buy problems [48,24,2,12,54,5,56], paging [33,23,13], graph problems [22,9,57,39], secretary problems [6,20], matching [4,37,38] and many more.…”