We study the design of computationally efficient randomized algorithms for the k-server problem. Existing randomized algorithms with the best known competitive ratios are, on the one hand, inherently implicit and, on the other hand, employ a rounding scheme that maintains a distribution over exponentially many configurations. In this work, we introduce a derandomization framework that transforms any randomized k-server algorithm on a hierarchically separated tree into one that uses only O(log k) random bits for request sequences of arbitrary length - hence maintaining a distribution over only polynomially many server configurations. Leveraging this black-box derandomization, we obtain the first polynomial-time randomized k-server algorithm on arbitrary n-point metrics with a polylogarithmic competitive ratio. Our results also have implications for the advice complexity of the k-server problem.

In non-clairvoyant scheduling, the goal is to minimize the total job completion time without prior knowledge of individual job processing times. This classical online optimization problem has recently gained attention through the framework of learning-augmented algorithms. We introduce a natural setting in which the scheduler receives continuous feedback in the form of progress bars—estimates of the fraction of each job completed over time. We design new algorithms for both adversarial and stochastic progress bars and prove strong competitive bounds. Our results in the adversarial case surprisingly induce improved guarantees for learning-augmented scheduling with job size predictions. We also introduce a general method for combining scheduling algorithms, yielding further insights in scheduling with predictions. Finally, we propose a stochastic model of progress bars as a more optimistic alternative to conventional worst-case models, and present an asymptotically optimal scheduling algorithm in this setting.

Introduced by Papadimitriou and Yannakakis in 1989, layered graph traversal is a central problem in online algorithms and mobile computing that has been studied for several decades, and which now is essentially resolved in its original formulation. In this paper, we demonstrate that what appears to be an innocuous modification of the problem actually leads to a drastic (exponential) reduction of the competitive ratio. Specifically, we present an algorithm that is O (log2 w )-competitive for traversing unweighted layered graphs of width w. Our algorithm chooses the agent’s position simply according to the probability distribution over the current layer that maximizes the sum of entropies of the induced distributions in the preceding layers.

International audience

We consider metrical task systems on general metric spaces with $n$ points, and show that any fully randomized algorithm can be turned into a randomized algorithm that uses only $2\log n$ random bits, and achieves the same competitive ratio up to a factor $2$. This provides the first order-optimal barely random algorithms for metrical task systems, i.e. which use a number of random bits that does not depend on the number of requests addressed to the system. We discuss implications on various aspects of online decision making such as: distributed systems, advice complexity and transaction costs, suggesting broad applicability. We put forward an equivalent view that we call collective metrical task systems where $k$ agents in a metrical task system team up, and suffer the average cost paid by each agent. Our results imply that such team can be $O(\log^2 n)$-competitive as soon as $k\geq n^2$. In comparison, a single agent is always $\Omega(n)$-competitive.

In collective tree exploration, a team of k mobile agents is assigned to go through all edges of an unknown tree as fast as possible. An edge of the tree is revealed to the team when one agent becomes adjacent to that edge. The agents start from the root and all move synchronously along one adjacent edge in each round. Communication between the agents is unrestricted, and they are, therefore, centrally controlled by a single exploration algorithm. The algorithm's guarantee is typically compared to the number of rounds required by the agents to go through all edges if they had known the tree in advance. This quantity is at least max{2n/k, 2D} where n is the number of nodes and D is the tree depth. Since the introduction of the problem by [11, 12], two types of guarantees have emerged: the first takes the form r(k)(n/k + D), where r(k) is called the competitive ratio, and the other takes the form 2n/k + f (k, D), where f (k, D) is called the competitive overhead. In this paper, we present the first algorithm with linear-in-D competitive overhead, thereby reconciling both approaches. Specifically, our bound is in 2n/k + O(klog2(k)-1 D) and leads to a competitive ratio in . This is the first improvement over O(k/In k) since the introduction of the problem, twenty years ago. Our algorithm is developed for an asynchronous generalization of collective tree exploration (ACTE). It belongs to a broad class of locally-greedy exploration algorithms that we define. We show that the analysis of locally-greedy algorithms can be seen through the lens of a 2-player game that we call the tree-mining game and which could be of independent interest.

We consider the problem of collaborative tree exploration posed by Fraigniaud, Gasieniec, Kowalski, and Pelc [8] where a team of k agents is tasked to collectively go through all the edges of an unknown tree as fast as possible and return to the root. Denoting by n the total number of nodes and by D the tree depth, the O(n/log(k) + D) algorithm of [8] achieves the best competitive ratio known with respect to the optimal exploration algorithm that knows the tree in advance, which takes order max {2n/k, 2D} rounds. Brass, Cabrera-Mora, Gasparri, and Xiao [1] consider an alternative performance criterion, the additive overhead with respect to 2n/k, and obtain a 2n/k + O((D + k)k) runtime guarantee. In this announcement, we present 'Breadth-First Depth-Next' (BFDN), a novel and simple algorithm that performs collaborative tree exploration in time 2n/k + O(D2 log(k)), thus outperforming [1] for all values of (n, D) and being order-optimal for fixed k and trees with depth D = o(√n). The proof of our result crucially relies on the analysis of a simple two-player game with balls in urns that could be of independent interest. We extend the guarantees of BFDN to: scenarios with limited memory and communication, adversarial setups where robots can be blocked, and exploration of classes of non-tree graphs. Finally, we provide a recursive version of BFDN with a runtime of Oℓ(n/k1/ℓ + log(k)D1+1/ℓ) for parameter ℓ ≥ 1, thereby improving performance for trees with large depth. A complete version of the paper is available online [2].

