{"id":"https://openalex.org/W2945159000","doi":"https://doi.org/10.48550/arxiv.1812.00456","title":"Revisiting the Softmax Bellman Operator: New Benefits and New Perspective","display_name":"Revisiting the Softmax Bellman Operator: New Benefits and New Perspective","publication_year":2018,"publication_date":"2018-12-02","ids":{"openalex":"https://openalex.org/W2945159000","doi":"https://doi.org/10.48550/arxiv.1812.00456","mag":"2945159000"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:1812.00456","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/1812.00456","pdf_url":"https://arxiv.org/pdf/1812.00456","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1812.00456","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5008391481","display_name":"Zhao Song","orcid":"https://orcid.org/0000-0002-0897-6724","observed_orcids":["https://orcid.org/0000-0002-0897-6724"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Song, Zhao","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5109839104","display_name":"Ronald Parr","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Parr, Ronald E.","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5016448581","display_name":"Lawrence Carin","orcid":"https://orcid.org/0000-0001-6277-7948","observed_orcids":["https://orcid.org/0000-0001-6277-7948"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Carin, Lawrence","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":46,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10462","display_name":"Reinforcement Learning in Robotics","score":1.0,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10462","display_name":"Reinforcement Learning in Robotics","score":1.0,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12794","display_name":"Adaptive Dynamic Programming Control","score":0.0,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12101","display_name":"Advanced Bandit Algorithms Research","score":0.0,"subfield":{"id":"https://openalex.org/subfields/1803","display_name":"Management Science and Operations Research"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/bellman-operator","display_name":"Bellman operator","score":0.6000000238418579},{"id":"https://openalex.org/keywords/deep-q-learning","display_name":"deep Q-learning","score":0.75},{"id":"https://openalex.org/keywords/double-q-learning","display_name":"double Q-learning","score":0.9269999861717224},{"id":"https://openalex.org/keywords/value-function-approximation","display_name":"value function approximation","score":0.8619999885559082},{"id":"https://openalex.org/keywords/contraction-property","display_name":"contraction property","score":0.6290000081062317},{"id":"https://openalex.org/keywords/inverse-temperature","display_name":"inverse temperature","score":0.7120000123977661},{"id":"https://openalex.org/keywords/convergence-rate","display_name":"convergence rate","score":0.527999997138977},{"id":"https://openalex.org/keywords/exploration","display_name":"exploration","score":0.5460000038146973}],"concepts":[{"id":"https://openalex.org/C188441871","wikidata":"https://www.wikidata.org/wiki/Q7554146","display_name":"Softmax function","level":3,"score":0.9613648653030396},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.7095911502838135},{"id":"https://openalex.org/C97541855","wikidata":"https://www.wikidata.org/wiki/Q830687","display_name":"Reinforcement learning","level":2,"score":0.5274931192398071},{"id":"https://openalex.org/C14646407","wikidata":"https://www.wikidata.org/wiki/Q1430750","display_name":"Bellman equation","level":2,"score":0.5145910978317261},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.47583556175231934},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.47554633021354675},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.46950626373291016},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.4628791809082031},{"id":"https://openalex.org/C2776291640","wikidata":"https://www.wikidata.org/wiki/Q2912517","display_name":"Value (mathematics)","level":2,"score":0.443136990070343},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.44164690375328064},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4102141857147217},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.34160739183425903},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.19457244873046875},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.14986860752105713},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.11102104187011719},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.07643204927444458},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"score":0.0},{"id":"https://openalex.org/C158448853","wikidata":"https://www.wikidata.org/wiki/Q425218","display_name":"Repressor","level":4,"score":0.0},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0},{"id":"https://openalex.org/C86339819","wikidata":"https://www.wikidata.org/wiki/Q407384","display_name":"Transcription