Optimal and learned algorithms for the online list update problem with Zipfian accesses

Piotr Indyk, Isabelle Quaye, Ronitt Rubinfeld, Sandeep Silwal
Proceedings of The 36th International Conference on Algorithmic Learning Theory, PMLR 272:611-648, 2025.

Abstract

The online list update problem is defined as follows: we are given a list of items and the cost to access any particular item is its position from the start of the list. A sequence of item accesses come online, and our goal is to dynamically reorder the list so that the aggregate access cost is small. We study the stochastic version of the problem where the items are accessed i.i.d. from an unknown distribution p. The study of the stochastic version goes back at least 60 years to McCabe. In this paper, we first consider the simple online algorithm which swaps an accessed item with the item right before it, unless it is at the very front. This algorithm is known as the Transposition rule. We theoretically analyze the stationary behavior of Transposition and prove that its performance is within 1+o(1) factor of the optimal offline algorithm for access sequences sampled from heavy-tailed distributions, proving a conjecture of Rivest from 1976. While the stationary behavior of the Transposition rule is theoretically optimal in the aforementioned i.i.d setting, it can catastrophically fail under adversarial access sequences where only the last and second to last items are repeatedly accessed. A desirable outcome would be a policy that performs well under both circumstances. To achieve this, we use reinforcement learning to design an adaptive policy that performs well for both the i.i.d. setting and the above-mentioned adversarial access. Unsurprisingly, the learned policy appears to be an interpolation between Move-to-Front and Transposition with its behavior closer to Move-to-Front for adversarial access sequences and closer to Transposition for sequences sampled from heavy tailed distributions suggesting that the policy is adaptive and capable of responding to patterns in the access sequence.

Cite this Paper

BibTeX
@InProceedings{pmlr-v272-indyk25a, title = {Optimal and learned algorithms for the online list update problem with Zipfian accesses}, author = {Indyk, Piotr and Quaye, Isabelle and Rubinfeld, Ronitt and Silwal, Sandeep}, booktitle = {Proceedings of The 36th International Conference on Algorithmic Learning Theory}, pages = {611--648}, year = {2025}, editor = {Kamath, Gautam and Loh, Po-Ling}, volume = {272}, series = {Proceedings of Machine Learning Research}, month = {24--27 Feb}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v272/main/assets/indyk25a/indyk25a.pdf}, url = {https://proceedings.mlr.press/v272/indyk25a.html}, abstract = {The online list update problem is defined as follows: we are given a list of items and the cost to access any particular item is its position from the start of the list. A sequence of item accesses come online, and our goal is to dynamically reorder the list so that the aggregate access cost is small. We study the stochastic version of the problem where the items are accessed i.i.d. from an unknown distribution $p$. The study of the stochastic version goes back at least 60 years to McCabe. In this paper, we first consider the simple online algorithm which swaps an accessed item with the item right before it, unless it is at the very front. This algorithm is known as the Transposition rule. We theoretically analyze the stationary behavior of Transposition and prove that its performance is within $1+o(1)$ factor of the optimal offline algorithm for access sequences sampled from heavy-tailed distributions, proving a conjecture of Rivest from 1976. While the stationary behavior of the Transposition rule is theoretically optimal in the aforementioned i.i.d setting, it can catastrophically fail under adversarial access sequences where only the last and second to last items are repeatedly accessed. A desirable outcome would be a policy that performs well under both circumstances. To achieve this, we use reinforcement learning to design an adaptive policy that performs well for both the i.i.d. setting and the above-mentioned adversarial access. Unsurprisingly, the learned policy appears to be an interpolation between Move-to-Front and Transposition with its behavior closer to Move-to-Front for adversarial access sequences and closer to Transposition for sequences sampled from heavy tailed distributions suggesting that the policy is adaptive and capable of responding to patterns in the access sequence.} }
Endnote
%0 Conference Paper %T Optimal and learned algorithms for the online list update problem with Zipfian accesses %A Piotr Indyk %A Isabelle Quaye %A Ronitt Rubinfeld %A Sandeep Silwal %B Proceedings of The 36th International Conference on Algorithmic Learning Theory %C Proceedings of Machine Learning Research %D 2025 %E Gautam Kamath %E Po-Ling Loh %F pmlr-v272-indyk25a %I PMLR %P 611--648 %U https://proceedings.mlr.press/v272/indyk25a.html %V 272 %X The online list update problem is defined as follows: we are given a list of items and the cost to access any particular item is its position from the start of the list. A sequence of item accesses come online, and our goal is to dynamically reorder the list so that the aggregate access cost is small. We study the stochastic version of the problem where the items are accessed i.i.d. from an unknown distribution $p$. The study of the stochastic version goes back at least 60 years to McCabe. In this paper, we first consider the simple online algorithm which swaps an accessed item with the item right before it, unless it is at the very front. This algorithm is known as the Transposition rule. We theoretically analyze the stationary behavior of Transposition and prove that its performance is within $1+o(1)$ factor of the optimal offline algorithm for access sequences sampled from heavy-tailed distributions, proving a conjecture of Rivest from 1976. While the stationary behavior of the Transposition rule is theoretically optimal in the aforementioned i.i.d setting, it can catastrophically fail under adversarial access sequences where only the last and second to last items are repeatedly accessed. A desirable outcome would be a policy that performs well under both circumstances. To achieve this, we use reinforcement learning to design an adaptive policy that performs well for both the i.i.d. setting and the above-mentioned adversarial access. Unsurprisingly, the learned policy appears to be an interpolation between Move-to-Front and Transposition with its behavior closer to Move-to-Front for adversarial access sequences and closer to Transposition for sequences sampled from heavy tailed distributions suggesting that the policy is adaptive and capable of responding to patterns in the access sequence.
APA
Indyk, P., Quaye, I., Rubinfeld, R. & Silwal, S.. (2025). Optimal and learned algorithms for the online list update problem with Zipfian accesses. Proceedings of The 36th International Conference on Algorithmic Learning Theory, in Proceedings of Machine Learning Research 272:611-648 Available from https://proceedings.mlr.press/v272/indyk25a.html.

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