We study the problem of collaborative tree exploration introduced by Fraigniaud, Gasieniec, Kowalski, and Pelc [Pierre Fraigniaud et al., 2006] where a team of k agents is tasked to collectively go through all the edges of an unknown tree as fast as possible and return to the root. Denoting by n the total number of nodes and by D the tree depth, the 𝒪(n/log(k)+D) algorithm of [Pierre Fraigniaud et al., 2006] achieves a 𝒪(k/log(k)) competitive ratio with respect to the cost of offline exploration which is at least max{{2n/k,2D}}. Brass, Cabrera-Mora, Gasparri, and Xiao [Peter Brass et al., 2011] study an alternative performance criterion, the competitive overhead with respect to the cost of offline exploration, with their 2n/k+𝒪((D+k)^k) guarantee. In this paper, we introduce "Breadth-First Depth-Next" (BFDN), a novel and simple algorithm that performs collaborative tree exploration in 2n/k+𝒪(D²log(k)) rounds, thus outperforming [Peter Brass et al., 2011] for all values of (n,D,k) and being order-optimal for trees of depth D = o(√n). Our analysis relies on a two-player game reflecting a problem of online resource allocation that could be of independent interest. We extend the guarantees of BFDN to: scenarios with limited memory and communication, adversarial setups where robots can be blocked, and exploration of classes of non-tree graphs. Finally, we provide a recursive version of BFDN with a runtime of 𝒪_𝓁(n/k^{1/𝓁}+log(k) D^{1+1/𝓁}) for parameter 𝓁 ≥ 1, thereby improving performance for trees with large depth.

We resolve an open problem of Hanneke (2021) on the subject of universally consistent online learning with non-i.i.d. processes and unbounded losses. The notion of an optimistically universal learning rule was defined by Hanneke in an effort to study learning theory under minimal assumptions. A given learning rule is said to be optimistically universal if it achieves a low long-run average loss whenever the data generating process makes this goal achievable by some learning rule. Hanneke (2021) posed as an open problem whether, for every unbounded loss, the family of processes admitting universal learning are precisely those having a finite number of distinct values almost surely. In this paper, we completely resolve this problem, showing that this is indeed the case. As a consequence, this also offers a dramatically simpler formulation of an optimistically universal learning rule for any unbounded loss: namely, the simple memorization rule already suffices. Our proof relies on constructing random measurable partitions of the instance space. This technique may be of independent interest in providing useful arguments towards solving the remaining open question of optimistically universal online learning for bounded losses.

We study universal consistency of non-i.i.d. processes in the context of online learning. A stochastic process is said to admit universal consistency if there exists a learner that achieves vanishing average loss for any measurable response function on this process. When the loss function is unbounded, [1] showed that the only processes admitting strong universal consistency are those taking a finite number of values almost surely. However, when the loss function is bounded, the class of processes admitting strong universal consistency is much richer and its characterization could be dependent on the response setting [2]. In this paper, we show that this class of processes is independent from the response setting thereby closing an open question of [3] (Open Problem 3). Specifically, we show that the class of processes that admit universal online learning is the same for binary classification as for multiclass classification with countable number of classes. Consequently, any output setting with bounded loss can be reduced to binary classification. Our reduction is constructive and practical. Indeed, we show that the nearest neighbor algorithm is transported by our construction. For binary classification on a process admitting strong universal learning, we prove that nearest neighbor successfully learns at least all finite unions of intervals.

Variational methods, such as mean-field (MF) and tree-reweighted (TRW), provide computationally efficient approximations of the log-partition function for generic graphical models but their approximation ratio is generally not quantified. As the primary contribution of this work, we provide an approach to quantify their approximation ratio for any discrete pairwise graphical model with non-negative potentials through a property of the underlying graph structure $G$. Specifically, we argue that (a variant of) TRW produces an estimate within factor $1/\sqrt{\kappa(G)}$ where $\kappa(G) \in (0,1]$ captures how far $G$ is from tree structure. As a consequence, the approximation ratio is $1$ for trees, $\sqrt{(d+1)/2}$ for graphs with maximum average degree $d$ and $1+1/(2\beta)+o_{\beta\to \infty}(1/\beta)$ for graphs with girth at least $\beta \log N$. The quantity $\kappa(G)$ is the solution of a min-max problem associated with the spanning tree polytope of $G$ that can be evaluated in polynomial time for any graph. We provide a near linear-time variant that achieves an approximation ratio depending on the minimal (across edges) effective resistance of the graph. We connect our results to the graph partition approximation method and thus provide a unified perspective.