A Regularization Approach to Metrical Task Systems

Abernethy, Jacob; Buchbinder, Niv; Stanton, Isabelle

doi:10.1007/978-3-642-16108-7_23

Cited by 13 publications

(30 citation statements)

References 22 publications

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“…For Equation (3), if we set α = 1 and η = ln(n) + ln ln n, we get the best known bound for MTS on uniform metrics (Bansal et al (2010); Abernethy et al (2010)). In particular, the bound is better than that obtained by the analysis of Blum and Burch (1997), who also interpolate between experts and MTS.…”

Section: Resultsmentioning

confidence: 99%

“…Blum et al (2003) discuss algorithms for decision making on lists and trees, for both a competitive analysis setting and an online learning setting, and show how they can be combined using the hedge algorithm (Freund and Schapire (1997)) to provide simultaneous guarantees. Papers such as Blum et al (1999) and Abernethy et al (2010) discuss competitive-analysis algorithms derived using tools from online learning, e.g., regularization. Other works attempt to strengthen the standard regret framework of online learning, such as learning with global cost functions (Even-Dar et al (2009)) and using more adaptive notions of regret as discussed above.…”

Section: Introductionmentioning

confidence: 99%

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Unified Algorithms for Online Learning and Competitive Analysis

et al. 2016

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Online learning and competitive analysis are two widely studied frameworks for online decisionmaking settings. Despite the frequent similarity of the problems they study, there are significant differences in their assumptions, goals and techniques, hindering a unified analysis and richer interplay between the two. In this paper, we provide several contributions in this direction. We provide a single unified algorithm which by parameter tuning, interpolates between optimal regret for learning from experts (in online learning) and optimal competitive ratio for the metrical task systems problem (MTS) (in competitive analysis), improving on the results of Blum and Burch (1997). The algorithm also allows us to obtain new regret bounds against "drifting" experts, which might be of independent interest. Moreover, our approach allows us to go beyond experts/MTS, obtaining similar unifying results for structured action sets and "combinatorial experts", whenever the setting has a certain matroid structure.

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Unified Algorithms for Online Learning and Competitive Analysis

et al. 2016

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“…The MMM problem is also a special case of classical Metrical Task Systems [6]; see [1,4] for more recent work. The best approximations for metrical task systems are poly-logarithmic in the size of the metric space.…”

Section: Related Workmentioning

confidence: 99%

“…The set S t is obtained by threshold rounding of the fractional base z t ∈ P B (M) as above. Instead, consider taking L different samples T (1) , T (2) , . .…”

Section: Proofmentioning

confidence: 99%

Changing Bases: Multistage Optimization for Matroids and Matchings

Gupta

Talwar

Wieder

2014

Automata, Languages, and Programming

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This paper is motivated by the fact that many systems need to be maintained continually while the underlying costs change over time. The challenge then is to continually maintain near-optimal solutions to the underlying optimization problems, without creating too much churn in the solution itself. We model this as a multistage combinatorial optimization problem where the input is a sequence of cost functions (one for each time step); while we can change the solution from step to step, we incur an additional cost for every such change.We first study the multistage matroid maintenance problem, where we need to maintain a base of a matroid in each time step under the changing cost functions and acquisition costs for adding new elements. The online version of this problem generalizes onine paging, and is a well-structured case of the metrical task systems. E.g., given a graph, we need to maintain a spanning tree Tt at each step: we pay ct(Tt) for the cost of the tree at time t, and also |Tt \ Tt−1| for the number of edges changed at this step. Our main result is a polynomial time O(log m log r)-approximation to the online multistage matroid maintenance problem, where m is the number of elements/edges and r is the rank of the matroid. This improves on results of Buchbinder et al. [7] who addressed the fractional version of this problem under uniform acquisition costs, and Buchbinder, Chen and Naor [8] who studied the fractional version of a more general problem. We also give an O(log m) approximation for the offline version of the problem. These bounds hold when the acquisition costs are non-uniform, in which case both these results are the best possible unless P=NP.We also study the perfect matching version of the problem, where we must maintain a perfect matching at each step under changing cost functions and costs for adding new elements. Surprisingly, the hardness drastically increases: for any constant ε > 0, there is no O(n 1−ε )-approximation to the multistage matching maintenance problem, even in the offline case.

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“…The algorithm allows a tradeoff, but does not simultaneously perform well for regret and CR. There is also work achieving simultaneous guarantees with respect to the static and dynamic optimal solutions in other settings, e.g., [3], and there have been some attempts to use algorithmic approaches from machine learning in the context of MTSs [4,5].…”

Section: Introductionmentioning

confidence: 99%