We study the classical linear bandit problem on \\emph{graphs} modeling arm rewards through an underlying graph structure $G$($V$,$E$) such that rewards of neighboring nodes are similar. Previous attempts along this line have primarily considered the arm rewards to be a smooth function over graph Laplacian, which however failed to characterize the inherent problem complexity in terms of the graph structure.%, where lies the primary motivation of this work. We bridge this gap by showing a regret guarantee of $\\tO(\\chi(\\overline{G})\\sqrt{T})$ \\footnote{$\\tO(\\cdot)$ notation hides dependencies on $\\log T$.} that scales only with the chromatic number of the complement graph $\\chi(\\overline{G})$, assuming the rewards to be a smooth function over a general class of graph embeddings—\\emph{Orthonormal Representations}. Our proposed algorithms yield a regret guarantee of $\\tilde O(r\\sqrt T)$ for any general embedding of rank $r$. Moreover, if the rewards correspond to a minimum rank embedding, the regret boils down to $\\tO(\\chi(\\overline{G})\\sqrt{T})$–none of the existing works were able to bring out such influences of graph structures over arm rewards. Finally, noting that computing the above minimum rank embedding is NP-Hard, we also propose an alternative $O(|V| + |E|)$ time computable embedding scheme—{\\it Greedy Embeddings}—based on greedy graph coloring, with which our algorithms perform optimally on a large family of graphs, e.g. \\hspace{-8pt} union of cliques, complement of $k$-colorable graphs, regular graphs, trees etc, and are also shown to outperform state-of-the-art methods on real datasets. Our findings open up new roads for exploiting graph structures on regret performance.<\/p>\n","protected":false},"excerpt":{"rendered":"
We study the classical linear bandit problem on \\emph{graphs} modeling arm rewards through an underlying graph structure $G$($V$,$E$) such that rewards of neighboring nodes are similar. Previous attempts along this line have primarily considered the arm rewards to be a smooth function over graph Laplacian, which however failed to characterize the inherent problem complexity in […]<\/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":null,"msr_publishername":"","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"","msr_number":"","msr_organization":"","msr_pages_string":"","msr_page_range_start":"595","msr_page_range_end":"605","msr_series":"","msr_volume":"","msr_copyright":"","msr_conference_name":"Uncertainty in Artificial Intelligence","msr_doi":"","msr_arxiv_id":"","msr_s2_paper_id":"","msr_mag_id":"2965678303","msr_pubmed_id":"","msr_other_authors":"","msr_other_contributors":"","msr_speaker":"","msr_award":"","msr_affiliation":"","msr_institution":"","msr_host":"","msr_version":"","msr_duration":"","msr_original_fields_of_study":null,"msr_release_tracker_id":"","msr_s2_match_type":"","msr_citation_count_updated":"","msr_published_date":"2018-12-31","msr_highlight_text":"","msr_notes":"","msr_longbiography":"","msr_publicationurl":"","msr_external_url":"","msr_secondary_video_url":"","msr_conference_url":"","msr_journal_url":"","msr_s2_pdf_url":"","msr_year":0,"msr_citation_count":0,"msr_influential_citations":0,"msr_reference_count":0,"msr_s2_match_confidence":0,"msr_microsoftintellectualproperty":true,"msr_s2_open_access":false,"msr_s2_author_ids":[],"msr_pub_ids":[],"msr_hide_image_in_river":0,"footnotes":""},"msr-research-highlight":[],"research-area":[13561,13556,13555],"msr-publication-type":[193716],"msr-publisher":[],"msr-focus-area":[],"msr-locale":[268875],"msr-post-option":[],"msr-field-of-study":[256714,250747,246691,251620,257161,250333,257158,257164,253219],"msr-conference":[],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-757006","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-algorithms","msr-research-area-artificial-intelligence","msr-research-area-search-information-retrieval","msr-locale-en_us","msr-field-of-study-chromatic-scale","msr-field-of-study-combinatorics","msr-field-of-study-computer-science","msr-field-of-study-graph-abstract-data-type","msr-field-of