Online Metric Allocation and Time-Varying Regularization

We introduce a general online allocation problem that connects several of the most fundamental problems in online optimization. Let be an -point metric space. Consider a resource that can be allocated in arbitrary fractions to the points of . At each time , a convex monotone cost function : [0, 1] → ℝ+ appears at some point ∈ . In response, an algorithm may change the allocation of the resource, paying movement cost as determined by the metric and service cost ( ), where is the fraction of the resource at at the end of time . For example, when the cost functions are () = , this is equivalent to randomized MTS, and when the cost functions are () = ∞·<1/, this is equivalent to fractional -server. Because of an inherent scale-freeness property of the problem, existing techniques for MTS and -server fail to achieve similar guarantees for metric allocation. To handle this, we consider a generalization of the online multiplicative update method where we decouple the rate at which a variable is updated from its value, resulting in interesting new dynamics. We use this to give an (log)-competitive algorithm for weighted star metrics. We then show how this corresponds to an extension of the online mirror descent framework to a setting where the regularizer is time-varying. Using this perspective, we further refine the guarantees of our algorithm. We also consider the case of non-convex cost functions. Using a simple ₂²-regularizer, we give tight bounds of Θ() on tree metrics, which imply deterministic and randomized competitive ratios of (2) and ( log ) respectively on arbitrary metrics

The Infinite Server Problem

We study a variant of the k-server problem, the infinite server problem, in which infinitely many servers reside initially at a particular point of the metric space and serve a sequence of requests. In the framework of competitive analysis, we show a surprisingly tight connection between this problem and the (h,k)-server problem, in which an online algorithm with k servers competes against an offline algorithm with h servers. Specifically, we show that the infinite server problem has bounded competitive ratio if and only if the (h,k)-server problem has bounded competitive ratio for some k=O(h). We give a lower bound of 3.146 for the competitive ratio of the infinite server problem, which implies the same lower bound for the (h,k)-server problem even when k>>h and holds also for the line metric; the previous known bounds were 2.4 for general metric spaces and 2 for the line. For weighted trees and layered graphs we obtain upper bounds, although they depend on the depth. Of particular interest is the infinite server problem on the line, which we show to be equivalent to the seemingly easier case in which all requests are in a fixed bounded interval away from the original position of the servers. This is a special case of a more general reduction from arbitrary metric spaces to bounded subspaces. Unfortunately, classical approaches (double coverage and generalizations, work function algorithm, balancing algorithms) fail even for this special case

Metrical Service Systems with Transformations

We consider a generalization of the fundamental online metrical service systems (MSS) problem where the feasible region can be transformed between requests. In this problem, which we call T-MSS, an algorithm maintains a point in a metric space and has to serve a sequence of requests. Each request is a map (transformation) : → between subsets and of the metric space. To serve it, the algorithm has to go to a point ∈ , paying the distance from its previous position. Then, the transformation is applied, modifying the algorithm’s state to ( ). Such transformations can model, e.g., changes to the environment that are outside of an algorithm’s control, and we therefore do not charge any additional cost to the algorithm when the transformation is applied. The transformations also allow to model requests occurring in the -taxi problem. We show that for -Lipschitz transformations, the competitive ratio is Θ()−2 on -point metrics. Here, the upper bound is achieved by a deterministic algorithm and the lower bound holds even for randomized algorithms. For the -taxi problem, we prove a competitive ratio of Õ(( log )2). For chasing convex bodies, we show that even with contracting transformations no competitive algorithm exists. The problem T-MSS has a striking connection to the following deep mathematical question: Given a finite metric space M, what is the required cardinality of an extension M̂ ⊇ M where each partial isometry on M extends to an automorphism? We give partial answers for special cases

Learning-Augmented Weighted Paging

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 (Lykouris and Vassilvitskii, 2018) to the weighted setting. For the case of perfect predictions, we provide an $\ell$-competitive deterministic and an $O(\log \ell)$-competitive randomized algorithm, where $\ell$ 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 $O(\ell)$ and $O(\log \ell)$ 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

Learning-augmented dynamic power management with multiple states via new ski rental bounds

No abstract availabl

Unbounded lower bound for k-server against weak adversaries

We study the resource augmented version of the $k$-server problem, also known as the $k$-server problem against weak adversaries or the $(h,k)$-server problem. In this setting, an online algorithm using $k$ servers is compared to an offline algorithm using $h$ servers, where $h\le k$. For uniform metrics, it has been known since the seminal work of Sleator and Tarjan (1985) that for any $\epsilon>0$, the competitive ratio drops to a constant if $k=(1+\epsilon) \cdot h$. This result was later generalized to weighted stars (Young 1994) and trees of bounded depth (Bansal et al. 2017). The main open problem for this setting is whether a similar phenomenon occurs on general metrics. We resolve this question negatively. With a simple recursive construction, we show that the competitive ratio is at least $\Omega(\log \log h)$, even as $k\to\infty$. Our lower bound holds for both deterministic and randomized algorithms. It also disproves the existence of a competitive algorithm for the infinite server problem on general metrics.Comment: To appear in STOC 202