factor","level":3,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"pmh:oai:arXiv.org:1812.00456","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/1812.00456","pdf_url":"https://arxiv.org/pdf/1812.00456","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"mag:2951589367","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/pdf/1812.00456v2","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"arXiv (Cornell University)","raw_type":null},{"id":"doi:10.48550/arxiv.1812.00456","doi":"https://doi.org/10.48550/arxiv.1812.00456","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1812.00456","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1812.00456","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/1812.00456","pdf_url":"https://arxiv.org/pdf/1812.00456","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G234683847","display_name":"RI: Small: Feature Encoding for Reinforcement Learning","funder_award_id":"1815300","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2945159000.pdf","grobid_xml":"https://content.openalex.org/works/W2945159000.grobid-xml"},"referenced_works_count":31,"referenced_works":["https://openalex.org/W51508254","https://openalex.org/W65193931","https://openalex.org/W1515891729","https://openalex.org/W1563348973","https://openalex.org/W2098432798","https://openalex.org/W2115211925","https://openalex.org/W2121863487","https://openalex.org/W2145060720","https://openalex.org/W2145339207","https://openalex.org/W2150468603","https://openalex.org/W2155968351","https://openalex.org/W2173564293","https://openalex.org/W2201581102","https://openalex.org/W2260756217","https://openalex.org/W2326740504","https://openalex.org/W2341171179","https://openalex.org/W2553109721","https://openalex.org/W2593044849","https://openalex.org/W2594103415","https://openalex.org/W2596758708","https://openalex.org/W2609650878","https://openalex.org/W2619268125","https://openalex.org/W2755631392","https://openalex.org/W2952264928","https://openalex.org/W2963169817","https://openalex.org/W2963170229","https://openalex.org/W2963211300","https://openalex.org/W2963267001","https://openalex.org/W2963423916","https://openalex.org/W2963757175","https://openalex.org/W2964291307"],"related_works":["https://openalex.org/W3035064526","https://openalex.org/W2561341241","https://openalex.org/W2626384641","https://openalex.org/W2905860909","https://openalex.org/W3193494395","https://openalex.org/W3175676778","https://openalex.org/W2903546694","https://openalex.org/W2921114252","https://openalex.org/W7096431683","https://openalex.org/W7203650104"],"abstract_inverted_index":{"The":[0],"impact":[1],"of":[2,33,43,83,110,159],"softmax":[3,49,85,135],"on":[4,46],"the":[5,30,34,48,84,95,102,108,115,134,139,157,173],"value":[6,23,160],"function":[7,113,161],"itself":[8],"in":[9,63,101,156],"reinforcement":[10],"learning":[11],"(RL)":[12],"is":[13,169],"often":[14],"viewed":[15],"as":[16],"problematic":[17],"because":[18],"it":[19,92],"leads":[20,57,152],"to":[21,58,94,153],"sub-optimal":[22,150],"(or":[24],"Q)":[25],"functions":[26],"and":[27,41,75,88,106],"interferes":[28],"with":[29,54,60],"contraction":[31],"properties":[32,82],"Bellman":[35,50,86,97,167],"operator.":[36],"Surprisingly,":[37],"despite":[38],"these":[39],"concerns,":[40],"independent":[42],"its":[44,67,111,126],"effect":[45],"exploration,":[47],"operator":[51,98,136,151],"when":[52,175],"combined":[53],"Deep":[55],"Q-learning,":[56],"Q-functions":[59],"superior":[61,127],"policies":[62],"practice,":[64],"even":[65],"outperforming":[66],"double":[68],"Q-learning":[69],"counterpart.":[70],"To":[71],"better":[72,154],"understand":[73],"how":[74],"why":[76,148],"this":[77],"occurs,":[78],"we":[79,130],"revisit":[80],"theoretical":[81],"operator,":[87],"prove":[89],"that":[90,133],"$(i)$":[91],"converges":[93],"standard":[96],"exponentially":[99],"fast":[100],"inverse":[103],"temperature":[104],"parameter,":[105],"$(ii)$":[107],"distance":[109],"Q":[112],"from":[114],"optimal":[116],"one":[117],"can":[118,137],"be":[119],"bounded.":[120],"These":[121],"alone":[122],"do":[123],"not":[124],"explain":[125],"performance,":[128],"so":[129],"also":[131],"show":[132],"reduce":[138],"overestimation":[140],"error,":[141],"which":[142],"may":[143],"give":[144],"some":[145],"insight":[146],"into":[147],"a":[149],"performance":[155],"presence":[158],"approximation.":[162],"A":[163],"comparison":[164],"among":[165],"different":[166],"operators":[168],"then":[170],"presented,":[171],"showing":[172],"trade-offs":[174],"selecting":[176],"them.":[177]},"counts_by_year":[{"year":2025,"cited_by_count":3},{"year":2024,"cited_by_count":10},{"year":2023,"cited_by_count":6},{"year":2022,"cited_by_count":10},{"year":2021,"cited_by_count":12},{"year":2020,"cited_by_count":5}],"updated_date":"2026-11-04T10:54:05.112534","created_date":"2019-05-29T00:00:00"}
