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Abstract

Abstract. The problem of combining expert advice, studied extensively in the Computational Learning Theory literature, and the Metrical Task System (MTS) problem, studied extensively in the area of On-line Algorithms, contain a number of interesting similarities. In this paper we explore the relationship between these problems and show how algorithms designed for each can be used to achieve good bounds and new approaches for solving the other. Specific contributions of this paper include:• An analysis of how two recent algorithms for the MTS problem can be applied to the problem of tracking the best expert in the "decision-theoretic" setting, providing good bounds and an approach of a much different flavor from the well-known multiplicative-update algorithms.• An analysis showing how the standard randomized Weighted Majority (or Hedge) algorithm can be used for the problem of "combining on-line algorithms on-line", giving much stronger guarantees than the results of Azar, Y., Broder, A., & Manasse, M. (1993). Proc ACM-SIAM Symposium on Discrete Algorithms (pp. 432-440) when the algorithms being combined occupy a state space of bounded diameter.• A generalization of the above, showing how (a simplified version of) Herbster and Warmuth's weight-sharing algorithm can be applied to give a "finely competitive" bound for the uniform-space Metrical Task System problem. We also give a new, simpler algorithm for tracking experts, which unfortunately does not carry over to the MTS problem.Finally, we present an experimental comparison of how these algorithms perform on a process migration problem, a problem that combines aspects of both the experts-tracking and MTS formalisms.

Online metric algorithms with untrusted predictions

“…We note that the proofs of these theorems are based on the powerful results by Blum and Burch (2000) and Fiat et al (1994). In Theorem 9, we show that the dependence on η/Off in the preceding theorems is tight up to constant factors for some MTS instance.…”

“…Metrical task systems (MTS), introduced by Borodin et al (1992), are a rich class of online problems capable of modeling certain aspects of computing and production systems, movements of emergency vehicles, and many others. MTS contain several fundamental problems in online optimization including caching, k-server, convex body chasing, and convex function chasing, and is also related to the experts problem in online learning (see Daniely and Mansour, 2019;Blum and Burch, 2000).…”

Playing against Hedge

“…If the greedy algorithm is to be optimal, even the 2M oscillatory steps should bring a cumulative reward greater than the original back and forth movement. On the other hand, if we prove this last inequality, this will also prove (14), whose last 2M steps bring more reward than the 2M oscillatory steps.…”

“…distributing jobs to parallel processors). A. Blum and C. Burch have used the following motivating scenario in [14]: A process runs on some machine in an environment with N machines in total. The process may move to a different machine at the end of a time interval.…”

Truth and Regret in Online Scheduling

“…distribution in every time period, a family of simple mechanisms achieves 1 The idea of combining several online algorithms into one that is nearly as good as the best of them has been explored previously, but in the limited context of metrical task systems and adaptive data structures. See, e.g., Blum and Burch (2000); Blum et al (2002).…”