{"id":"https://openalex.org/W7166586870","doi":"https://doi.org/10.48550/arxiv.2606.27721","title":"Learning to Reason with Curriculum II: Compositional Generalization","display_name":"Learning to Reason with Curriculum II: Compositional Generalization","publication_year":2026,"publication_date":"2026-06-26","ids":{"openalex":"https://openalex.org/W7166586870","doi":"https://doi.org/10.48550/arxiv.2606.27721"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2606.27721","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.27721","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,"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.2606.27721","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5064641699","display_name":"Nived Rajaraman","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Rajaraman, Nived","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5130237001","display_name":"Audrey Huang","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Huang, Audrey","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5089372170","display_name":"Miroslav Dud\u0131\u0301k","orcid":"https://orcid.org/0009-0001-8565-0621"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Dudik, Miroslav","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5127786042","display_name":"Robert Schapire","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Schapire, Robert","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5139630283","display_name":"Dylan Foster","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Foster, Dylan","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5015082848","display_name":"Akshay Krishnamurthy","orcid":"https://orcid.org/0000-0002-5738-2383"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Krishnamurthy, Akshay","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/T12072","display_name":"Machine Learning and Algorithms","score":0.8626000285148621,"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/T12072","display_name":"Machine Learning and Algorithms","score":0.8626000285148621,"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/T12002","display_name":"Computability, Logic, AI Algorithms","score":0.024399999529123306,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"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/T12535","display_name":"Machine Learning and Data Classification","score":0.014299999922513962,"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/generalization","display_name":"Generalization","score":0.7552000284194946},{"id":"https://openalex.org/keywords/sequence-learning","display_name":"Sequence learning","score":0.6283000111579895},{"id":"https://openalex.org/keywords/sequence","display_name":"Sequence (biology)","score":0.6028000116348267},{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.5031999945640564},{"id":"https://openalex.org/keywords/modular-design","display_name":"Modular design","score":0.4672999978065491},{"id":"https://openalex.org/keywords/block","display_name":"Block (permutation group theory)","score":0.4629000127315521},{"id":"https://openalex.org/keywords/outcome","display_name":"Outcome (game theory)","score":0.4510999917984009},{"id":"https://openalex.org/keywords/reinforcement-learning","display_name":"Reinforcement learning","score":0.4350000023841858},{"id":"https://openalex.org/keywords/algorithmic-learning-theory","display_name":"Algorithmic learning theory","score":0.40450000762939453},{"id":"https://openalex.org/keywords/state","display_name":"State (computer science)","score":0.38269999623298645}],"concepts":[{"id":"https://openalex.org/C177148314","wikidata":"https://www.wikidata.org/wiki/Q170084","display_name":"Generalization","level":2,"score":0.7552000284194946},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.6482999920845032},{"id":"https://openalex.org/C40506919","wikidata":"https://www.wikidata.org/wiki/Q7452469","display_name":"Sequence learning","level":2,"score":0.6283000111579895},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.6028000116348267},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.5723999738693237},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.5031999945640564},{"id":"https://openalex.org/C101468663","wikidata":"https://www.wikidata.org/wiki/Q1620158","display_name":"Modular design","level":2,"score":0.4672999978065491},{"id":"https://openalex.org/C2777210771","wikidata":"https://www.wikidata.org/wiki/Q4927124","display_name":"Block (permutation group theory)","level":2,"score":0.4629000127315521},{"id":"https://openalex.org/C148220186","wikidata":"https://www.wikidata.org/wiki/Q7111912","display_name":"Outcome (game theory)","level":2,"score":0.4510999917984009},{"id":"https://openalex.org/C97541855","wikidata":"https://www.wikidata.org/wiki/Q830687","display_name":"Reinforcement learning","level":2,"score":0.4350000023841858},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.43389999866485596},{"id":"https://openalex.org/C32254414","wikidata":"https://www.wikidata.org/wiki/Q4724364","display_name":"Algorithmic learning theory","level":3,"score":0.40450000762939453},{"id":"https://openalex.org/C48103436","wikidata":"https://www.wikidata.org/wiki/Q599031","display_name":"State (computer science)","level":2,"score":0.38269999623298645},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.37279999256134033},{"id":"https://openalex.org/C50292564","wikidata":"https://www.wikidata.org/wiki/Q2462783","display_name":"Computational learning theory","level":3,"score":0.3646000027656555},{"id":"https://openalex.org/C85847156","wikidata":"https://www.wikidata.org/wiki/Q59015987","display_name":"Verifiable secret sharing","level":3,"score":0.34860000014305115},{"id":"https://openalex.org/C26517878","wikidata":"https://www.wikidata.org/wiki/Q228039","display_name":"Key (lock)","level":2,"score":0.3463999927043915},{"id":"https://openalex.org/C176248197","wikidata":"https://www.wikidata.org/wiki/Q458526","display_name":"Probably