-study-independent-set","msr-field-of-study-kernel-method","msr-field-of-study-linear-bandits","msr-field-of-study-orthogonal-embedding","msr-field-of-study-regret-minimization"],"msr_publishername":"","msr_edition":"","msr_affiliation":"","msr_published_date":"2018-12-31","msr_host":"","msr_duration":"","msr_version":"","msr_speaker":"","msr_other_contributors":"","msr_booktitle":"","msr_pages_string":"","msr_chapter":"","msr_isbn":"","msr_journal":"","msr_volume":"","msr_number":"","msr_editors":"","msr_series":"","msr_issue":"","msr_organization":"","msr_how_published":"","msr_notes":"","msr_highlight_text":"","msr_release_tracker_id":"","msr_original_fields_of_study":"","msr_download_urls":"","msr_external_url":"","msr_secondary_video_url":"","msr_longbiography":"","msr_microsoftintellectualproperty":1,"msr_main_download":"","msr_publicationurl":"","msr_doi":"","msr_publication_uploader":[{"type":"url","viewUrl":"false","id":"false","title":"http:\/\/auai.org\/uai2019\/proceedings\/papers\/206.pdf","label_id":"243132","label":0},{"type":"url","viewUrl":"false","id":"false","title":"http:\/\/proceedings.mlr.press\/v115\/s20a.html","label_id":"243109","label":0},{"type":"url","viewUrl":"false","id":"false","title":"https:\/\/dblp.uni-trier.de\/db\/conf\/uai\/uai2019.html#SahaSB19","label_id":"243109","label":0}],"msr_related_uploader":"","msr_citation_count":0,"msr_citation_count_updated":"","msr_s2_paper_id":"","msr_influential_citations":0,"msr_reference_count":0,"msr_arxiv_id":"","msr_s2_author_ids":[],"msr_s2_open_access":false,"msr_s2_pdf_url":null,"msr_attachments":[],"msr-author-ordering":[{"type":"user_nicename","value":"Aadirupa Saha","user_id":39835,"rest_url":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/microsoft-research\/v1\/researchers?person=Aadirupa Saha"},{"type":"text","value":"Shreyas Sheshadri","user_id":0,"rest_url":false},{"type":"text","value":"Chiranjib Bhattacharyya","user_id":0,"rest_url":false}],"msr_impact_theme":[],"msr_research_lab":[],"msr_event":[],"msr_group":[],"msr_project":[],"publication":[],"video":[],"msr-tool":[],"msr_publication_type":"inproceedings","related_content":[],"_links":{"self":[{"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item\/757006","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item"}],"about":[{"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/types\/msr-research-item"}],"version-history":[{"count":3,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item\/757006\/revisions"}],"predecessor-version":[{"id":757015,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item\/757006\/revisions\/757015"}],"wp:attachment":[{"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/media?parent=757006"}],"wp:term":[{"taxonomy":"msr-research-highlight","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-research-highlight?post=757006"},{"taxonomy":"msr-research-area","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/research-area?post=757006"},{"taxonomy":"msr-publication-type","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-publication-type?post=757006"},{"taxonomy":"msr-publisher","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-publisher?post=757006"},{"taxonomy":"msr-focus-area","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-focus-area?post=757006"},{"taxonomy":"msr-locale","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-locale?post=757006"},{"taxonomy":"msr-post-option","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-post-option?post=757006"},{"taxonomy":"msr-field-of-study","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-field-of-study?post=757006"},{"taxonomy":"msr-conference","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-conference?post=757006"},{"taxonomy":"msr-journal","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-journal?post=757006"},{"taxonomy":"msr-impact-theme","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-impact-theme?post=757006"},{"taxonomy":"msr-pillar","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-pillar?post=757006"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}