{"id":"https://openalex.org/W7165664883","doi":"https://doi.org/10.48550/arxiv.2606.21497","title":"Breaking chains with trees: Deep learning with $\\mathcal{O}(\\log N)$ parallel time complexity","display_name":"Breaking chains with trees: Deep learning with $\\mathcal{O}(\\log N)$ parallel time complexity","publication_year":2026,"publication_date":"2026-06-19","ids":{"openalex":"https://openalex.org/W7165664883","doi":"https://doi.org/10.48550/arxiv.2606.21497"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2606.21497","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.21497","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.48550/arxiv.2606.21497","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5078701018","display_name":"Neeraj Mohan Sushma","orcid":"https://orcid.org/0000-0003-1068-1463"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Sushma, Neeraj Mohan","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5120278161","display_name":"Aditya Nagarsekar","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Nagarsekar, Aditya","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5139201445","display_name":"Cabrel Teguemne Fokam","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Fokam, Cabrel Teguemne","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5037634838","display_name":"Robin Schiewer","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Schiewer, Robin","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5069676305","display_name":"A Pal","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Pal, Amit Kumar","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5016175782","display_name":"Anand Subramoney","orcid":"https://orcid.org/0000-0002-7333-9860"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Subramoney, Anand","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5052732444","display_name":"David Kappel","orcid":"https://orcid.org/0000-0001-5942-4033"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Kappel, David","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":false,"cited_by_count":0,"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/T11273","display_name":"Advanced Graph Neural Networks","score":0.2906000018119812,"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/T11273","display_name":"Advanced Graph Neural Networks","score":0.2906000018119812,"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/T10036","display_name":"Advanced Neural Network Applications","score":0.1932000070810318,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"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/T10028","display_name":"Topic Modeling","score":0.12860000133514404,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/backpropagation","display_name":"Backpropagation","score":0.7696999907493591},{"id":"https://openalex.org/keywords/deep-learning","display_name":"Deep learning","score":0.7401999831199646},{"id":"https://openalex.org/keywords/scalability","display_name":"Scalability","score":0.677299976348877},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.6255999803543091},{"id":"https://openalex.org/keywords/inference","display_name":"Inference","score":0.5365999937057495},{"id":"https://openalex.org/keywords/sequence","display_name":"Sequence (biology)","score":0.5246999859809875},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.515500009059906},{"id":"https://openalex.org/keywords/types-of-artificial-neural-networks","display_name":"Types of artificial neural networks","score":0.4415999948978424}],"concepts":[{"id":"https://openalex.org/C155032097","wikidata":"https://www.wikidata.org/wiki/Q798503","display_name":"Backpropagation","level":3,"score":0.7696999907493591},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.7631999850273132},{"id":"https://openalex.org/C108583219","wikidata":"https://www.wikidata.org/wiki/Q197536","display_name":"Deep learning","level":2,"score":0.7401999831199646},{"id":"https://openalex.org/C48044578","wikidata":"https://www.wikidata.org/wiki/Q727490","display_name":"Scalability","level":2,"score":0.677299976348877},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.6725000143051147},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.6255999803543091},{"id":"https://openalex.org/C2776214188","wikidata":"https://www.wikidata.org/wiki/Q408386","display_name":"Inference","level":2,"score":0.5365999937057495},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.5246999859809875},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.515500009059906},{"id":"https://openalex.org/C177973122","wikidata":"https://www.wikidata.org/wiki/Q7860946","display_name":"Types of artificial neural