{"id":"https://openalex.org/W7164201854","doi":"https://doi.org/10.48550/arxiv.2606.09883","title":"TD-Grokking: Learning from Zero-Reward Problems by Training-Time Decomposition","display_name":"TD-Grokking: Learning from Zero-Reward Problems by Training-Time Decomposition","publication_year":2026,"publication_date":"2026-06-03","ids":{"openalex":"https://openalex.org/W7164201854","doi":"https://doi.org/10.48550/arxiv.2606.09883"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2606.09883","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.09883","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":null,"license_id":null,"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.2606.09883","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5004687734","display_name":"Ningyuan Xi","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Xi, Ningyuan","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5138379275","display_name":"Hao Xu","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Xu, Hao","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5039197077","display_name":"Hongsheng Xin","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Xin, Hongsheng","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5136468357","display_name":"Ning Miao","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Miao, Ning","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/T12026","display_name":"Explainable Artificial Intelligence (XAI)","score":0.20160000026226044,"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/T12026","display_name":"Explainable Artificial Intelligence (XAI)","score":0.20160000026226044,"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/T13702","display_name":"Machine Learning in Healthcare","score":0.15889999270439148,"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/T10462","display_name":"Reinforcement Learning in Robotics","score":0.1152999997138977,"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/verifiable-secret-sharing","display_name":"Verifiable secret sharing","score":0.7738999724388123},{"id":"https://openalex.org/keywords/decomposition","display_name":"Decomposition","score":0.7106999754905701},{"id":"https://openalex.org/keywords/usable","display_name":"USable","score":0.6776000261306763},{"id":"https://openalex.org/keywords/reinforcement-learning","display_name":"Reinforcement learning","score":0.6705999970436096},{"id":"https://openalex.org/keywords/bottleneck","display_name":"Bottleneck","score":0.6531000137329102},{"id":"https://openalex.org/keywords/task","display_name":"Task (project management)","score":0.5756000280380249}],"concepts":[{"id":"https://openalex.org/C85847156","wikidata":"https://www.wikidata.org/wiki/Q59015987","display_name":"Verifiable secret sharing","level":3,"score":0.7738999724388123},{"id":"https://openalex.org/C124681953","wikidata":"https://www.wikidata.org/wiki/Q339062","display_name":"Decomposition","level":2,"score":0.7106999754905701},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.7016000151634216},{"id":"https://openalex.org/C2780615836","wikidata":"https://www.wikidata.org/wiki/Q2471869","display_name":"USable","level":2,"score":0.6776000261306763},{"id":"https://openalex.org/C97541855","wikidata":"https://www.wikidata.org/wiki/Q830687","display_name":"Reinforcement learning","level":2,"score":0.6705999970436096},{"id":"https://openalex.org/C2780513914","wikidata":"https://www.wikidata.org/wiki/Q18210350","display_name":"Bottleneck","level":2,"score":0.6531000137329102},{"id":"https://openalex.org/C2780451532","wikidata":"https://www.wikidata.org/wiki/Q759676","display_name":"Task (project management)","level":2,"score":0.5756000280380249},{"id":"https://openalex.org/C98045186","wikidata":"https://www.wikidata.org/wiki/Q205663","display_name":"Process (computing)","level":2,"score":0.5465999841690063},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.5162000060081482},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.414000004529953},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.40860000252723694},{"id":"https://openalex.org/C12725497","wikidata":"https://www.wikidata.org/wiki/Q810247","display_name":"Baseline (sea)","level":2,"score":0.38690000772476196},{"id":"https://openalex.org/C2776760102","wikidata":"https://www.wikidata.org/wiki/Q5139990","display_name":"Code (set theory)","level":3,"score":0.3447999954223633},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.33660000562667847},{"id":"https://openalex.org/C84945661","wikidata":"https://www.wikidata.org/wiki/Q7366567","display_name":"Root cause","level":2,"score":0.30160000920295715},{"id":"https://openalex.org/C171078966","wikidata":"https://www.wikidata.org/wiki/Q111029","display_name":"Root (linguistics)","level":2,"score":0.2815999984741211},{"id":"https://openalex.org/C125411270","wikidata":"https://www.wikidata.org/wiki/Q18653","display_name":"Encoding (memory)","level":2,"score":0.273499995470047},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.26179999113082886}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2606.09883","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.09883","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":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2606.09883","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.09883","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":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Large":[0],"language":[1],"models":[2],"(LLMs)":[3],"have":[4],"made":[5],"remarkable":[6],"progress":[7],"in":[8],"reasoning":[9,38],"tasks,":[10],"largely":[11],"driven":[12],"by":[13],"post-training":[14],"paradigms,":[15],"especially":[16],"reinforcement":[17],"learning":[18],"with":[19,82,143],"verifiable":[20,113],"rewards":[21],"(RLVR).":[22],"However,":[23],"a":[24,98],"critical":[25],"bottleneck":[26],"persists:":[27],"RLVR":[28],"fails":[29],"on":[30,125],"highly":[31],"challenging":[32],"zero-reward":[33,103,154],"problems,":[34],"where":[35,118],"all":[36,139],"sampled":[37],"trajectories":[39],"yield":[40],"uniformly":[41],"failed":[42],"outcomes,":[43],"providing":[44],"no":[45],"optimization":[46],"signal":[47],"to":[48,54,86],"drive":[49],"model":[50,81],"improvement.":[51],"Prior":[52],"efforts":[53],"address":[55,93],"this":[56],"limitation,":[57],"such":[58],"as":[59,136,138],"dense":[60],"process":[61],"supervision,":[62],"partial":[63],"reward":[64],"assignment,":[65],"or":[66,74],"prefix-guided":[67],"exploration,":[68],"suffer":[69],"from":[70],"inherent":[71],"task":[72],"constraints":[73],"do":[75],"not":[76],"fully":[77],"equip":[78],"the":[79,83,88],"policy":[80],"capabilities":[84],"necessary":[85],"solve":[87],"original":[89],"intractable":[90,108],"problems.":[91,104],"To":[92],"this,":[94],"we":[95],"propose":[96],"TD-Grokking,":[97],"training-time":[99,150],"decomposition":[100,151],"framework":[101],"for":[102],"It":[105],"recursively":[106],"decomposes":[107],"root":[109],"problems":[110],"into":[111,156],"self-contained,":[112],"subproblems,":[114],"forming":[115],"hierarchical":[116],"trees":[117],"solvable":[119],"leaves":[120],"provide":[121],"non-zero":[122],"rewards.":[123],"Evaluations":[124],"mathematical":[126],"and":[127,166],"medical":[128],"tasks":[129],"show":[130],"that":[131,149],"TD-Grokking":[132],"outperforms":[133],"vanilla":[134],"GRPO":[135],"well":[137],"baseline":[140],"approaches.":[141],"Together":[142],"detailed":[144],"analysis,":[145],"these":[146],"results":[147],"confirm":[148],"effectively":[152],"converts":[153],"examples":[155],"usable":[157],"training":[158],"signals,":[159],"enabling":[160],"consistent":[161],"performance":[162],"gains.":[163],"Our":[164],"code":[165],"datasets":[167],"are":[168],"available":[169],"at":[170],"https://anonymous.4open.science/r/TD-Grokking-6567/.":[171]},"counts_by_year":[],"updated_date":"2026-07-01T06:00:48.157686","created_date":"2026-06-11T00:00:00"}