approximately correct learning","level":4,"score":0.3409999907016754},{"id":"https://openalex.org/C48145219","wikidata":"https://www.wikidata.org/wiki/Q1335365","display_name":"Security token","level":2,"score":0.33320000767707825},{"id":"https://openalex.org/C77618280","wikidata":"https://www.wikidata.org/wiki/Q1155772","display_name":"Scheme (mathematics)","level":2,"score":0.3222000002861023},{"id":"https://openalex.org/C24138899","wikidata":"https://www.wikidata.org/wiki/Q17141258","display_name":"Instance-based learning","level":3,"score":0.3165000081062317},{"id":"https://openalex.org/C98045186","wikidata":"https://www.wikidata.org/wiki/Q205663","display_name":"Process (computing)","level":2,"score":0.3131999969482422},{"id":"https://openalex.org/C77967617","wikidata":"https://www.wikidata.org/wiki/Q4677561","display_name":"Active learning (machine learning)","level":2,"score":0.305400013923645},{"id":"https://openalex.org/C168167062","wikidata":"https://www.wikidata.org/wiki/Q1117970","display_name":"Component (thermodynamics)","level":2,"score":0.30149999260902405},{"id":"https://openalex.org/C2780451532","wikidata":"https://www.wikidata.org/wiki/Q759676","display_name":"Task (project management)","level":2,"score":0.28119999170303345},{"id":"https://openalex.org/C175154964","wikidata":"https://www.wikidata.org/wiki/Q380077","display_name":"Task analysis","level":3,"score":0.2750999927520752},{"id":"https://openalex.org/C2776608160","wikidata":"https://www.wikidata.org/wiki/Q4785462","display_name":"Natural (archaeology)","level":2,"score":0.2696000039577484},{"id":"https://openalex.org/C203313322","wikidata":"https://www.wikidata.org/wiki/Q5158394","display_name":"Concept class","level":2,"score":0.2669000029563904},{"id":"https://openalex.org/C47177190","wikidata":"https://www.wikidata.org/wiki/Q207137","display_name":"Curriculum","level":2,"score":0.26510000228881836},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.26190000772476196},{"id":"https://openalex.org/C136389625","wikidata":"https://www.wikidata.org/wiki/Q334384","display_name":"Supervised learning","level":3,"score":0.2603999972343445},{"id":"https://openalex.org/C114289077","wikidata":"https://www.wikidata.org/wiki/Q3284399","display_name":"Statistical model","level":2,"score":0.25870001316070557},{"id":"https://openalex.org/C124304363","wikidata":"https://www.wikidata.org/wiki/Q673661","display_name":"Abstraction","level":2,"score":0.2554999887943268}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2606.27721","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.27721","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,"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.2606.27721","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.27721","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,"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":[{"score":0.7949337363243103,"id":"https://metadata.un.org/sdg/4","display_name":"Quality Education"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Compositional":[0],"generalization,":[1],"the":[2,32,63,72,118,138,147,162,167,188,204,207,213],"ability":[3],"to":[4,11,68,113,218],"solve":[5,114],"complex":[6],"problems":[7],"by":[8,133,172,180],"combining":[9],"solutions":[10],"simpler":[12],"sub-problems,":[13,111],"is":[14],"a":[15,25,46,80,130,177,191,221],"fundamental":[16],"capability":[17],"of":[18,35,66,74,77,102,146,157],"both":[19],"natural":[20],"and":[21,24,42,89,116],"artificial":[22],"intelligence,":[23],"key":[26],"mechanism":[27],"underlying":[28],"chain-of-thought":[29],"reasoning.":[30],"However,":[31],"theoretical":[33],"underpinnings":[34],"compositional":[36],"generalization":[37],"remain":[38],"poorly":[39],"understood:":[40],"when":[41],"why":[43],"does":[44],"decomposing":[45,106],"problem":[47,65],"into":[48,109],"parts":[49],"yield":[50],"more":[51],"efficient":[52],"learning":[53,67,112,151,182],"than":[54,125],"solving":[55],"it":[56],"directly?":[57],"We":[58,92],"study":[59],"this":[60,103],"question":[61],"through":[62],"canonical":[64],"simulate":[69],"semiautomata":[70],"(predicting":[71],"outcome":[73,197],"$T$":[75,217],"steps":[76],"sequential":[78],"computation),":[79],"model":[81,194,209],"that":[82,94,201],"captures":[83],"state":[84],"tracking,":[85],"regular":[86],"language":[87],"recognition,":[88],"modular":[90],"arithmetic.":[91],"show":[93,200],"an":[95,196,228],"autocurriculum-based":[96],"approach":[97],"building":[98],"on":[99,143,206],"Part":[100],"I":[101],"series,":[104],"recursively":[105],"longer":[107],"sequences":[108],"shorter":[110,222],"them,":[115],"composing":[117],"solutions,":[119],"achieves":[120],"dramatically":[121],"better":[122],"statistical":[123],"complexity":[124],"direct":[126,173],"methods.":[127],"(i)":[128],"For":[129,176],"setting":[131,178],"inspired":[132,179],"supervised":[134],"fine-tuning":[135],"(SFT)":[136],"where":[137,187],"learner":[139,189],"receives":[140],"interactive":[141],"feedback":[142],"intermediate":[144],"states":[145],"computation,":[148],"curriculum":[149,202],"facilitates":[150],"from":[152,210],"only":[153],"$2^{\\mathcal{O}(\\sqrt{\\log":[154],"T})}$":[155],"tokens":[156],"supervision;":[158],"i.e.,":[159],"subpolynomial":[160],"in":[161],"sequence":[163,215],"length":[164,216,224],"$T$,":[165],"overcoming":[166],"$\u03a9(T)$":[168],"token":[169],"barrier":[170],"required":[171],"simulation.":[174],"(ii)":[175],"reinforcement":[181],"with":[183],"verifiable":[184],"rewards":[185],"(RLVR),":[186],"improves":[190],"pre-trained":[192],"reference":[193,208],"using":[195],"verifier,":[198],"we":[199],"reduces":[203],"requirement":[205],"coverage":[211,219],"at":[212,220],"full":[214],"block":[223],"$B":[225],"\\ll":[226],"T$,":[227],"exponentially":[229],"weaker":[230],"condition.":[231]},"counts_by_year":[],"updated_date":"2026-07-28T07:46:37.118299","created_date":"2026-06-30T00:00:00"}