networks","level":4,"score":0.4415999948978424},{"id":"https://openalex.org/C101738243","wikidata":"https://www.wikidata.org/wiki/Q786435","display_name":"Autoencoder","level":3,"score":0.42820000648498535},{"id":"https://openalex.org/C147168706","wikidata":"https://www.wikidata.org/wiki/Q1457734","display_name":"Recurrent neural network","level":3,"score":0.3984000086784363},{"id":"https://openalex.org/C40506919","wikidata":"https://www.wikidata.org/wiki/Q7452469","display_name":"Sequence learning","level":2,"score":0.3939000070095062},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.37770000100135803},{"id":"https://openalex.org/C2984842247","wikidata":"https://www.wikidata.org/wiki/Q197536","display_name":"Deep neural networks","level":3,"score":0.36250001192092896},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.30959999561309814},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.29760000109672546},{"id":"https://openalex.org/C2777472644","wikidata":"https://www.wikidata.org/wiki/Q16968992","display_name":"Approximate inference","level":3,"score":0.29440000653266907},{"id":"https://openalex.org/C125411270","wikidata":"https://www.wikidata.org/wiki/Q18653","display_name":"Encoding (memory)","level":2,"score":0.2736999988555908},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.2655999958515167},{"id":"https://openalex.org/C71559656","wikidata":"https://www.wikidata.org/wiki/Q671298","display_name":"Divide and conquer algorithms","level":2,"score":0.26109999418258667},{"id":"https://openalex.org/C179799912","wikidata":"https://www.wikidata.org/wiki/Q205084","display_name":"Computational complexity theory","level":2,"score":0.26089999079704285},{"id":"https://openalex.org/C106515295","wikidata":"https://www.wikidata.org/wiki/Q26806595","display_name":"Parallel processing","level":2,"score":0.25540000200271606}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2606.21497","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.21497","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2606.21497","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.21497","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Modern":[0],"deep":[1,84,125],"neural":[2,85,126],"network":[3,140],"architectures":[4],"are":[5,32],"trained":[6,91],"via":[7],"backpropagation,":[8],"which":[9,45],"requires":[10,46],"errors":[11],"to":[12,123,153,186,191],"be":[13,22],"sequentially":[14],"propagated":[15],"through":[16,198],"all":[17],"layers":[18],"before":[19],"parameters":[20],"can":[21],"updated.":[23],"This":[24],"introduces":[25],"two":[26],"limitations:":[27],"locking,":[28],"where":[29,134],"layer-wise":[30],"updates":[31],"strictly":[33],"interdependent":[34],"and":[35,40,49,62,65,175],"cannot":[36],"proceed":[37],"in":[38,128],"parallel,":[39],"the":[41,101,113,117,137],"weight":[42],"transport":[43],"problem,":[44],"symmetric":[47],"forward":[48],"backward":[50],"pathways":[51],"for":[52,68,103],"exact":[53],"gradient":[54],"computation.":[55],"These":[56],"constraints":[57],"restrict":[58],"parallelism,":[59],"increase":[60],"memory":[61],"communication":[63],"overhead,":[64],"pose":[66],"challenges":[67],"scalable":[69],"learning.":[70],"In":[71],"this":[72],"work,":[73],"we":[74],"propose":[75],"Hierarchical":[76],"Block-Local":[77],"Learning":[78],"(HBLL),":[79],"a":[80,148,170],"framework":[81],"that":[82,120,144,193],"decomposes":[83],"networks":[86,127],"into":[87],"hierarchically":[88],"linked":[89],"blocks":[90],"using":[92],"local":[93],"learning":[94],"objectives":[95],"derived":[96],"from":[97],"variational":[98],"principles,":[99],"eliminating":[100],"need":[102],"full":[104],"end-to-end":[105],"backpropagation":[106,197],"while":[107],"maintaining":[108],"effective":[109,162],"information":[110],"propagation":[111],"across":[112],"network.":[114],"HBLL":[115,145,168,185],"is":[116,121,136],"first":[118],"algorithm":[119],"able":[122],"train":[124],"$\\mathcal{O}(\\log":[129],"N)$":[130],"parallel":[131],"time":[132],"complexity,":[133],"$N$":[135],"number":[138],"of":[139,150,164,172],"layers.":[141,165],"We":[142,166,182],"show":[143],"implicitly":[146],"defines":[147],"family":[149],"subnetworks":[151],"corresponding":[152],"different":[154,161],"hierarchical":[155],"paths,":[156],"enabling":[157],"flexible":[158],"inference":[159],"with":[160],"numbers":[163],"evaluate":[167],"on":[169,196],"set":[171],"challenging":[173],"vision":[174],"language":[176],"modeling":[177],"tasks,":[178],"achieving":[179],"competitive":[180],"performance.":[181],"also":[183],"extend":[184],"recurrent":[187],"sequence":[188],"architectures,":[189],"applying":[190],"settings":[192],"otherwise":[194],"rely":[195],"time.":[199]},"counts_by_year":[],"updated_date":"2026-07-01T06:00:48.157686","created_date":"2026-06-24T00:00:00"}