[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82776-en":3,"doc-seo-82776-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},82776,137441390410,"Hazel","https://ap-avatar.wpscdn.com/avatar/2000252f4ab5702993?_k=1776741390130283984",8,"Research & Report","Stochastic Caching via Subset Entropy","The work studies beyond worst-case analysis for online caching by moving from generic stochastic assumptions to a distribution-sensitive information measure. It shows that standard min-max effects make worst-case input performance reappear under randomization, motivating the need to distinguish “easy” from “hard” distributions. For stochastic caching, the paper defines subset entropy and uses it to obtain a fine-grained characterization of competitive ratios, including a new analysis of LRU on stochastic inputs.","arXiv :2607 .03947v 1 [ cs .DS] 4 Jul 2026  \nStochastic Caching via Subset Entropy  \nRavi Kumar \\# Ñ  \nGoogle Research, Mountain View CA, USA  \nRoie Levin \\# Ñ  \nRutgers University, Piscataway NJ, USA  \nJoseph (Seffi) Naor \\# Ñ  \nTechnion – Israel Institute of Technology, Haifa, Israel  \nDebmalya Panigrahi \\# Ñ Duke University, Durham NC, USA  \n~~ Abstract ~~  \nA classic approach to beyond worst-case algorithm design is to impose stochastic assumptions on the input. However, a limiting feature of such stochastic analyses is that, by the min-max principle, performance on worst-case distributions mirrors that of randomized algorithms on worst-case inputs. In other words, the same shortcoming of worst-case analysis—its inability to distinguish between “easy” and “hard’ instances—reappearsas an inability to distinguish between “easy” and “hard” distributions. This raises a natural question: Can we characterize “easy” input distributions with useful beyond worst-case bounds?  \nA canonical example is the stochastic caching problem (Aho, Denning, and Ullman, 1971) . When the page requests are drawn i.i.d. from the uniform distribution, the best achievable competitive ratio is 􀀤 (log􀀺) , matching the performance of the best randomized algorithm on worst-case instances (Fiat, Karp, Luby, McGeoch, Sleator, and Young, 1991) . However, when the input distribution has less entropy, intuition suggests that we should be able to do better by exploiting the information provided by the distribution. In this paper, we formalize this intuition by defining a new information-theoretic parameter of probability distributions called subset entropy. We then use this parameter to give a fine-grained characterization of the competitive ratio of stochastic caching, including a new analysis for the well-known LRU algorithm on stochastic inputs.  \nWhile our technical results are for the caching problem, we believe the broader principle—parameterizing algorithmic performance by an entropy measure of the input—is of independent interest and might apply to other online and stochastic optimization problems. Indeed, for many other fundamental problems such as (comparison-based) sorting, online matching, online load balancing, etc., etc., the hardest stochastic instances involve high-entropy distributions. We hope our work is a step toward a broader theory of fine-grained algorithmic performance for this class of problems.  \n2012 ACM Subject Classification Theory of computation → Online algorithms  \nKeywords and phrases Online Algorithms, Beyond Worst-Case Analysis, Caching, Paging, Entropy Digital Object Identifier 10.4230/LIPIcs.APPROX.2026.1  \nFunding Joseph (Seffi) Naor: Supported in part by ISF grant 3001/24 and United States – Israel BSF grant 2022418.  \nDebmalya Panigrahi: Supported in part by NSF grants CCF-2329230 and CCF-1955703 .  \n© Ravi Kumar, Roie Levin, Joseph Naor, and Debmalya Panigrahi;  \nlicensed under Creative Commons License CC-BY 4.0  \n29th International Conference on Approximation Algorithms for Combinatorial Optimization Problems (APPROX 2026) .  \nLeibniz International Proceedings in Informatics  \nSchloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany  \nR. Kumar, R. Levin, J. Naor, and D. Panigrahi 1:1  \n 1  Introduction  \nStochastic optimization is a well-known and extensively-studied framework for beyond worst-case analysis of algorithmic problems. The basic premise is to make an assumption about the input being drawn from a probability distribution, rather than being generated adversarially, thereby enabling algorithms that exploit structural properties of the distribution. However, a common shortfall of this strategy is that algorithmic performance on the worst-case input distribution often mirrors that of the best randomized algorithm for worst-case inputs. This equivalence, a consequence of themin-max principle (Yao [35]), is often useful for proving randomized lower bounds for algorithms but, conversely, als","cbCaio0aXLeoB5uv","https://ap.wps.com/l/cbCaio0aXLeoB5uv","pdf",879352,1,23,"English","en",105,"# Introduction\n## The Caching Problem\n## Subset Entropy and Beyond Worst-Case Characterization","[{\"question\":\"What problem does the paper address in beyond worst-case algorithm design?\",\"answer\":\"It addresses the limitation that min-max-based stochastic analyses often make performance on worst-case distributions mirror randomized algorithms on worst-case inputs, failing to separate “easy” from “hard” instances or distributions.\"},{\"question\":\"What is the subset entropy parameter used for?\",\"answer\":\"The paper introduces subset entropy as a new information-theoretic parameter of probability distributions and uses it to characterize the competitive ratio of stochastic caching more precisely.\"},{\"question\":\"How does the paper relate its results to the caching algorithms and scenarios discussed?\",\"answer\":\"It focuses on stochastic caching, contrasts known competitive ratios under uniform (high-entropy) i.i.d. requests, and provides a new analysis for the well-known LRU algorithm on stochastic 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problem does the paper address in beyond worst-case algorithm design?","Question",{"text":75,"@type":76},"It addresses the limitation that min-max-based stochastic analyses often make performance on worst-case distributions mirror randomized algorithms on worst-case inputs, failing to separate “easy” from “hard” instances or distributions.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What is the subset entropy parameter used for?",{"text":80,"@type":76},"The paper introduces subset entropy as a new information-theoretic parameter of probability distributions and uses it to characterize the competitive ratio of stochastic caching more precisely.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the paper relate its results to the caching algorithms and scenarios discussed?",{"text":84,"@type":76},"It focuses on stochastic caching, contrasts known competitive ratios under uniform (high-entropy) i.i.d. requests, and provides a new analysis for the well-known LRU algorithm on 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