{"id":1178140,"date":"2026-07-07T10:12:13","date_gmt":"2026-07-07T17:12:13","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/?post_type=msr-research-item&#038;p=1178140"},"modified":"2026-07-07T10:13:47","modified_gmt":"2026-07-07T17:13:47","slug":"online-algorithms-via-minimax-and-posterior-matching","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/online-algorithms-via-minimax-and-posterior-matching\/","title":{"rendered":"Online Algorithms via Minimax and Posterior Matching"},"content":{"rendered":"\n\n\n<p class=\"wp-block-paragraph\">Competitive analysis is central to the study of online algorithms, but upper bounds are often highly problem-specific. We develop a more unifying methodology via the minimax viewpoint. Guided by Yao&#8217;s principle, we reduce worst-case competitive analysis to Bayesian online design under an arbitrary correlated prior over arrival sequences. For such a prior, let <math><mrow><msup><mi>X<\/mi><\/msup><mi>e<\/mi><mi>m<\/mi><mo>><\/mo><\/mrow><\/math> be the hindsight-optimal fractional solution for the realized instance, and let <math><semantics><mrow><mtext>X^{(t)}=mathbb E[X^<\/em>mid mathcal F_t]<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">X^{(t)}=mathbb E[X^<\/em>mid mathcal F_t]<\/annotation><\/semantics><\/math> be its posterior process. Our guiding rule is emph{posterior matching}: at each time <math><mi>t<\/mi><\/math>, choose the feasible online action that tracks the current posterior <math><mrow><msup><mi>X<\/mi><mrow><mrow><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mrow><\/msup><\/mrow><\/math> as closely as the online constraints permit.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">We show that this single principle yields optimal or near-optimal guarantees for several classical online fractional problems, including set cover, load balancing, matching and more general resource-allocation problems, recovering or improving state-of-the-art bounds in these settings with norm\/concave objectives. Via known rounding reductions, it also yields randomized integral guarantees for weighted paging, MTS on star metrics, and ski-rental. At a technical level, our analysis reduces competitive guarantees to key probabilistic inequalities for the vector martingales generated by the posterior of the offline optimum. The resulting framework gives a reusable route from Bayesian online design under arbitrary correlated priors to information-theoretic worst-case competitive guarantees.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Competitive analysis is central to the study of online algorithms, but upper bounds are often highly problem-specific. We develop a more unifying methodology via the minimax viewpoint. Guided by Yao&#8217;s principle, we reduce worst-case competitive analysis to Bayesian online design under an arbitrary correlated prior over arrival sequences. For such a prior, let Xem> be [&hellip;]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":[{"type":"text","value":"Thomas Kesselheim","user_id":0},{"type":"user_nicename","value":"Marco Molinaro","user_id":"42204"},{"type":"text","value":"Kalen Patton","user_id":0},{"type":"text","value":"Sahil 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