Online Metric Allocation and Time-Varying Regularization

Authors Nikhil Bansal, Christian Coester

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Nikhil Bansal
  • University of Michigan, Ann Arbor, MI, USA
Christian Coester
  • University of Sheffield, UK


We thank Ravi Kumar, Manish Purohit and Erik Vee for many useful discussions that inspired this work.

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Nikhil Bansal and Christian Coester. Online Metric Allocation and Time-Varying Regularization. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 13:1-13:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


We introduce a general online allocation problem that connects several of the most fundamental problems in online optimization. Let M be an n-point metric space. Consider a resource that can be allocated in arbitrary fractions to the points of M. At each time t, a convex monotone cost function c_t: [0,1] → ℝ_+ appears at some point r_t ∈ M. In response, an algorithm may change the allocation of the resource, paying movement cost as determined by the metric and service cost c_t(x_{r_t}), where x_{r_t} is the fraction of the resource at r_t at the end of time t. For example, when the cost functions are c_t(x) = α x, this is equivalent to randomized MTS, and when the cost functions are c_t(x) = ∞⋅1_{x < 1/k}, this is equivalent to fractional k-server. Because of an inherent scale-freeness property of the problem, existing techniques for MTS and k-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 O(log n)-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 𝓁₂²-regularizer, we give tight bounds of Θ(n) on tree metrics, which imply deterministic and randomized competitive ratios of O(n²) and O(nlog n) respectively on arbitrary metrics.

Subject Classification

ACM Subject Classification
  • Theory of computation → Online algorithms
  • Theory of computation → K-server algorithms
  • Online algorithms
  • competitive analysis
  • k-server
  • metrical task systems
  • mirror descent
  • regularization


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  1. C. J. Argue, Anupam Gupta, Guru Guruganesh, and Ziye Tang. Chasing convex bodies with linear competitive ratio (invited paper). In STOC '21, 2021. URL:
  2. Nikhil Bansal, Niv Buchbinder, Aleksander Madry, and Joseph Naor. A polylogarithmic-competitive algorithm for the k-server problem. J. ACM, 62(5), 2015. URL:
  3. Nikhil Bansal, Niv Buchbinder, and Joseph Naor. Metrical task systems and the k-server problem on HSTs. In ICALP' 10, 2010. URL:
  4. Nikhil Bansal, Niv Buchbinder, and Joseph Naor. Towards the randomized k-server conjecture: A primal-dual approach. In Symposium on Discrete Algorithms, SODA, pages 40-55, 2010. Google Scholar
  5. Nikhil Bansal, Niv Buchbinder, and Joseph Naor. A primal-dual randomized algorithm for weighted paging. J. ACM, 59(4):19:1-19:24, 2012. Google Scholar
  6. Yair Bartal. Probabilistic approximations of metric spaces and its algorithmic applications. In FOCS '96, 1996. URL:
  7. Yair Bartal, Avrim Blum, Carl Burch, and Andrew Tomkins. A polylog(n)-competitive algorithm for metrical task systems. In STOC '97, 1997. URL:
  8. Avrim Blum, Howard J. Karloff, Yuval Rabani, and Michael E. Saks. A decomposition theorem for task systems and bounds for randomized server problems. SIAM J. Comput., 30(5), 2000. URL:
  9. Allan Borodin and Ran El-Yaniv. Online computation and competitive analysis. Cambridge University Press, 1998. Google Scholar
  10. Allan Borodin, Nathan Linial, and Michael E. Saks. An optimal on-line algorithm for metrical task system. J. ACM, 39(4), 1992. URL:
  11. Sébastien Bubeck, Michael B. Cohen, James R. Lee, and Yin Tat Lee. Metrical task systems on trees via mirror descent and unfair gluing. In SODA '19, 2019. URL:
  12. Sébastien Bubeck, Michael B. Cohen, Yin Tat Lee, James R. Lee, and Aleksander Madry. k-server via multiscale entropic regularization. In STOC '18, 2018. URL:
  13. Sébastien Bubeck, Bo'az Klartag, Yin Tat Lee, Yuanzhi Li, and Mark Sellke. Chasing nested convex bodies nearly optimally. In SODA '20, 2020. URL:
  14. Sébastien Bubeck, Yin Tat Lee, Yuanzhi Li, and Mark Sellke. Competitively chasing convex bodies. In STOC '19, 2019. URL:
  15. Niv Buchbinder, Shahar Chen, and Joseph (Seffi) Naor. Competitive analysis via regularization. In Symposium on Discrete Algorithms, SODA, pages 436-444, 2014. Google Scholar
  16. Niv Buchbinder, Anupam Gupta, Marco Molinaro, and Joseph (Seffi) Naor. k-servers with a smile: Online algorithms via projections. In Symposium on Discrete Algorithms, SODA, pages 98-116, 2019. Google Scholar
  17. Niv Buchbinder and Joseph Naor. The design of competitive online algorithms via a primal-dual approach. Found. Trends Theor. Comput. Sci., 3(2-3):93-263, 2009. Google Scholar
  18. Niv Buchbinder and Joseph Naor. Online primal-dual algorithms for covering and packing. Math. Oper. Res., 34(2):270-286, 2009. Google Scholar
  19. Christian Coester and James R. Lee. Pure entropic regularization for metrical task systems. In COLT '19, 2019. URL:
  20. Aaron Cote, Adam Meyerson, and Laura J. Poplawski. Randomized k-server on hierarchical binary trees. In STOC '08, 2008. URL:
  21. Farzam Ebrahimnejad and James R. Lee. Multiscale entropic regularization for MTS on general metric spaces. In ITCS '22, 2022. Google Scholar
  22. Amos Fiat and Manor Mendel. Better algorithms for unfair metrical task systems and applications. SIAM J. Comput., 32(6), 2003. URL:
  23. Joel Friedman and Nathan Linial. On convex body chasing. Discret. Comput. Geom., 9, 1993. URL:
  24. Anupam Gupta, Amit Kumar, and Debmalya Panigrahi. A hitting set relaxation for k-server and an extension to time-windows. In FOCS '21, 2021. Google Scholar
  25. James R. Lee. Fusible HSTs and the randomized k-server conjecture. In FOCS '18, pages 438-449, 2018. URL:
  26. Steve Seiden. Unfair problems and randomized algorithms for metrical task systems. Inf. Comput., 148(2), 1999. URL:
  27. Mark Sellke. Chasing convex bodies optimally. In Shuchi Chawla, editor, SODA '20, 2020. URL: