| | | When Can | LLMs | Learn | to Reason | with | Weak | Supervision? | | | --- | --- | -------- | ---- | ----- | --------- | ---- | ---- | ------------ | --- | SalmanRahman12* JingyanShen2* AnnaMordvina2 HamidPalangi3 SaadiaGabriel1 PavelIzmailov2 | | | ❖Projectpage: | | salmanrahman.net/rlvr-weak-supervision | | | | | | | ----- | -------- | ------------- | ------------- | -------------------------------------- | --- | --------------------------------------------------- | --- | ---------------------- | --- | | | | Abstract | | | | pabilitiesinlargelanguagemodels(Guoetal.,2025;Jaech | | | | | | | | | | | etal.,2024;Teametal.,2025). | | Withonlybinaryfeedback | | | Large | language | models | have achieved | signifi- | | | | | | 6202 rpA 02 ]GL.sc[ 1v47581.4062:viXra cantreasoningimprovementsthroughreinforce- oncorrectness,RLVRhasenabledsubstantialgainsacross diversereasoningtaskswithoutrequiringdensesupervision. | ment | learning | with verifiable | rewards | (RLVR). | | | | | | | ---- | -------- | --------------- | ------- | ------- | --- | --- | --- | --- | --- | Yetasmodelcapabilitiesgrow,constructinghigh- However,recentfindingssuggesttheseimprovementsmay qualityrewardsignalsbecomesincreasinglydif- bedrivenbyfactorsotherthantheintegrationofcorrectness | | | | | | | signals. | SomestudiesreportthatRLVRsucceedsevenun- | | | | ------- | ------ | ------------ | ------------- | ---- | --- | -------- | ---------------------------------------- | --- | --- | | ficult, | making | it essential | to understand | when | | | | | | RLVR can succeed under weaker forms of su- derextremeconditions: trainingonjustasingleexample canyieldsignificantgains(Wangetal.,2025a),andrandom | pervision. | | We conduct | a systematic | empirical | | | | | | | ---------- | --- | ---------- | ------------ | --------- | --- | --- | --- | --- | --- | studyacrossdiversemodelfamiliesandreasoning orincorrectrewardssometimesmatchground-truthperfor- domains under three weak supervision settings: mance (Shao et al., 2025). Other work shows that proxy signalssuchasself-certainty(Zhaoetal.,2025;Prabhudesai | scarce | data, | noisy rewards, | and self-supervised | | | | | | | | ------ | ----- | -------------- | ------------------- | --- | --- | --- | --- | --- | --- | proxy rewards. We find that generalization is etal.,2025),entropyminimization(Agarwaletal.,2025), majorityvoting(Zuoetal.,2025),orself-generatedtraining governedbytrainingrewardsaturationdynamics: models that generalize exhibit a prolonged pre- data(Huangetal.,2025)canreplaceverifiablerewards. saturationphaseduringwhichtrainingrewardand Furthermore,techniquesthatsucceedononemodelfamily downstream performance climb together, while oftenfailonothers(Shaoetal.,2025),underreportedbase- modelsthatsaturaterapidlymemorizeratherthan linesmayinflateperceivedbenefits(Chandaketal.,2025), learn. Weidentifyreasoningfaithfulness,defined and prolonged training with proxy rewards (i.e., reward astheextenttowhichamodel’sintermediatesteps | | | | | | | signals derived | from | model outputs without | ground-truth | | --- | --- | --- | --- | --- | --- | --------------- | ---- | --------------------- | ------------ | logicallysupportitsfinalanswer,asthepre-RL verification)canleadtorewardhackingandperformance propertythatpredictswhichregimeamodelfalls collapse(Shafayatetal.,2025). Thesemixedresultsleavea into,whileoutputdiversityaloneisuninformative. | | | | | | | fundamentalquestion: | | WhencanRLVRgeneralize1under | | | --- | --- | --- | --- | --- | --- | -------------------- | --- | --------------------------- | --- | Motivatedbythesefindings,wedisentanglethe weaksupervision,andwhatdeterminessuccessorfailure? contributionsofcontinualpre-trainingandsuper- visedfine-tuning,findingthatSFTonexplicitrea- UnderstandingwhenRLVRworksunderweaksupervision mattersforpractice.Ground-truthverifiersareoftenlimited: soningtracesisnecessaryforgeneralizationunder weaksupervision,whilecontinualpre-trainingon labelsmaybenoisyorunavailable,andasmodelsbecome domaindataamplifiestheeffect.Appliedtogether strongerthantheirsupervisors,alternativerewardsignals becomenecessary(Burnsetal.,2023). toLlama3.2-3B-Base,theseinterventionsenable generalizationacrossallthreesettingswherethe WeconductasystematicempiricalstudyofRLVRunder basemodelpreviouslyfailed. weak supervision across two model families (Qwen and | | | | | | | Llama), | and three | reasoning domains (MATH, | SCIENCE, | | --- | --- | --- | --- | --- | --- | ---------- | --------------------------------------- | ------------------------ | -------- | | | | | | | | andGRAPH). | Ourworkisorganizedaroundthreequestions: | | | 1.Introduction | | | | | | | • RQ1(WeakSupervision):DoesRLVRgeneralizeacross | | | | | --- | --- | --- | --- | --- | --- | ----------------------------------------------- | --- | --- | --- | Reinforcementlearningwithverifiablerewards(RLVR)has modelfamiliesanddomainsunderscarcedata,noisyre- emergedasapowerfulparadigmforimprovingreasoningca- wards,andself-supervisedproxyrewards? *Equal 1University • RQ2(ModelProperties): Whatpre-RLmodelproper- | | contribution | | of | California, | Los An- | | | | | | ----- | ------------ | --------------- | -------- | -------------- | ------- | --- | --- | --- | --- | | | 2New | | 3Google. | | | | | | | | geles | | York University | | Correspondence | | | | | | 1Throughout,weusegeneralizationtomeanimprovementon | to: Salman | Rahman | , | | Jingyan | Shen | | | | | | ---------- | ------ | --------------------- | --- | ------- | ---- | --- | --- | --- | --- | downstreamevaluationbenchmarks,bothin-domainheld-outsets . andout-of-domaintransfer,followingRLtraining. Preprint.April21,2026. 1 LLMReasoningwithWeakSupervision tiesdeterminewhetheramodelgeneralizesunderweak 2.ExperimentalSetup supervision? | | | | | | | | We evaluate | the | following | model families: | | (1) Qwen2.5- | | | -------------------- | --- | --- | ---------------------------- | --- | --- | --- | -------------- | --- | ----------------------------------- | --------------- | --- | ------------ | --- | | • RQ3(Intervention): | | | Howcanweenablegeneralization | | | | | | | | | | | | | | | | | | | 1.5B/3B(Base): | | General-purposemodelspretrainedon18 | | | | | inmodelsthatfailunderweaksupervision? trilliontokens(Team,2024);(2)Qwen2.5-Math-1.5B/7B Ourinvestigationuncoversthreefindings. First,general- (Math-specialized): BuiltuponQwen2.5withanadditional izationunderweaksupervisionisgovernedbytraining 1trillionmath-relatedtokens(Yangetal.,2024);(3)Llama- rewardsaturationdynamics. Modelsthatgeneralizeex- 3.2-3B/8B-Instruct(Instruction-tuned): Pretrainedon9 hibitaprolongedpre-saturationphaseduringwhichtraining trilliontokensandalignedviaSFT,rejectionsampling,and | | | | | | | | DPO(Dubeyetal.,2024). | | | WeusetheInstructvariantsfor | | | | | --- | --- | --- | --- | --- | --- | --- | --------------------- | --- | --- | --------------------------- | --- | --- | --- | rewardclimbssteadilyandthemodellearnstransferablerea- soningpatterns;modelsthatfailsaturaterapidlyandentera Llamabecausethebasemodelsdonotreliablyfollowthe post-saturationphasewherefurthertrainingyieldsdiminish- required formatfor on-policy rollouts. We revisit Llama- ingreturns. Whichregimeamodelfallsintodependsonits Basein§4,whereSFThandlestheformat-followingissue. | pretrainingpriors: | | modelswithstrongdomain-alignedpre- | | | | | | | | | | | | | ------------------ | --- | ---------------------------------- | --- | --- | --- | --- | ------------------- | --- | ----------------------------- | --- | --- | --- | --- | | | | | | | | | DomainsandDatasets. | | Weselectthreedomainswithvary- | | | | | training(QwenonMATHandSCIENCE)sustainextended | | | | | | | | inglevelsofpretrainingexposure: | | | | MATH(highexposure), | | | | --- | --- | --- | --- | --- | --- | --- | ------------------------------- | --- | --- | --- | ------------------- | --- | --- | pre-saturationphasesandgeneralizeunderscarcedata,noisy SCIENCE(moderatecoverage)andGRAPHtasks(underrep- rewards,andself-supervisedproxyrewards,whilemodels | | | | | | | | resentedintypicalpretrainingcorpora). | | | | WeuseSkywork- | | | | --- | --- | --- | --- | --- | --- | --- | ------------------------------------- | --- | --- | --- | ------------- | --- | --- | withoutsuchpriors(Llamaacrossalldomains,andQwenon OR1(Heetal.,2025a)forMATH,SCPdatasets(Liuetal., GRAPH)saturaterapidlyandfailtogeneralizeevenunder | | | | | | | | 2025a; Lu | et al., | 2025) spanning | physics, | | chemistry, | and | | ------------------- | --- | -------------------------------- | --- | --- | --- | --- | --------- | ------- | -------------- | -------- | --- | ---------- | --- | | moderatelabelnoise. | | Wetreatthemodel-familycontrastas | | | | | | | | | | | | biologyforSCIENCE,andtasksfromReasoningGym(Sto- aproxyforpretraining-priorstrengthratherthananintrin- janovskietal.,2025)involvingdiscretealgorithmicreason- sicpropertyofeitherfamily,areadingthat§4confirmsby | | | | | | | | ingforGRAPH. | ForMathandScience,weusethe1.5B/3B | | | | | | | --- | --- | --- | --- | --- | --- | --- | ------------ | --------------------------------- | --- | --- | --- | --- | --- | showingthatcontinualpre-trainingonmathdatatransforms modelsasourprimaryexperimentsandadditionallyevalu- Llama’sRLbehaviortoresembleQwen’s. ate7B/8Bmodelstoverifythatourfindingsholdatlarger Second,reasoningfaithfulness,notoutputdiversity,dis- scale. For GRAPH, we only use the 7B/8B variants be- tinguishesmodelsthatgeneralizefrommodelsthatmem- cause the smaller models achieve solve@16 = 0, leaving orize. Anaturalhypothesisforrapidsaturationisthatfail- noinformativesignalforRL.Moredetailsareprovidedin | ing models | lack | exploratory | | capacity. | We find | the oppo- | AppendixB. | | | | | | | | ---------- | ---- | ----------- | --- | --------- | ------- | --------- | ---------- | --- | --- | --- | --- | --- | --- | site: Llamamodelsreachperfecttrainingrewardfasterthan | | | | | | | | Model-AwareDataFiltering. | | | Toensureinformativetrain- | | | | | --- | --- | --- | --- | --- | --- | --- | ------------------------- | --- | --- | ------------------------- | --- | --- | --- | Qwenandmaintainhigheroutputdiversitythroughouttrain- ingsignals,weimplementmodel-specificdifficultyfiltering. | ing, yet | they generalize | | poorly. | The | missing | property is | | | | | | | | | -------- | --------------- | --- | ------- | --- | ------- | ----------- | --- | --- | --- | --- | --- | --- | --- | Foreachproblem,wesample16responsesandcountcorrect reasoningfaithfulness,definedbywhetheramodel’sinter- | | | | | | | | solutions | (solve@16 | ∈ [0,16]). | We | retain | only problems | | | ------------------------------------------- | --- | --- | --- | --- | --- | ---------- | --------- | --------- | ---------- | --- | ------ | ------------- | --- | | mediatestepslogicallysupportitsfinalanswer. | | | | | | Modelsthat | | | | | | | | wheresolve@16∈[1,15],effectivelydiscardinginstances | saturate | rapidly | produce | correct | answers | through | reason- | | | | | | | | | -------- | ------- | ------- | ------- | ------- | ------- | ------- | --- | --- | --- | --- | --- | --- | --- | thatareeithertrivialorintractableforthemodel,stratified ingchainsthatdonotjustifythem,memorizingratherthan | | | | | | | | equally | across difficulty | levels | (details | in | Appendix | B.2). | | --------- | --------- | --- | ---------------- | --- | ---- | ---------- | ------- | ----------------- | ------ | -------- | --- | -------- | ----- | | learning. | Diversity | is | only informative | | when | considered | | | | | | | | Thisfilteredsetservesasthecandidatepoolforallweak jointlywithfaithfulness. supervisionsettingsstudiedinthiswork;wedescribehow Third,SFTonexplicitreasoningtracesisnecessaryfor trainingdataisconstructedfromthispoolforallsettings | generalizationunderweaksupervision, | | | | | andcontinual | | in§3. | | | | | | | | ---------------------------------------------- | --------- | --- | --- | ------- | ------------ | ---------- | ---------------------- | --- | ----------------------- | --- | --- | --- | --- | | pre-training | amplifies | | the | effect. | We run a | controlled | | | | | | | | | | | | | | | | TrainingConfiguration. | | WeuseGRPO(GroupRelative | | | | | | comparisonthatdisentanglesthetwointerventions, | | | | | | train- | | | | | | | | PolicyOptimization)asourRLalgorithm(Shaoetal.,2024). | ing Llama3.2-3B | | Base, | a continually | | pre-trained | variant | | | | | | | | | --------------- | --- | ----- | ------------- | --- | ----------- | ------- | --- | --- | --- | --- | --- | --- | --- | ForeachqueryqsampledfromtrainingdatasetsD,agroup | (CPT, ours), | and | Instruct, | each | with | either Thinking | SFT | | | | | | | | | --------------------------------------------------- | --- | --------- | ---- | ---- | --------------- | --- | ----------------------- | ----------- | ---- | ----------------------- | --- | ------------- | --- | | | | | | | | | ofindividualresponses{o | | }G | aresampledfromthepolicy | | | | | (explicitreasoningtraces)orNon-ThinkingSFT(finalso- | | | | | | | | | i | i=1 | | | | | | | | | | | | π before | the update. | GRPO | maximizes | | the following | | θold | lutionsonly). | ThinkingSFTisnecessary: | | | | itimprovesrea- | | | | | | | | | | ------------- | ----------------------- | --- | --- | --- | -------------- | --- | --- | --- | --- | --- | --- | --- | --- | objective: soningfaithfulness,extendsthepre-saturationphase,and (cid:104) (θ)=E enablesgeneralizationunderallthreeweaksupervisionset- J GRPO (q,a)∼D,{oi}G i=1 ∼πθold (·|q) | | | | | | | | G | |oi| | | | | | | | ------------------------------------------------ | --- | --- | --- | --- | --- | --- | ---------- | ---------- | ------------- | ---------- | ----------- | --- | ------- | | tings,whileNon-ThinkingSFTonthesamepromptsfails. | | | | | | | 1 (cid:88) | 1 (cid:88) | | | | | | | | | | | | | | | | min (cid:0) ρ | Aˆ ,clip(ρ | ,1−ϵ,1+ϵ)Aˆ | | (cid:1) | Continualpre-trainingisamultiplierratherthanasubstitute. i,t i i,t i | | | | | | | | G | |o | | | | | | | | ---------------------------------------------- | --- | --- | --- | --- | --- | --- | --- | ----- | --- | ------ | --------- | --- | --- | | | | | | | | | i=1 | i t=1 | | | | | | | CPTcombinedwithThinkingSFTproducesthestrongest | | | | | | | | | | | (cid:105) | | | | | | | | | | | | | −βD | (π ||π | ) , | | | generalization, recovering performance in settings where KL θ ref Llamapreviouslyfailed. πθ(oi,t|q,oi,0 1 1 sat (effectivepre-saturationlearning)whileredmarkrapidsaturationt(8) < 100. Thelarge-smallgapatsaturationstepsG(n1,8) and sat sat,in G(n1,8) aregenerallysmall.Resultsonmorebenchmarksandpass@kmetricsarereportedinTable3-7inAppendix. sat,ood In-domainBenchmarks OODBenchmark Model t(8) sat ∆( s 8 a ) t ∆∗ p ( o 8 s ) t ∆( s 8 a ) t ∆∗ p ( o 8 s ) t Gsat,in ∆( s 8 a ) t ∆∗ p ( o 8 s ) t Gsat,ood TrainingDomain:Math MATH500 AMC G(2048,8) SCP-Hard G(2048,8) sat,in sat,ood Qwen2.5-Math-1.5B 302 29.7 1.5 18.7 0.6 -1.1 10.5 2.1 2.4 Qwen2.5-1.5B 170 32.1 0.9 12.7 3.3 -0.5 7.0 0.3 -0.4 Llama3.2-3B-Instruct 55 10.8 -1.9 8.8 -2.1 -0.9 3.9 0.0 1.5 TrainingDomain:Science SCP-Hard GPQA-Diamond G(2048,8) MATH500 G(2048,8) sat,in sat,ood Qwen2.5-Math-1.5B 268 14.5 1.1 16.9 1.6 1.1 25.3 0.8 1.1 Qwen2.5-1.5B 161 6.4 0.2 13.3 1.7 1.8 32.3 2.1 1.2 Llama3.2-3B-Instruct 61 1.8 1.7 11.9 3.0 5.1 7.3 2.2 0.6 TrainingDomain:Graph QuantumLock LargestIsland G(n1,8)† MATH500 G(n1,8)† sat,in sat,ood Qwen2.5-Math-7B 150 8.3 4.9 19.8 1.9 -1.8 21.0 2.1 -3.7 Llama3.1-8B-Instruct 29 10.1 7.1 1.8 1.0 3.0† 9.1 3.8 0.0† acrossdifficultylevelsdefinedin§2. ForN < 64,were- tional gain after saturation, defined as ∆∗(n)(M) := post peatpromptsuniformlytoreachbatchsize64(e.g.,N =8 max M(n)(t)−M(n)(cid:0) t(n)(cid:1) . Values near zero implies8repeats). t∈[t( s n at ),T] sat indicatenegligiblepost-saturationgains. Tostudytrainingdynamics,weleveragerewardsaturationto • Large-small gap G(n′,n)(M): we define this gap as distinguishperiodswherethepolicyimprovesonthetrain- sat M(n′)(t(n))−M(n)(t(n))forn′ > n,whichcompares ingdatasetfromthosewhereitplateaus. Intuitively,once sat sat performancebetweenlarger(n′)andsmaller(n)datasets trainingrewardsaturates,furtherupdatesyieldlittlenewsig- (cid:104) (cid:105) atthesaturationstepofthesmallerrun. Atthesmaller nal.Wedefiner¯ :=E 1 (cid:80)G r as t q∼D,{oi}G i=1 ∼πold(·|q) G i=1 i run’ssaturationstep,howmuchbetterdoesthelargerrun theexpectedtrainingrewardatupdatestept∈{1,...,T}, perform? Largerpositivevaluesindicatesubstantialben- andletr¯ max :=max 1≤t≤T r¯ t bethemaximumrewardob- efitfrommoredata;valuesnearzerosuggestlimitedad- servedduringtraining. Weidentifytraininghassaturated vantagefromincreasingdatasetsize. WedenoteG(n1,8) sat,in oncetherewardisclosetothismaximum,anddefinethe astheaveragegapoverthein-domainbenchmarks,and saturationstepastheearliestupdatewherethisoccurs: G(n1,8) astheaveragegapoverOODbenchmarks. sat,ood (cid:110) t sat :=inf t∈{1,...,T eff }:r¯ t ≥ϵ max r¯ max }. Pre-saturationphasedominatessmall-samplelearning, and its length predicts generalization. Table 1 sum- Weuseϵ =0.99andsetT =T−50,i.e.,wesearchfor max eff marizes the proposed metrics across model families and t onlyuptothefirstT updatestoavoidboundaryeffects sat eff training domains when training on 8 examples. Results neartheendoftraining.Wedefinethepre-saturationphase onmorebenchmarksandpass@kmetricsareprovidedin asallstepst∈{1,...,t −1}andpost-saturationphase sat Appendix C.2 and Tables 3-7. All model-domain pairs asallstepst∈{min(t sat ,T),...,T}. showclearlypositive∆(8)forallmetrics(i.e.,bothavg@16 sat To quantify data efficiency, we introduce three metrics. andpass@k,k ∈ {4,8,16})acrossin-domainandout-of- LetM(n)(t)denoteanevaluationmetric(e.g.,avg@16on domain benchmarks, indicating that as few as 8 training MATH-500)attrainingsteptfortrainingwithnsamples, examples can trigger measurable learning during the pre- andt(n)bethecorrespondingsaturationstep. saturation phase. Neither G(2048,8) nor G(2048,8) is sig- sat sat,in sat,out nificantly greater than zero on 7 out of 8 model-domain • Pre-saturationgain∆(n)(M): performancegainfrom sat pairs,indicatingthatthepre-saturationimprovementsare initializationtosaturationas∆(n)(M):=M(n)(cid:0) t(n)(cid:1) − oftencomparabletothoseobtainedwithlargertrainingsets. sat sat M(n)(0). Largerpositivevaluesindicateeffectivelearn- Thissuggeststhatearlylearningisnotstronglydata-limited. ingbeforesaturation. Incontrast,thepost-saturationresidual∆∗(8) istypically post • Post-saturation residual ∆∗(n)(M): maximum addi- smallerthan∆(8),indicatingdiminishingreturnsoncethe post sat 4 LLMReasoningwithWeakSupervision Qwen2.5-Math-7B Llama-3.2-3B-Instruct gestthatevenformodelswithstrongmathematicalpriors, | | | Graph | | | | Math | | | | | | | | | | --- | --- | ----- | --- | --- | --- | ---- | --- | --- | --- | --- | --- | --- | --- | --- | thelackofdomain-specificpre-trainingacceleratessatura- | | draweR gniniarT 1.0 | | | draweR gniniarT 0.75 | | | | | | | | | | | | --- | ------------------- | --- | --- | -------------------- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 0.8 | | | | | 0.60 | | | | tionandnecessitateshigherdatavolumetodrivelearning. | | | | | | | | --- | --- | --- | --- | ---- | --- | --- | --- | --------------------------------------------------- | --- | --- | --- | --- | --- | --- | 0.6 | | | | | 0.45 | | | | Wefurtherprovideillustrationsfor7Band8Bmodelson | | | | | | | | --- | --- | --- | --- | ---- | --- | --- | --- | ----------------------------------------------- | --- | --- | --- | --- | --- | --- | 0.4 | | | | | 0.30 | | | | MATHandSCIENCEdomainsinAppendixC.3. | | | | | | | | --- | --- | --- | --- | ---- | --- | --- | --- | ----------------------------------- | --- | --- | --- | --- | --- | --- | | | 0.2 | | | 0.15 | | | | | | | | | | | 0.0 )%( kcoL mutnauQ Extendedpre-saturationenablesout-of-domaintransfer. | | 40 | | | )%( 005-HTAM 52 | | | | | | | | | | | | --- | --- | --- | --- | --------------- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | Positive∆(8) | | 30 | | | 48 | | | | | valuesinTable1indicatethatthereasoning | | | | | | | --- | --- | --- | --- | --- | --- | --- | --- | -------- | -------------------------------------- | --- | -------------- | --- | ----- | -------- | | | 20 | | | | | | | | sat | | | | | | | | | | | 44 | | | | patterns | learned during | the | pre-saturation | | phase | transfer | 10 | | | | | 40 | | | | acrossdomains,particularlyforQwenmodels. | | | | | Withonly | | | --- | --- | --- | --- | --- | --- | --- | --- | ---------------------------------------- | --- | --- | --- | --- | -------- | --- | 0 36 | | | | | 28 | | | | 8samples,Qwen2.5-1.5BtrainedonMATHachievescon- | | | | | | | | --- | --- | --- | --- | --- | --- | --- | --- | ---------------------------------------------- | --- | --- | --- | --- | --- | --- | )%( 005-HTAM 78 | | | | | 24 | | | | sistent gains | on the | out-of-domain | | SCIENCE | benchmark | | | --- | --- | --- | --- | ------- | --- | --- | --- | ------------- | ------ | ------------- | --- | ------- | --------- | --- | | | 72 | | | )%( CMA | | | | | | | | | | | | | 66 | | | 20 | | | | | | | | | | | (SCP-Hard),whileQwen2.5-Math-7BtrainedonGRAPH | | 60 | | | 16 | | | | | | | | | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | improvesout-of-domainMATH-500performanceby21.0% | | 54 | | | 12 | | | | | | | | | | | | --- | --- | --- | --- | ----- | --- | --- | --- | --------- | ------------ | ----- | ------ | ---- | ------- | ------- | | | | | | | | | | (Fig. 1). | In contrast, | Llama | models | show | limited | out-of- | | | 0 | 150 | 300 | 450 0 | 150 | 300 | 450 | | | | | | | | Training Steps Training Steps domain transfer even when in-domain performance im- | | | =0 | | =0.3 | | =0.7 | | | | | | | | | | --- | --- | --- | --- | ---- | --- | ---- | --- | --- | --- | --- | --- | --- | --- | --- | proves;theirgainsremainlocalizedtothespecifictraining | | | =0.1 | | =0.5 | | =0.9 | | | | | | | | | | --- | --- | ---- | --- | ---- | --- | ---- | --- | --- | --- | --- | --- | --- | --- | --- | distribution. Figure2.Effectofrewardlabelcorruptionontrainingdynam- icsandgeneralization.γdenotesthefractionoftrainingprompts Takeaway: | withcorruptedlabels,rangingfromclean(γ | | | | | | = 0)tomostlyin- | | | | | | | | | | -------------------------------------- | --- | --- | --- | --- | --- | --------------- | --- | --- | --- | --- | --- | --- | --- | --- | (1)RLVRcangeneralizefromasfewas8samples | correct(γ | | = 0.9). | | | | | | | | | | | | | | --------- | --- | ------- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ForQwenonGRAPHandLlamaonMATH, generalizationdegradeatγ ≥ 0.5. when models remain in an extended pre-saturation ForLlama,trainingreward curvesstaycloseacrossallγ,suggestingoverfittingtonoise. phase,whereasrapidlysaturatingmodelsrequiresub- stantiallymoredata.(2)Whetherscarce-datalearning 8-samplerunreachest(8). succeedsismodel-anddomain-dependent,reflecting | | | | sat | | | | | theinfluenceofpretrainingpriors. | | | | (3)Inthelow-data | | | | ---- | ------- | --- | -------- | ------ | ------ | ------------ | --- | -------------------------------- | --- | --- | --- | ---------------- | --- | --- | | Fig. | 1 shows | the | training | curves | across | data scales. | The | | | | | | | | regime,Llamamodelscanachieveperfecttrainingre- lengthofthepre-saturationphaseistheprimarydeterminant wardsmuchfasterthanQwenbyrapidlymemorizing ofwhetheramodelcangeneralize. With8trainingsamples, trainingexamplesbutachievelittlemeaningfultask | Qwen2.5-Math-1.5B | | | on | MATH increases | | reward | steadily | | | | | | | | | ----------------- | --- | --- | --- | -------------- | --- | ------ | -------- | --- | --- | --- | --- | --- | --- | --- | learning. forover300steps;thissustainedascentallowsthemodelto extractgeneralizablereasoningpatternsthattransfertoheld- | out | evaluation | benchmarks | | such | as MATH-500 | and | SCP- | 3.2.NoisyRewards | | | | | | | | --- | ---------- | ---------- | --- | ---- | ----------- | --- | ---- | ---------------- | --- | --- | --- | --- | --- | --- | Hard. Awithin-familycomparisonisolatesthepretraining effect: Qwen2.5-Math-1.5B,whichsharesarchitecturewith Whenground-truthverifiersareavailablebutimperfect,re- | | | | | | | | | wardlabelsmaycontainerrors. | | | ToevaluateRLVRrobust- | | | | | --- | --- | --- | --- | --- | --- | --- | --- | --------------------------- | --- | --- | --------------------- | --- | --- | --- | Qwen2.5-1.5Bbuthasadditionalmath-specificpretraining, nesstosuchnoisysupervision,wevarythefractionofincor- saturatesmoreslowlyandtransfersfurther(Table1). | | | | | | | | | rectlabelsγ | byrandomlyreplacingground-truthanswers | | | | | | | --- | --- | --- | --- | --- | --- | --- | --- | ----------- | -------------------------------------- | --- | --- | --- | --- | --- | Figs. 13, 14, and 15 (Appendix C.1) show the full range with the most frequent incorrect answer produced by the | N | ∈ {8,32,64,512,2048} | | | across | MATH, | SCIENCE, | and | | | | | | | | | ---------------------------------------- | -------------------- | -------- | ------ | ------- | ----- | ------------ | --- | ---------------------------------- | ------------ | ---------- | --- | --------------- | ------ | --- | | | | | | | | | | modelitself(detailsinAppendixD.1). | | | | Unlessotherwise | | | | GRAPH. | | For Qwen | models | on MATH | | and SCIENCE, | in- | | | | | | | | | | | | | | | | | noted,experimentsuseN | | =2048. | | | | | | domainperformanceisnearlyindependentofN. | | | | | | ForLlama | | | | | | | | | | | | | | | | | | RLVR | demonstrates | robustness | | to reward | noise, | but | acrossalldomains,andforQwenonGRAPH,differentN | | | | | | | | | generalizationvariesacrossmodels. | | | | Fig.2andAppendix | | | | --- | --- | --- | --- | --- | --- | --- | --- | --------------------------------- | --- | --- | --- | ---------------- | --- | --- | producesvisiblydifferentdynamicsonsomeoftheevals, Fig.26summarizeperformanceacrosssevenmodel–domain withsmallerdatasetssaturatingearlierandatlowerdown- streamperformance. pairsundervaryingγ. Atγ ≤0.3,testperformanceacross | | | | | | | | | mostsettingsremainsclosetothecleanrewards(γ | | | | | | =0),in- | | --- | --- | --- | --- | --- | --- | --- | --- | ------------------------------------------- | --- | --- | --- | --- | --- | ------- | Modelswithoutdomain-alignedpriorssaturaterapidly dicating robustness to moderate label noise. On MATH | andfailtogeneralize. | | | Incontrast,Llamamodelsacrossall | | | | | | | | | | | | | -------------------- | --- | --- | ------------------------------- | --- | --- | --- | --- | ------------ | ---- | ------ | -------- | ----- | ----- | ---- | | | | | | | | | | and SCIENCE, | Qwen | models | maintain | gains | under | sub- | domains,andQwenonGRAPH(Fig.1)exhibitcleardepen- stantial corruption (up to γ = 0.7). In contrast, Qwen | denceondatascale. | | | ForLlama,trainingon8samplesleads | | | | | | | | | | | | | ----------------- | --- | --- | -------------------------------- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | onGRAPHandLlamaonMATHandSCIENCEdegradeat torapidsaturation,witht(8)occurringwithinthefirst100: | | | | | | | | | γ ≥ 0.5. | Higherγ | leadstoconsistentlylowertrainingre- | | | | | | --- | --- | --- | --- | --- | --- | --- | --- | -------- | ------- | ----------------------------------- | --- | --- | --- | --- | sat itmaximizesthetrainingrewardmuchfasterthantheQwen wardsthroughouttraining,butforLlamaonMATH,training models. Thesemodelsrequirelargerdatasets(N ≥512)to rewardcurvesremainnearlyidenticalacrossallγ despite achievemeaningfulgeneralization(detailsinAppendixC.1 severecorruption,indicatingLlamafitsincorrectanswers | Fig.13andFig.14). | | | TheresultsintheGRAPHdomainsug- | | | | | | | | | | | | | ----------------- | --- | --- | ------------------------------ | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 5 LLMReasoningwithWeakSupervision | Qwen2.5-3B | Llama-3.2-3B-Instruct | | | | | | | ------------------- | --------------------- | --- | --- | --- | --- | --- | | Science | Science | | | | | | | draweR gniniarT 1.0 | 1.0 | | | | | | | 0.8 | 0.8 | | | | | | | 0.6 | 0.6 | | | | | | 0.4 0.4 0.2 0.2 )%( 005-HTAM | 65 | 48 | | | | | | | --- | --- | -------------------------------------------------------- | --- | --- | --- | --- | | | 40 | Figure4.Evolutionofsemanticdiversityduring8-sampletrain- | | | | | 60 | 55 | 32 | ingonMATH.Llamashowssignificantlyhigherpost-saturation | | | | | | --- | --- | ------------------------------------------------------ | --- | --- | --- | --- | | 50 | 24 | | | | | | diversitythanQwen,albeitwithlowerperformanceoutcomes. | 45 | 16 | | | | | | | --- | --- | --- | --- | --- | --- | --- | 40 )%( draH-PCS 40 SCIENCE)showimprovementwithmajorityvoting,while 20 | 32 | | othermodelsfailentirely. | | ForQwen2.5-3Bon | | SCIENCE, | | --- | --- | ----------------------------------------------------- | --- | --------------- | --- | -------- | | 24 | 15 | | | | | | | | 10 | majorityvotingyieldstemporarygainsbeforecollapseafter | | | | | 16 | 8 | 5 | | | | | | | --- | --- | --------- | --------------------------------------- | --- | --- | --- | | | | 500steps, | asthepolicyconvergestowardasingleoutput | | | | | 0 | 0 | | | | | | 0 150300450600750 0 150300450600750 tomaximizeagreement. Self-certaintyrewardsleadtoper- | Training Steps | Training Steps | | | | | | | -------------- | -------------- | --- | --- | --- | --- | --- | RLVR Majority Vote Self Certainty formancecollapseacrossallsettings. Theseresultsshow thatcurrentself-supervisedproxyrewardsareinsufficient | | | toreplaceverifiablefeedbackinmostsettings. | | | | (detailsin | | --- | --- | ------------------------------------------ | --- | --- | --- | ---------- | Figure3.Comparisonofrewardvariants(RLVR,self-certainty, AppendixD.2andFig.27). majorityvote)with1024trainingsamples.Proxyrewardswith- outverifiersexhibitfailuremodesunderprolongedtraining:train- ingcollapse(self-certainty)andrewardspikesfollowedbyperfor- Takeaway: Self-supervisedproxyrewardssucceed mancedrops(majorityvote)(moreresultsareinAppendixD.2). onlyformath-specializedmodels(Qwen-Mathunder | | | majorityvoting). | Othermodelsexhibitthesamepat- | | | | | --- | --- | ---------------- | ----------------------------- | --- | --- | --- | moreeasily. Wealsoobservethatmodel-domainpairswith ternweobservedin§3.1and§3.2: fastersaturation faster saturation (§3.1) are generally less robust to label andweakerpretrainingpriorscoincidewithbrittle- noise,aconnectionwedevelopin§3.4and§4. ness. Under prolonged training, the failure mode | | | isrewardhacking: | | policiesconvergetowardoutputs | | | | --- | --- | ---------------- | --- | ----------------------------- | --- | --- | Takeaway: Robustnesstolabelnoisevariessharply thatmaximizetheproxywithoutcorrespondingdown- | acrossmodel-domainpairs:QwenonMATHandSCI- | | streamgains. | | | | | | ----------------------------------------- | --- | ------------ | --- | --- | --- | --- | ENCE toleratesupto70%corruption, whileLlama andQwenonGRAPHdegradeat50%.Model-domain pairsthatsaturatefasterundercleanrewardsareless 3.4.WhyDoModelsFailUnderWeakSupervision? robust,andLlamafitscorruptedlabelsnearlyasfast | | | Theresultsin§3.1–§3.3showaconsistentpattern: | | | | models | | --- | --- | -------------------------------------------- | --- | --- | --- | ------ | ascleanones—evidencethatrapidsaturationreflects | | | withstrongdomain-alignedpretraining(Qwenon | | | | MATH | | --- | --- | ------------------------------------------ | --- | --- | --- | ---- | memorizationcapacityratherthanlearningefficiency. | | | and SCIENCE) | generalize | under weak | supervision, | while | | --- | --- | -------------------------------- | ------------------- | ---------- | ------------ | ---------- | | | | thosewithout(Llamaacrossdomains, | | | Qwenon | GRAPH) | | | | fail. A | natural hypothesis, | motivated | by prior | work link- | 3.3.Self-SupervisedProxyRewards ingdiminishedexploratorycapacitytorapidpolicysatura- Whenground-truthverifiersareentirelyunavailable,models tion(Cuietal.,2025),isthatfailingmodelsproduceless mustrelyonalternativerewardsignals(Burnsetal.,2023; diverse outputs. To test this, we analyze model behavior Rahmanetal.,2025;Bowmanetal.,2022). Recentwork alongtwocomplementaryaxes: responsediversityandrea- hasproposedself-supervisedproxyrewardsderivedfrom soningfaithfulness. Formaldefinitionsandimplementation model outputs, but whether these approaches work well detailsareprovidedinAppendixF. acrossmodelfamiliesandtaskdomainsremainsunexplored. | | | To quantify | response | diversity, we | quantify semantic | di- | | ------------------------- | ------------------------- | ----------- | -------- | ------------- | ----------------- | --- | | Weevaluatetwosuchrewards: | self-certainty(Zhaoetal., | | | | | | versitytocharacterizemeaningfulpatternsinthemodel’s 2025)andmajorityvote(Zuoetal.,2025)(implementation reasoningratherthansurface-levelvariation(Farquharetal., detailsinAppendixD.2). 2024;Lietal.,2025).Wemeasurediversityonthe8-sample Proxyrewardstriggerrewardhackingandpolicycol- subsetoftheMATH,SCIENCEandGRAPHtrainingdatasets, lapse. WhileRLVRtoleratesmoderatelabelnoiseinsome aswellasontheMATH-500evaluationdataset,overase- model-domainpairs(§3.2),Fig.3showsthatfullyreplacing lectionofpromptsatvariousstepsthroughouttraining. For verifiablefeedbackwithself-supervisedproxysignalsintro- each prompt, we cluster model responses using pairwise ducesseverefailuresunderprolongedtraining. Onlymath- similarity judgments from an LLM judge and define the specialized models (Qwen2.5-Math-1.5B on MATH and diversityscoreastheShannondiversityindexoverthere- 6 LLMReasoningwithWeakSupervision | | | | | | | | faithfulness. | Fig. 5 (right) | reports | faithful | diversity: | di- | | --- | --- | --- | --- | --- | --- | --- | ----------------------------------------- | -------------- | ------- | -------- | ---------- | --------- | | | | | | | | | versitycomputedonlyoverfaithfulresponses. | | | | | Thisjoint | measurerevealsaconsistentpatternacrossallthreedomains. | | | | | | | | On MATH,Llama’sapparentdiversityadvantage(Fig. | | | | | 4) | | --- | --- | --- | --- | --- | --- | --- | ---------------------------------------------- | --- | --- | --- | --- | --- | disappears—mostdiverseresponsesareunfaithful,andthe | | | | | | | | faithfulsubsetisnarrow. | | OnSCIENCE,alignedproportions | | | | | --- | --- | --- | --- | --- | --- | --- | ----------------------- | --- | ---------------------------- | --- | --- | --- | areuniformlyhighacrossmodels,maskingrealdifferences inreasoningquality;faithfuldiversityseparatesthem,with Qwen-Mathmaintainingthehighestvaluesthroughouttrain- | | | | | | | | ing. OnGRAPH,Qwen-MathandLlamashowcomparable | | | | | | | --- | --- | --- | --- | --- | --- | --- | -------------------------------------------- | --- | --- | --- | --- | --- | alignedproportions,butQwen-Mathsustainshigherfaithful | | | | | | | | diversity. | Ineverycase,themodelthatgeneralizesbestin | | | | | | --- | --- | --- | --- | --- | --- | --- | ---------- | ----------------------------------------- | --- | --- | --- | --- | §3.1istheoneexploringthewidestrangeoffaithfulrea- Figure5.Evolutionofreasoningfaithfulness(oncorrectsam- soningpaths—nottheonewiththehighestrawdiversity, ples)andfaithfuldiversityonmodelsthroughoutRLusing8 nor the one with the highest aligned proportion. Raw di- samplesfromavarietyofdatasets.LlamamodelsintheMATH versityoverstatesexploratorycapacity;alignedproportion domainexhibitsignificantlylowerfaithfulnesscomparedtoQwen. saturatesoneasierdomains;onlytheirintersectionpredicts generalization. | sultingclusters. | | SeeFigure31forthejudgemodelprompt. | | | | | | | | | | | | ------------------------------------------- | --- | ---------------------------------- | --- | --- | --- | ----- | --------- | --- | --- | --- | --- | --- | | Highdiversitydoesnotpreventrapidsaturation. | | | | | | Fig.4 | Takeaway: | | | | | | reportstheevolutionofdiversityscoresformodelstrained | | | | | | | | Low reasoning | faithfulness, | | not low | diversity, | ex- | | --- | --- | --- | --- | --- | --- | --- | ------------- | ------------- | --- | ------- | ---------- | --- | on 8 samples from the MATH training dataset, computed plains why models fail under weak supervision: on the corresponding training set. Llama reaches reward rapidlysaturatingmodelsmemorizeanswersrather saturation earlier and retains higher diversity than Qwen, than acquire transferable reasoning. Raw diversity theoppositeofwhattheexploration-saturationhypothesis metricsaremisleading—Llamaexhibitshigherout- predicts. DiversitycomputedontheMATH-500evaluation | | | | | | | | put diversity | than Qwen | while | generalizing | | worse. | | --- | --- | --- | --- | --- | --- | --- | ------------- | --------- | ----- | ------------ | --- | ------ | datasetispresentedintheappendix(Fig.30). Diversitybecomesinformativeonlywhencomputed overfaithfulresponses. Sincediversityalonedoesnotexplainfailureunderweak | supervision, | we | investigate | the faithfulness | | of a model’s | | | | | | | | | ------------ | --------------------------------------- | ----------- | ---------------- | --- | ------------ | --- | --- | --- | --- | --- | --- | --- | | reasoning. | Inspiredbypriorwork(Bakeretal.,2025),we | | | | | | | | | | | | definearesponseasfaithfulifitsreasoningtracecontains Insummary,§3showsthatthesurprisingcapabilitiesoften the information needed to justify the final answer and is attributedtoRLVR,suchaslearningfromscarcedata,tol- logicallyconsistentwithit. Atagiventrainingstepandfor eratingnoisyrewards,succeedingwithoutverification,are notuniversalbutdependonpre-RLreasoningfaithfulness. agivenprompt,wecategorizeeachpolicyrolloutasaligned, partiallyaligned,ormisalignedbasedonrubricsprovided §4takesupthenaturalquestion: canpre-RLinterventions to an LLM-as-a-judge (see prompt in Fig. 32). We then targetingfaithfulnessextendthepre-saturationphaseand computethepolicyfaithfulnessrateF (l)asthefractionof recovergeneralizationunderweaksupervision? π | responsesassignedtolabell. | | | AppendixFoutlinesresults | | | | | | | | | | | -------------------------- | --- | --- | ------------------------ | --- | --- | --- | --- | --- | --- | --- | --- | --- | forinter-modelagreementonalignmentcategorizationto 4.ImprovingRLVRUnderWeakSupervision evaluatethereliabilityofourLLM-as-a-judge. viaPre-RLTraining Modelswithrapidsaturationexhibitlowreasoningfaith- Section3showedthatrapidsaturationandlowreasoning fulness. Fig.5(left)showsthefractionofcorrectresponses | | | | | | | | faithfulnessarelinked: | modelsthatgeneralizepoorlyunder | | | | | | --- | --- | --- | --- | --- | --- | --- | ---------------------- | ------------------------------- | --- | --- | --- | --- | thatarealignedoverRLtrainingacrossmodelsanddomains weaksupervisionproducecorrectanswersthroughreason- | studiedin§3.1. | | Onthe | MATH domain, | theLlamamodel | | | | | | | | | | --------------------------------------- | ----- | --------- | -------------------------------- | ------------- | ------------ | --- | ------------------------------------------------------- | ------------ | --------------- | ---------------------- | ---- | ---------- | | | | | | | | | ingthatdoesnotsupportthem. | | | Thisraisesacausalques- | | | | shows much | lower | reasoning | faithfulness | during | training | | | | | | | | | | | | | | | | tion. Iffaithfulnessdrivesthepre-saturationphase,andthe | | | | | | | thantheQwenmodels. | | | ThisindicatesthatLlama’srapidre- | | | | | | | | | | | | | | | | | | pre-saturation | phase drives | generalization, | | then | instilling | | wardgainsdonotreflectimprovedreasoning: | | | | | asubstantial | | | | | | | | faithfulnessbeforeRLshouldextendthephaseandrecover fractionofcorrectanswersarememorized,withreasoning | | | | | | | | generalization. | Wetestthisbyrunningacontrolledcom- | | | | | | ----------- | ------ | ------- | ---------- | ----- | -------- | --- | --------------- | ---------------------------------- | --- | --- | --- | --- | | traces that | do not | support | them. Fig. | 33 in | Appendix | F | | | | | | | parisonofpre-RLinterventionsonLlama3.2-3B,themodel | includes | additional | faithfulness | results | on these | domains, | | | | | | | | | -------- | ---------- | ------------ | ------- | -------- | -------- | --- | --- | --- | --- | --- | --- | --- | coveringproportionalignedandproportionmisalignedon thatfailedmostconsistentlyin§3. correct,incorrectandallresponses. Westudytwoaxesofpre-RLtraining. Thefirstiscontinual Reasoning diversity should be considered jointly with pre-training(CPT),extendedtrainingondomain-specific 7 LLMReasoningwithWeakSupervision | | draweR gniniarT | | | )%( 005-HTAM | | | 32 | | | | | | | ------ | --------------- | --- | --- | ------------ | --- | --- | ------- | --- | --- | ------------ | --- | --- | | | | | | | 56 | | | | | )%( draH-PCS | | | | | | | | | | | )%( CMA | | | 15 | | | | ecracS | | | | | | | 24 | | | | | | | ataD | 0.6 | | | | 48 | | | | | | | | | | | | | | 40 | | 16 | | | | | | 12 18 6 | | 0.0 | | | | 12 | | 0 | | | 0 | | | | -------- | --------------- | --- | --- | ------------ | ----- | --- | ------- | --- | --- | ------------ | --- | ------- | | | 0 | 150 | 300 | 450 | 0 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 450 | | | draweR gniniarT | | | )%( 005-HTAM | 64 | | | | | | | | | | | | | | | | 30 | | | )%( draH-PCS | | | | ytirojaM | | | | | 56 | | )%( CMA | | | | | | etoV | | 0.6 | | | | 48 | | | | | 15 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 15 20 | | 0.0 | | | | 0 | | | | | 0 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 0 150 300 450 600 0 150 300 450 600 0 150 300 450 600 0 150 300 450 600 60 | draweR ysioN | draweR gniniarT | | | )%( 005-HTAM | | | | | | | | | | ------------ | --------------- | --- | --- | ------------ | --- | --- | ------- | --- | --- | ------------ | --- | --- | | | 0.6 | | | | | | 25 | | | )%( draH-PCS | | | | )7.0= | | | | | 54 | | )%( CMA | | | | | | 15 | | | | | | 48 | | 20 | | | | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 0.3 | ( | | | | | | | 15 | | | | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 20 | | | | | | 15 | | 5 | | | | | | | --- | --- | -------------- | --- | --- | -------------- | --- | ----- | -------------- | --- | ----- | -------------- | ------- | | | | | | | 10 | | 0 | | | 0 | | | | | 0 | 150 | 300 | 450 | 0 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 450 | | | | Training Steps | | | Training Steps | | | Training Steps | | | Training Steps | | Base + Thinking SFT Base + Non-Thinking SFT CPT + Thinking SFT CPT + Non-Thinking SFT Instruct Figure6.RLtrainingdynamicsandgeneralizationonMATHforLlama3.2-3BBase,CPT,andInstructvariantsunderdifferent SFTinitializationsacrossthreeweaksupervisionsettings:scarcedata(N =8,top),majorityvote(middle),andnoisyreward (γ =0.7,bottom).ThinkingSFT(solidlines)consistentlyprolongsthepre-saturationphaseandimprovesgeneralizationforbothCPT andBasemodelscomparedtotheirNon-ThinkingSFTcounterparts(dashedlines)andtheInstructbaseline(dash-dot).CPT+Thinking SFTachievesthestrongestperformanceacrossallsettings. pretrainingtokenstostrengthenthepretrainingprior. The explicitreasoningtracesinfluencesubsequentRLdynamics. second is supervised fine-tuning (SFT), with the specific WecomparetwoSFTregimesthatdifferonlyinwhether questionofwhetherSFTonexplicitreasoningtracesdiffers thesupervisionincludesexplicitreasoning. Bothregimes in its effect from SFT on final answers alone. Crossing use the same 43.5K math prompts and differ only in the these axes gives a 2×2 design: two initializations (Base, targetoutput. Specifically,wesamplethesepromptsfrom CPT)eachfollowedbytwoSFTregimes(Thinking,Non- OpenThoughts-114K (Guha et al., 2025), retaining only Thinking).WeadditionallyincludeLlama3.2-3B-Instructas thosewhosereasoningtraceshavecorrectfinalanswersand areference: itsharesthearchitectureofLlama3.2-3B-Base totallengthbelow8192tokens. | but | has undergone | | extensive instruction | | tuning, | rejection | | | | | | | | --- | ------------- | --- | --------------------- | --- | ------- | --------- | ------------------------------------------------- | --- | --- | --- | --- | --- | | | | | | | | | • Non-thinkingSFT:Themodelissupervisedtooutputthe | | | | | | sampling,andDPO,providingastrongoff-the-shelfbase- finalsolutionwithoutgeneratingintermediatereasoning | lineagainstwhichtojudgeourtargetedinterventions. | | | | | | We | | | | | | | | ------------------------------------------------ | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | traces. thenrunRLunderallthreeweaksupervisionsettingsfrom | | | | | | | | • ThinkingSFT:Themodelistrainedonexplicit,verified | | | | | | | --- | --- | --- | --- | --- | --- | --- | -------------------------------------------------- | --- | --- | --- | --- | --- | §3: scarcedata,noisyrewards,andself-supervisedproxy long-formreasoningtraces. rewards. Wefocusonthe MATH domainfortworeasons: Llama’s AtrainingexampleisshowninFig.12intheAppendix.The | | | | | | | | SFTregimesarenear-iso-compute: | | | | ThinkingSFTtrainson | | | --- | --- | --- | --- | --- | --- | --- | ------------------------------ | --- | --- | --- | ------------------- | --- | baselinefailureissharpestthere,providingthecleanesttest roughly1Btokens,Non-ThinkingSFTonroughly0.27B, ofwhetherpre-RLinterventionscanrecovergeneralization; | | | | | | | | bothnegligiblerelativetothe52B-tokenCPTstage. | | | | | Differ- | | --- | --- | --- | --- | --- | --- | --- | --------------------------------------------- | --- | --- | --- | --- | ------- | andhigh-qualitymathpretrainingcorpora(Nemotron-CC- encesbetweenThinkingandNon-ThinkingSFTtherefore Math)andreasoning-tracedatasets(OpenThoughts-114K) areavailable,enablingtheinterventionsatsufficientscale. reflectthecontentofthesupervisionratherthanitscost. We | | | | | | | | reporttheCPTlosscurveinAppendixFig. | | | | | 10andtheSFT | | --- | --- | --- | --- | --- | --- | --- | ----------------------------------- | --- | --- | --- | --- | ----------- | ContinualPre-Training(CPT).Wecontinuallypre-train losscurvesinFig. 11. | Llama3.2-3B-Base | | | for one epoch | on | approximately | 52B | | | | | | | | ---------------- | --- | --- | ------------- | --- | ------------- | --- | --- | --- | --- | --- | --- | --- | ImplementationandtrainingdetailsofSFTareprovidedin | math | tokens | from | the Nemotron-CC-Math | | dataset | (Ma- | | | | | | | | ------------------- | ------ | ---- | ------------------------------- | --- | ------- | ---- | -------------------------------------------------- | --- | ---------------------------------- | --- | --- | --- | | | | | | | | | AppendixB.6. | | ForthesubsequentRLphase,weevaluate | | | | | habadietal.,2025)2. | | | TrainingdetailsareprovidedinAp- | | | | | | | | | | | | | | | | | | acrossallthreeweaksupervisionsettings:scarcedata(N | | | | | = | pendixB.5. | | | | | | | | 8), noisy | rewards | (γ = | 0.7), and | self-supervised | proxy | | ------------------- | --- | --- | ----------------------------- | --- | --- | --- | ---------------------- | ------- | ----------------------------- | --------- | --------------- | ----- | | SFTTrainingRegimes. | | | FollowingCPTorBaseinitializa- | | | | | | | | | | | | | | | | | | rewards(majorityvote). | | Allotherhyperparametersfollow | | | | tion,weapplysupervisedfine-tuningtodeterminewhether theconfigurationsin§2,withthemaximumresponselength duringRLextendedto8192tokenstoaccommodatelong- 2Nemotron-CC-Math-v1 8 LLMReasoningwithWeakSupervision formreasoningtraces. 4.1.Results | Fig. 6 reports | RL | training | dynamics | for | the five | pre-RL | | | | | | | | | -------------- | --- | -------- | -------- | --- | -------- | ------ | --- | --- | --- | --- | --- | --- | --- | configurations(Base,CPT,andInstruct,withThinkingSFT | or Non-Thinking | | SFT | applied to | Base | and CPT) | across | | | | | | | | | -------------------------------- | ------ | --------- | ---------- | --------------- | -------- | -------- | --- | --- | --- | --- | --- | --- | --- | | thethreeweaksupervisionsettings. | | | | Foreachsetting, | | we | | | | | | | | | plot training | reward | alongside | three | downstream | | metrics: | | | | | | | | Figure7.EvolutionofreasoningfaithfulnessoftheLlama3.2- twoin-domain(MATH-500,AMC)andoneout-of-domain 3Bfamilyonweaksupervisiondomainswhencombinedwith (SCP-Hard);additionalbenchmarksandpass@kresultsare continualpretrainingandSFTvariants.Whencombinedwith Fig.34andFig.35inAppendixG.Wedrawthreefindings Thinking-SFT and CPT, the Llama3.2-3B-Base model exhibits higherreasoningfaithfulness. fromthisfigure,developedintheparagraphsbelow. ThinkingSFTisnecessaryforsubstantiallearningunder | | | | | | | | highest faithfulness | | among | all configurations, | | | consistent | | --- | --- | --- | --- | --- | --- | --- | -------------------- | --- | ----- | ------------------- | --- | --- | ---------- | weaksupervision.TheInstructbaselineisflatordecreasing | | | | | | | | with its strongest | generalization | | | across | all weak | supervi- | | --- | --- | --- | --- | --- | --- | --- | ------------------ | -------------- | --- | --- | ------ | -------- | -------- | acrossallthreesettingsonalldownstreamevaluations— | | | | | | | | sion settings. | Together | with | the | extended | pre-saturation | | | --- | --- | --- | --- | --- | --- | --- | -------------- | -------- | ---- | --- | -------- | -------------- | --- | RLproducesnomeaningfulimprovementfromthisstarting | | | | | | | | dynamics | visible in | Fig. | 6 (leftmost | column), | | this result | | --- | --- | --- | --- | --- | --- | --- | -------- | ---------- | ---- | ----------- | -------- | --- | ----------- | point. ThinkingSFTistheonlyinterventionthatenables | | | | | | | | supportsourhypothesisin§3.4: | | | | pre-RLinterventionsthat | | | | --- | --- | --- | --- | --- | --- | --- | ---------------------------- | --- | --- | --- | ----------------------- | --- | --- | substantialdownstreamgainsonscarcedataandmajority instillfaithfulnessproducelongerpre-saturationphasesand vote,anditdoessoforbothBaseandCPTinitializations recoveredgeneralization,inmodelsthatpreviouslyfailed. | (solidblueandsolidred). | | | Non-ThinkingSFTshowsmodest | | | | | | | | | | | | ----------------------- | ------------ | ------ | -------------------------- | ------- | ---------------- | ----- | --------- | ---------------------------------- | --- | --- | --- | --- | --- | | gains only | when | paired | with CPT, | and | only under | noisy | | | | | | | | | | | | | | | | Takeaway: | SFTonexplicitreasoningtraces,noton | | | | | | | rewards; | Non-Thinking | | SFT on | Base is | flat or degrades | | | | | | | | | finalanswers,isnecessaryforLlamatolearnsubstan- acrossallthreesettings. | | | | | | | | tiallyfromRLunderweaksupervision. | | | | | Itraisesrea- | | | --- | --- | --- | --- | --- | --- | --- | --------------------------------- | --- | --- | --- | --- | ------------ | --- | soningfaithfulness,extendsthepre-saturationphase, | CPTamplifiestheThinkingSFTeffect. | | | | | ThinkingSFT | | | | | | | | | | --------------------------------- | --- | --- | --- | --- | ----------- | --- | --- | --- | --- | --- | --- | --- | --- | onBasealoneproducesmodestgains. CombinedwithCPT, andrecoversgeneralizationunderscarcedata,noisy | | | | | | | | rewards, | and self-supervised | | | proxy | rewards. | Con- | | --- | --- | --- | --- | --- | --- | --- | -------- | ------------------- | --- | --- | ----- | -------- | ---- | itproducessubstantiallylargergainsoneveryevaluation: CPT+ThinkingSFTisthetop-performingcurveacrossall tinualpre-trainingamplifiestheeffectbutdoesnot threeweaksupervisionsettingsandallthreeevals.TheCPT substituteforit: CPT+Non-ThinkingSFTfailsde- | | | | | | | | spitematchedcompute. | | | Thestrongestconfiguration, | | | | | --- | --- | --- | --- | --- | --- | --- | -------------------- | --- | --- | -------------------------- | --- | --- | --- | +Non-ThinkingSFTcomparisonrulesoutacompute-based explanation:thesame52BCPTtokens,pairedwithSFTtar- CPT + Thinking SFT, recovers performance in set- tingswhereLlamahadpreviouslycollapsedentirely. getsthatstripreasoningtraces,failtoenablegeneralization | onscarcedataandmajorityvote. | | | | Theamplificationisspe- | | | | | | | | | | | --------------------------------------------------- | --- | --- | ------------------------------- | ---------------------- | --- | --- | ------------- | --- | --- | --- | --- | --- | --- | | cifictothecombination: | | | extrapre-trainingcomputealoneis | | | | | | | | | | | | insufficient;ThinkingSFTalonehelpsbutislimited,only | | | | | | | 5.RelatedWork | | | | | | | thecombinationrecoversfullgeneralization. | | | | | | | | RLVR for | Reasoning. | Reinforcement | | | learning | with ver- | | --- | --- | --- | --- | --- | --- | --- | -------- | ---------- | ------------- | --- | --- | -------- | --------- | Baseinitializationfailsundermostweaksupervisionset- ifiable rewards has emerged as an effective post-training tingsregardlessofSFT.TheBasemodelshowsmeaningful | | | | | | | | method for | improving | reasoning | | in large | language | mod- | | --------------------------------- | --- | --- | --- | --- | ------------- | --- | ----------- | ---------- | --------- | ------- | -------- | -------- | ---------- | | improvementonlyintwocombinations: | | | | | Base+Thinking | | | | | | | | | | | | | | | | | els (Guo et | al., 2025; | Olmo | et al., | 2025; | Yu et | al., 2025; | SFT under scarce data and majority vote, and even there Zengetal.,2025). RecentworkhasexploredwhenRLVR | gains are | modest. | Under | noisy | rewards, | neither | Base + | | | | | | | | | --------- | ------- | ----- | ----- | -------- | ------- | ------ | --- | --- | --- | --- | --- | --- | --- | yieldsimprovements(Liuetal.,2025b;a;Huetal.,2025). Thinking SFT nor Base + Non-Thinking SFT produces Wangetal.(2025a)demonstratethattrainingonasingleex- | meaningfuldownstreamimprovement. | | | | ThisisolatesCPT’s | | | | | | | | | | | -------------------------------- | --- | --- | --- | ----------------- | --- | --- | ----------------------------------------- | --- | --- | --- | --- | --------- | --- | | | | | | | | | amplecanprovidemeaningfullearningsignals. | | | | | Otherwork | | contribution:ThinkingSFTisnecessarybutnotsufficient— exploresalternativerewards,includingself-certainty(Zhao domain-alignedpretrainingisrequiredfortheintervention etal.,2025),majorityvoting(Zuoetal.,2025),negativesig- togeneralizeacrossallthreeweaksupervisionsettings. nals(Zhuetal.,2025),self-generatedtrainingdataHuang | | | | | | | | etal.(2025),andspuriousrewards(Shaoetal.,2025). | | | | | | How- | | ----------------------------------------- | --- | --- | --- | --- | --- | ------- | ----------------------------------------------- | --- | --- | --- | --- | --- | ---- | | ThinkingSFTimprovesreasoningfaithfulness. | | | | | | In§3.4, | | | | | | | | ever,thesefindingsoftendonottransferacrossmodelfam- weidentifiedlowreasoningfaithfulnessasthepre-RLprop- | | | | | | | | ilies, with | studies | reporting | inconsistent | | results | between | | --------- | ------------- | --- | ------------ | ---------- | --- | ------- | ----------- | ------- | --------- | ------------ | --- | ------- | ------- | | erty that | distinguished | | failing from | succeeding | | models. | | | | | | | | QwenandLlama(Zengetal.,2025;Gandhietal.,2025; | Fig. 7 shows | that | Thinking | SFT | raises | aligned-response | | | | | | | | | | ------------ | ---- | -------- | --- | ------ | ---------------- | --- | --- | --- | --- | --- | --- | --- | --- | ratethroughoutthepre-saturationphase,relativetotheNon- Shao et al., 2025). Moreover, most prior work focuses onimprovingperformanceonnarrowdomains(primarily | ThinkingSFTbaseline. | | | CPT+ThinkingSFTachievesthe | | | | | | | | | | | | -------------------- | --- | --- | -------------------------- | --- | --- | --- | ------------------------------------ | --- | --- | --- | --- | ------------- | --- | | | | | | | | | math)withoutexamininggeneralization. | | | | | Recentwork(He | | 9 LLMReasoningwithWeakSupervision et al., 2026; Yang et al., 2026; Plesner et al., 2026) has gesttwoconcretepracticesforRLfromweaksupervision. concurrentlystudiedwhenandhowRLVRcanlearnunder First, monitor training reward saturation as a diagnostic: self-supervisionornoisysupervision.Ourworkextendsthis plateauedrewardwithflatdownstreamperformanceindi- literatureintwoways. First,wecharacterizetheconditions catesthemodelhasexhaustedwhatRLcanextractfromits underwhichRLVRgeneralizesacrossmodelfamiliesand priors,andfurtherRLcomputeisunlikelytohelp. Second, domains, focusing on saturation dynamics and reasoning whenweaksupervisionfails, allocatecomputetopre-RL faithfulness. Second, we identify a concrete intervention interventionsthatinstallstrongpriorsratherthantolonger thatrestoresgeneralizationinmodelswhereweaksupervi- RL training. Taken together, our findings argue that RL sionwouldotherwisefail. underweaksupervisionisbestunderstoodnotasatraining techniqueappliedtoafixedmodel,butasthefinalstageof | Role of | Pre-Training | and | Fine-Tuning | | in RL. | Recent | | | | | | | | | ------- | ------------ | --- | ----------- | --- | ------ | ------ | --- | --- | --- | --- | --- | --- | --- | apipelinewhosesuccessislargelydeterminedbeforeRL workemphasizesthatpre-trainingandmid-trainingshape begins. | RL generalization | | (Qi et | al., 2025; | Wang | et | al., 2025b; | | | | | | | | | ----------------- | --- | ------ | ---------- | ---- | --- | ----------- | --- | --- | --- | --- | --- | --- | --- | Zhangetal.,2025;Akteretal.,2025),butfocusesoncom- puteallocationanddistributionalignmenttoimproveperfor- Acknowledgements mance.Ourworkspecificallyfocusesonunderstandinghow WewouldliketothankLeonLi,VatsalBaherwani,Rohun | base model | priors | shaped | from continual | | pretraining | and | | | | | | | | | ---------- | ------ | ------ | -------------- | --- | ----------- | --- | --- | --- | --- | --- | --- | --- | --- | Agrawal,SiyanZhao,LiweiJiang,andAndyHanfortheir | reasoning | SFT can | enable | generalization | | across | different | | | | | | | | | --------- | ------- | ------ | -------------- | --- | ------ | --------- | ---------- | ----------- | --- | -------- | --- | ---------- | ----- | | | | | | | | | insightful | discussions | and | feedback | on | the draft. | Pavel | weaksupervisionsettings. | | | | | | | | Izmailov | was supported | | by a grant | from | the | Alignment | | --- | --- | --- | --- | --- | --- | --- | -------- | ------------- | --- | ---------- | ---- | --- | --------- | Diversity and Faithfulness in Reasoning. Maintaining Project,fundedbytheUKAISecurityInstitute(grantAP- | outputdiversityduringRLhasbeenproposedtopromote | | | | | | | S2-100141). | | | | | | | | ----------------------------------------------- | --- | --- | --- | --- | --- | --- | ----------- | --- | --- | --- | --- | --- | --- | explorationandmitigatemodelcollapse(Kirketal.,2024; Casperetal.,2023;Rafailovetal.,2023;Yuetal.,2025), References | but prior | work has | not explored | | what | types | of diversity | | | | | | | | | --------- | -------- | ------------ | --- | ---- | ----- | ------------ | --- | --- | --- | --- | --- | --- | --- | benefitgeneralization. Separately,researchhashighlighted Agarwal,S.,Zhang,Z.,Yuan,L.,Han,J.,andPeng,H. The | | | | | | | | unreasonable | effectiveness | | of | entropy | minimization | in | | --- | --- | --- | --- | --- | --- | --- | ------------ | ------------- | --- | --- | ------- | ------------ | --- | mismatchesbetweenchain-of-thoughttracesandmodelpre- dictions(Turpinetal.,2023;Chenetal.,2025b;Bakeretal., llmreasoning. arXivpreprintarXiv:2505.15134,2025. 2025;Tuteketal.,2025)andemphasizedtheimportanceof | | | | | | | | AI-MO. | Aime 2024. | https://huggingface.co | | | | | | ------------------------------------------- | --- | --- | --- | --- | --- | ---------- | ------ | ---------- | ---------------------- | --- | --- | --- | --- | | ensuringfaithfulreasoningthroughouttraining | | | | | | (Guietal., | | | | | | | | /datasets/AI-MO/aimo-validation-aime, 2026). Wenetal.(2025)arguesthatRLVRcanincentivize 2024a. correctreasoninginbaseLLMsaslongaspriorshavebeen | established. | Our | work connects | | these | lines of | research, | | | | | | | | | ------------ | -------------- | ------------- | ---- | ----- | -------- | ----------- | ------ | --------- | ---------------------- | --- | --- | --- | --- | | | | | | | | | AI-MO. | Amc 2023. | https://huggingface.co | | | | | | showing | that diversity | alone | does | not | ensure | generaliza- | | | | | | | | /datasets/AI-MO/aimo-validation-amc, tionandthatreasoningfaithfulnessdistinguishesmodels’ 2024b. | trainingdynamics. | | Wefurtherdemonstratethatpre-RLin- | | | | | | | | | | | | | ----------------- | --- | --------------------------------- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | terventioncanimprovereasoningfaithfulnessandimprove Akter, S. N., Prabhumoye, S., Nyberg, E., Patwary, M., generalizationunderweaksupervision. Shoeybi,M.,Choi,Y.,andCatanzaro,B. 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Successunderscarcedata,noisy itoringreasoningmodelsformisbehaviorandtherisksof promotingobfuscation.arXivpreprintarXiv:2503.11926, rewards,andself-supervisedproxyrewardsdependsonpre- | RLproperties,pretrainingpriorsandreasoningfaithfulness, | | | | | | | 2025. | | | | | | | | ------------------------------------------------------- | ----- | -------- | ------ | ------ | ---- | -------- | ------- | ------------ | --- | ---------- | --------- | --- | ----------- | | rather than | on RL | dynamics | alone. | Models | that | saturate | | | | | | | | | | | | | | | | Bowman, | S. 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On | the interplay of | | ---------- | --------------- | ---------- | ---------------- | pre-training,mid-training,andrlonreasoninglanguage | models. | arXivpreprintarXiv:2512.07783,2025. | | | | --------- | ----------------------------------- | ----------------- | ------------ | | Zhao, X., | Kang, Z., Feng, | A., Levine, S., | and Song, D. | | Learning | to reason without | external rewards. | arXiv | preprintarXiv:2505.19590,2025. Zhu,X.,Xia,M.,Wei,Z.,Chen,W.-L.,Chen,D.,andMeng, Y. Thesurprisingeffectivenessofnegativereinforcement | inllmreasoning. | arXivpreprintarXiv:2506.01347,2025. | | | | --------------- | ----------------------------------- | --- | --- | Zuo,Y.,Zhang,K.,Sheng,L.,Qu,S.,Cui,G.,Zhu,X.,Li, | H.,Zhang,Y.,Long,X.,Hua,E.,etal. | | | Ttrl: Test-time | | -------------------------------- | --- | --- | --------------- | reinforcementlearning.arXivpreprintarXiv:2504.16084, 2025. 14 LLMReasoningwithWeakSupervision A.LimitationsandFutureWork Weacknowledgeseverallimitations. First,duetocomputationalconstraints,ouranalysisisrestrictedtospecificmodel families and scales. Validating these findings across larger architectures and broader task suites remains an important direction.Second,ouranalysisofdiversityandfaithfulnessreliesonanLLM-as-a-judgeframework.Althoughweconducted small-scalehumanverificationtovalidatelabelquality,wecurrentlyrestrictthisevaluationtoasmallscaletoallowfor reasonablelabelingcosts.Consequently,thedevelopmentofscalablemetricsforreasoningfaithfulnessanddiversityremains animportantdirectionforfutureresearch. B.ImplementationDetails B.1.TrainingandEvaluationDatasets WeinvestigateRLtrainingdynamicsacrosstwomodelfamilies: Qwen(comprisingQwen2.5-1.5B/3BandQwen2.5-Math- 1.5B/7B)andLlama(Llama-3.2-3B/8B-Instruct). Ouranalysisspansthreedistinctreasoningdomains,MATH,SCIENCE, andGRAPH,allowingforaholisticinvestigationofRLVRunderweaksupervisionacrossdifferentdomainsandmodel families. ForMATH,wesampletrainingpromptsfromtheSkywork-OR1(Heetal.,2025b)dataset. ForSCIENCE,wedraw problemsfromtheSCPdatasetcuratedbypriorwork(Liuetal.,2025a;Luetal.,2025),byselectingPhysics,Chemistry, andBiologysubjects. ForGRAPH,wegeneratetwosyntheticalgorithmictasks,QuantumLockandLargestIsland,using thecurriculumspecificationsprovidedbytheReasoningGymbenchmark(Stojanovskietal.,2025). Foreachtask,we instantiatefivedifficultylevelsfollowingthebenchmark’scurriculum,withabalancednumberofsamplesperlevel. Weincludethefollowingdomain-specificbenchmarksforevaluations: • MATH500(Lightmanetal.,2023): AwidelyusedsubsetoftheMATHtestsplit(Hendrycksetal.). • AMC(AI-MO,2024b): 40competition-levelmathquestions. • AIME2024(AI-MO,2024a): 30competition-levelmathquestions. • AIME2025(Opencompass,2024): 30competition-levelmathquestions. • MinervaMath(Lewkowyczetal.,2022): Asetof272undergraduate-levelscienceandmathquestionsfromMIT OpenCourseWare. • OlympiadBench(Heetal.,2024): Abenchmarkof675problemsfrominternationalmatholympiadsandphysics contests. • GPQA-Diamond(Reinetal.,2024): 198expert-levelquestionsfromGPQAspanningphysics,chemistry,andbiology; wepreprocessthedatafollowingpreviouspractice(Chengetal.,2025). • SCP-Hard(Luetal.,2025;Liuetal.,2025a): Aheld-outsetof50SCPquestionsfilteredsuchthatthebasemodels (Qwen2.5-1.5BseriesmodelsandLlama3.2-3B-Instructmodel)achievesolve@16=1,containingdisjointquestions fromtheSCPtrainingdatasets. • SuperGPQA (Du et al., 2025): a subset constructed from the original SuperGPQA which contains 319 science questionsand250non-sciencequestions. • MMLUSCI(Wangetal.,2024): asubsetofMMLUProbenchmarkcontainingallcollege-levelchemistry,physics andbiologyquestions. • ScienceBench(Wangetal.,2023): 692college-levelsciencequestions. • GraphTest: Aheld-outsetof50algorithmicallygeneratedinstancesfromtheQuantumLock andLargestIsland tasks using Reasoning Gym (Stojanovski et al., 2025), disjoint from training, filtered such that the base models (Qwen2.5-1.5BseriesandLlama3.2-3B-Instruct)achievePass@16=1. WealsonotethatGPQA-Diamond,MMLUSCI,andSuperGPQAaremultiple-choicebenchmarks,forwhich pass@kmaybealessreliablemetric. Table2detailsthetrainingandevaluationdatasetsacrossthethreereasoningdomains. 15 LLMReasoningwithWeakSupervision Table2.Trainingdatasetsandevaluationbenchmarksacrossthreereasoningdomains. Domain TrainingSource In-DistributionEval Out-of-DistributionEval MATH Skywork-OR1 MATH-500,AMC,AIME-2024,AIME- Science Bench, SuperGPQA, GPQA- 2025,MinervaMath,OlympiadBench Diamond,SCP-Hard,MMLUSCI SCIENCE SCP-116K SCP-Hard,GPQA-Diamond,MMLUSCI, MATH-500, AMC, Minerva Math, SuperGPQA,ScienceBench OlympiadBench GRAPH ReasoningGym QuantumLock,LargestIsland MATH-500,AIME-2024,MinervaMath, GPQA-Diamond,SCP-Hard PrompttemplateforMATHandGRAPH. system YouareahelpfulAIAssistant,designedtoprovidewell-reasonedanddetailedresponses. YouFIRSTthinkaboutthereasoningprocessstepbystepandthenprovidetheuserwiththeanswer. Pleaseencloseyourfinalanswerinthebox: \boxed{YourAnswer}. user assistant Figure8.PrompttemplateusedforRLtrainingandevaluationonMATHandGRAPH.Theplaceholderisreplaced withtheactualmathematicalquestionduringfine-tuningandevaluation.Specialtokensareomittedforclarity. B.2.TrainingDataPreparationDetails Wedescribeourprocedureforconstructingfilteredtrainingdatasetstailoredtoeachmodel’scapabilities. DifficultyEstimation. Foreachprobleminthesourcedataset,wesample16responsesfromthebasemodelandcount thenumberofcorrectsolutions,yieldingsolve@16∈[0,16]. Weretainonlyproblemswithsolve@16∈[1,15],excluding problemsthataretoodifficult(solve@16=0)ortriviallyeasy(solve@16=16)forthemodel. PrompttemplateforSCIENCE. system Let’sthinkstepbystepandoutputthefinalanswerwithin\boxed{}. user assistant Figure9.PrompttemplateusedforRLtrainingandevaluationonSCIENCE.Theplaceholderisreplacedwiththe actualmathematicalquestionduringfine-tuningandevaluation.Specialtokensareomittedforclarity. Stratified Sampling. We use a stratified round-robin selection method to construct training subsets of size N ∈ {8,32,64,512,2048}. Filteredproblemsarepartitionedinto15bins{B }15 accordingtotheirsolve@16values. Toselect i i=1 N problems: 1. Initialization: Setthecurrentcountofselectedproblemsn =0. total 2. Round-RobinSelection: Whilen 1 during RL training, we observe that ∆(8) keeps the same sign for all k ∈ {1,4,8,16} across most sat model-benchmarkpairs,indicatingconsistentimprovementinbothpass@1andpass@k. Thisindicatesthatduringthe pre-saturationperiod,themodelisnotjustclosingpass@kandpass@1gap. C.3.AdditionalExperimentalResultsonLargeModels Inthissection,wewillreportthefullevaluationresultson7Band8Bmodels. Fig. 20 and Fig. 21 show the results of Qwen2.5-Math-7B and Llama3.1-8B-Instruct models on MATH domain with 19 LLMReasoningwithWeakSupervision Table7.Graph-domaintraining(7B/8B):in-distributionbenchmarks. | Model | Metric t(8) | QuantumLock | | LargestIsland | | ----- | ----------- | ----------- | --- | ------------- | sat | | | ∆ ∆∗ | G ∆ | ∆∗ G | | -------------------- | ---------- | --------- | --------- | --------- | | | | sat post | sat sat | post sat | | Qwen2.5-Math-7B | Avg@16 151 | 8.0 4.9 | 7.3 19.8 | 1.9 -10.9 | | | Pass@4 | 22.6 1.5 | -1.3 16.5 | 6.1 16.3 | | | Pass@8 | 26.8 2.8 | -2.4 10.7 | 9.0 29.9 | | | Pass@16 | 30.6 5.5 | -4.1 6.8 | 8.6 41.3 | | LLama3.1-8B-Instruct | Avg@16 29 | 10.1 7.1 | 3.2 1.8 | 1.0 -0.2 | | | Pass@4 | 15.3 -0.0 | 6.2 1.8 | 0.0 1.6 | | | Pass@8 | 15.4 7.4 | 12.5 1.8 | 0.0 2.8 | | | Pass@16 | 20.1 16.0 | 25.0 1.8 | 0.0 3.1 | in-domainandout-of-domainbenchmarks,respectively. Fig.22andFig.23presenttheresultsofQwen2.5-Math-7BandLlama3.1-8B-InstructmodelsonSCIENCEdomainwith in-domainandout-of-domainbenchmarks,respectively. Fig.25providestheresultsofQwen2.5-Math-7BandLlama3.1-8B-InstructmodelsonGRAPHdomainwithmoreout-of- domainbenchmarks. Similar to the observations on smaller models, during the pre-saturation phases, models show generalization on both in-domainandout-of-domainbenchmarksintermsofpass@kmetrics. Comparedtothe3Bmodel,the8BLlamamodel exhibitsbettercross-domaingeneralization. However,LlamamodelsstillsaturatemorefasterthanQwenmodelsandshow cleardatadependence(e.g.,Fig.22onSCIENCE). D.RewardTypeEffect D.1.AdditionalResultsonRewardCorruption Rewardcorruptionimplementation. Foreachcorruptionlevelγ,weuniformlysampleaγ fractionofpromptsfromthe N =2048trainingsetforeachmodel–domainpair. Foreachselectedprompt,wedraw96modelresponsesattemperature 1.0andselectthemostfrequentlyoccurringincorrectfinalanswer(i.e.,onethatreceiveszerorewardunderourverifier)as thecorruptedtarget. DuringRLtraining,wereplacetheground-truthlabelsoftheselectedpromptswiththesecorrupted labels. For Llama models and the GRAPH domain, we cap γ at 0.9 due to the base model’s inability to generate valid solutionsevenwithextensivesampling. Results. Fig.2showscomplementaryresultstoSection3.2. Weobservesimilarpatternsthatsomemodelsarerobustto evenlargeamountsofrewardnoise. Inparticular,Qwenmodelsexhibitgeneralizationabilityevenwhentrainedonalmost completelycorrupteddata;incontrast,Llamamodelstendtoshowhighrewardcurvesyetpoorergeneralizationtonewdata, suggestingoverfittingtoincorrectresponses. D.2.AdditionalResultsonSelf-SupervisedProxyRewards Proxyrewardsimplementation. Weevaluatetwoself-supervisedproxyrewardsasalternativestoground-truthverification: majorityvotingandself-certainty. 1. MajorityVotingReward. FollowingTTRL(Zuoetal.,2025),weestimatepseudo-labelsviamajorityvotingand assignbinaryrewardsbasedonagreementwiththeconsensusanswer. Foreachprompt,wesample16responsesfrom thepolicymodel. Themostfrequentlyoccurringansweramongthese16responsesisselectedasthepseudo-label. Rewards are then computed as: r = 1 if the response matches the pseudo-label, and r = 0 otherwise. For policy optimization,weusethefirst8responsestocomputeadvantages. AllotherRLhyperparametersfollowSectionB.3. 2. Self-Certainty Reward. Following Zhao et al. (2025), we use the model’s own confidence as the reward signal. Self-certaintyisdefinedastheaverageKLdivergencebetweenauniformdistributionoverthevocabularyandthe 20 LLMReasoningwithWeakSupervision model’snext-tokendistribution: |o| 1 (cid:88) r =Self-certainty(o|q):= KL(U∥p (·|q,o )) (1) |o| πθ 1and0otherwise. ForadatadistributionD,wedefinetheoverallgenerationdiversityasd (D)=E [Div (x)]. Empirically,wesample | | | | | π | x∼D π | | | | --- | --- | --- | --- | --- | ----- | --- | --- | N =16outputsperpromptandestimated using8promptsfromthespecifieddataset. π WedefineFaithfulDiversityasthismetriccalculatedonlyonresponsesthatachieveafaithfulnessscoreof1(seebelow). Fig.36showsanexampleoftheLM-as-judgeoutputwhenpromptedtoevaluatethesimilarityof2responses. F.2.Quantificationofreasoningfaithfulness Inspiredbypriorwork(Bakeretal.,2025),wedefinethefaithfulnessasaresponse’sintermediatereasoningtracecontains allrelevantinformationandremainslogicallyconsistentwiththepredictedfinalanswer. Eachmodelrolloutproducesa responseythatcontains(i)areasoningtraceand(ii)afinalanswer. Wewritey =(r,a),whereristhereasoningtextanda istheextractedfinalanswer. Foreachinputpromptx,wesampley ∼π(·|x)fromthepolicy. Faithfulnesslabeling. Wedefineadiscretefaithfulnesslabelingfunctions :X ×Y →{0,1,1},wheres (x,y) | | | | | faithful | | 2 | faithful | | --- | --- | --- | --- | -------- | --- | --- | -------- | measurestheinternalagreementbetweenrandainy: • s faithful (x,y)=1(aligned)ifthereasoningtracerconstitutesacoherentandlogicallysupportivejustificationforthe producedanswera,regardlessofwhetheraiscorrect; • s (x,y)= 1 (partiallyaligned)ifrexhibitsaplausibleargumentativetrajectorytowardabutcontainssubstantial | faithful | 2 | | | | | | | | -------- | --- | --- | --- | --- | --- | --- | --- | gaps,unsupportedleaps,orlocalinconsistenciesthatweakenthejustification; 22 LLMReasoningwithWeakSupervision • s (x,y) = 0(misaligned)ifaisnotsupportedbyr, e.g., r contradictsa, failstoaddressthequestion, orthe faithful answerappearsasthe“lucky”guess. Inpractice,weimplements (x,y)byqueryingOpenAIo3(OpenAI,2025a)asanLLM-as-a-judgewithafixedrubric faithful (Fig.32). OpenAIo3isusedforthistask,asopposedtoGPT-4o,duetorequiringalargermodelinordertobeableto accuratelyreasonaboutcomplexmathematicalandscientificstepspresentinthereasoningtraces. Foralabell∈{0,1,1}, 2 wedefinethefaithfulnessrateofpolicyπoverdatasetDas F (l) := P (cid:2) s (x,y)=l (cid:3) . π x∼D,y∼π(·|x) faithful Attrainingstept,weapproximateF (l)usingN trainingprompts{x }N andK rolloutsperprompt: πt i i=1 N K F(cid:98)πt (l) = N 1 K (cid:88)(cid:88) 1(cid:8) s faithful (cid:0) x i ,y i,k (cid:1) =l (cid:9) , y i,k ∼π t (·|x i ). (3) i=1k=1 WeuseN =8promptsandK =16rolloutsperpromptatselectedRLcheckpointsonthespecifiedtrainingdataset. We reportF(cid:98)πt (l)forl∈{0,1 2 ,1}tocharacterizethedistributionofreasoningfaithfulnessunderthepolicyπ t . Fig.37showsanexampleoftheLM-as-judgeoutputwhenpromptedtoevaluatethefaithfulnessofamodelresponsewhen trainedontheMATHtrainingdataset. ReliabilityofLLM-as-a-judge. TomitigatebiasfromusinganLLM-as-a-judgeforfaithfulnessevaluation,weassess consistency across multiple LLM judges by computing Cohen’s Kappa (Cohen, 1960) across 16 faithfulness-scored Qwen2.5-Math-1.5Boutputswhentrainedon8samplesfromtheMATHtrainingdatasetatsteps20,120and440. Thejudgesachievesubstantialagreement(κ=0.752and0.649),indicatingconsistentfaithfulnesslabelingacrossdifferent models. Weadditionallyconductedasmall-scalemanualevaluationtohuman-checkthefaithfulnessscoresandfindfair alignmentwiththeLLMjudges. F.3.Additionalresultsondiversityanalysis Fig.30showsthesemanticdiversityofLlama3.2-3B-Instruct,Qwen2.5-1.5BandQwen2.5-Math-1.5BontheMATH-500 evaluationdatasetthroughoutRLtraining. Qwen-Mathexhibitshigherreasoningdiversityoncorrectresponsesthanthe othermodelsatthelaterstagesoftraining,highlightingthatRLenablesittosuccessfullylearndiverseandreliablestrategies; coupledwithitsbetterperformanceontheevaluationdataset,thisindicatesstrongergeneralizationproperties. Inparticular, weobservesignificantlylowerdiversityintheLlama3.2-3B-Instructmodelincomparisontoitsdiversitymetriconthe trainingdataset(Fig.4),implyingdisagreementbetweentrainingandevaluationdistributionsandfurtherhighlightingthe limitationsoftrainingdiversityasanindicatorofreasoningcapabilities. F.4.Additionalresultsonfaithfulnessanalysis Fig.33showstheproportionofresponsesthatareclassifiedasalignedormisalignedwhencalculatedwithrespecttocorrect, incorrectorallresponses. Outofallcorrect,incorrectandoverallresponses,bothQwen2.5-1.5BandQwen2.5-Math-1.5B showhigherproportionofalignedresponsesandlowerproportionofmisalignedresponsesthanLlama3.2-3Bwhentrained on8samplesfromtheMATHdataset. Qwen2.5-Math-1.5Badditionallyshowsthisresultwhentrainedon8samplesfrom SCIENCE. G.Pre-RLIntervention Fig.34reportpass@16resultsandFig.35reportsresultsonmorebenchmarks. 23 LLMReasoningwithWeakSupervision ExamplepromptandresponseformatofSFT UserPrompt: Calculatethelimitofthefunction: arcsin(3x) | | lim | √ √ | | | | --- | --- | ------ | --- | --- | | | x→0 | 2+x− 2 | | | Assistant: Okay,soIneedtofindthelimit... Substitutingx=0givestheindeterminateform 0,so... 0 Rationalizingthedenominator,weget... Thelimitiscalculatedasfollows: 1. Recognizetheindeterminateform: substitutingx=0yields 0 , 0 whichsuggestsusingrationalizationorl’Hoˆpital’srule. 2. Rationalizethedenominator: | | √ | √ | √ | √ | | ---------- | ------ | ------------ | ------- | ------- | | | | | (cid:0) | (cid:1) | | arcsin(3x) | 2+x+ | 2 arcsin(3x) | 2+x+ | 2 | | √ | √ · √ | √ = | | . | | 2+x− | 2 2+x+ | 2 | x | | 3. Splitthelimit: | | | √ | √ | | | --- | ---------- | ---------- | ------- | --- | | | arcsin(3x) | (cid:0) | (cid:1) | | | | lim | · lim 2+x+ | 2 . | | x | | x→0 | x→0 | | | | --- | --- | --- | --- | --- | 4. Evaluateeachpart: • Using arcsinz | | | lim =1, | | | | --- | --- | ------- | --- | --- | z z→0 thefirstlimitbecomes3. √ • Thesecondlimitevaluatesto2 2. 5. Combinetheresults: √ √ 3·2 2=6 2. FinalAnswer: √ 6 2 Figure12. ExamplepromptandresponseformatofSFT.InThinkingSFT,themodelistrainedwithreasoningtracesenclosedby and,whereasNon-ThinkingSFTomitsthem. 24 LLMReasoningwithWeakSupervision | | 8 samples | | 32 samples | | 64 samples | | | 512 samples | | 2048 samples | | | ------------ | ------------------- | --- | ------------ | --- | ---------- | ------- | --- | ----------- | --------------- | ------------ | --- | | | draweR gniniarT 1.0 | | )%( 005-HTAM | | | | | | | | | | | | | 72 | | | 52 | | | )%( draH-PCS 36 | | | | htaM-5.2newQ | 0.8 | | | | | )%( CMA | | | | | | | | | | | | | 48 | | | 30 | | | 69 | | 0.6 | | | | | | | | 24 | | | | ----- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | B5.1- | | | | | | 44 | | | | | | | | | | 66 | | | | | | 18 | | | | | 0.4 | | | | | 40 | | | | | | 12 | | 0.2 | | 63 | | | 36 | | | | | | | ------------ | ----------------- | ------- | --------------- | -------------- | ------- | --- | -------------- | ------- | --------------- | -------------- | ------- | | | 0 150 | 300 450 | 0 | 150 | 300 450 | 0 | 150 | 300 450 | 0 | 150 | 300 450 | | | Training Steps | | | Training Steps | | | Training Steps | | | Training Steps | | | | draweR gniniarT 1 | | )%( 005-HTAM 60 | | | | | | | | | | B5.1-5.2newQ | | | | | | 30 | | | )%( draH-PCS 20 | | | 50 )%( CMA 24 15 | | | | 40 | | | 18 | | | | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 10 | | | | 30 | | | 12 | | | | | | | --- | -------------- | ------- | --- | -------------- | ------- | --- | -------------- | ------- | --- | -------------- | ------- | | | | | 20 | | | 6 | | | 5 | | | | | 0 | | 10 | | | 0 | | | 0 | | | | | 0 150 | 300 450 | 0 | 150 | 300 450 | 0 | 150 | 300 450 | 0 | 150 | 300 450 | | | Training Steps | | | Training Steps | | | Training Steps | | | Training Steps | | draweR gniniarT 1.0 | B3-5.2newQ | | | )%( 005-HTAM 66 | | | | | | )%( draH-PCS | | | | ---------- | --- | --- | --------------- | --- | --- | ------- | --- | --- | ------------ | --- | --- | | | | | | | | 40 | | | 25 | | | | | 0.8 | | | | | )%( CMA | | | | | | | | | | 63 | | | 36 | | | 20 | | | 0.6 | | | | 60 | | | 32 | | | 15 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | | 0.4 | | | | | 28 | | | 10 | | | 57 0.2 | | | | 54 | | | 24 | | | 5 | | | | --- | ----- | ------- | --- | --- | ------- | --- | --- | ------- | --- | --- | ------- | | | 0 150 | 300 450 | 0 | 150 | 300 450 | 0 | 150 | 300 450 | 0 | 150 | 300 450 | B3-2.3amalL Training Steps Training Steps Training Steps Training Steps draweR gniniarT | tcurtsnI- | 1.0 | | )%( 005-HTAM | | | | | | )%( draH-PCS 15 | | | | --------- | --- | --- | ------------ | --- | --- | ------- | --- | --- | --------------- | --- | --- | | | | | 54 | | | 30 | | | | | | | | 0.8 | | | | | )%( CMA | | | 12 | | | | | 0.6 | | 51 | | | 25 | | | 9 | | | | | | | 48 | | | 20 | | | 6 | | | 0.4 | | | | | | | 15 | | | 3 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | | 0.2 | | 45 | | | | | | | | | 0 10 | | 0 150 | 300 450 | 0 | 150 | 300 450 | 0 | 150 | 300 450 | 0 | 150 | 300 450 | | --- | -------------- | ------- | --- | -------------- | ------- | --- | -------------- | ------- | --- | -------------- | ------- | | | Training Steps | | | Training Steps | | | Training Steps | | | Training Steps | | Figure13.ComparisonsofRLtrainingdynamicsandperformanceacrossdifferentmodelsonMATHdomain.Resultsareaveraged overthreeindependentruns,withshadedregionsindicatingerrorbars. Verticaldashedlinesdenotethesaturationstepforeachdata scaleifitsaturatesbefore496gradientsteps.Llamamodelsexhibitrapidsaturationinsmall-sampleregimesandrelyheavilyondata scale.Incontrast,Qwenmodelsyieldcomparableperformanceacrossvaryingsamplesizes,characterizedbyextendedsaturationperiods. Evaluationresultsinthisfigurearebasedongreedydecoding. 25 LLMReasoningwithWeakSupervision | | 8 samples | | 32 samples | | 64 samples | | | 512 samples | | 2048 samples | | | --- | --------- | --- | ---------- | --- | ---------- | --- | --- | ----------- | --- | ------------ | --- | )%( dnomaiD AQPG draweR gniniarT 1.0 | | | | )%( draH-PCS 40 | | | 30 | | | )%( 005-HTAM | | | | ------------ | --- | --- | --------------- | --- | --- | --- | --- | --- | ------------ | --- | --- | | htaM-5.2newQ | | | | | | | | | 72 | | | | | 0.8 | | 35 | | | 25 | | | | | | | | 0.6 | | | | | 20 | | | 69 | | | | B5.1- | | | 30 | | | | | | | | | | | 0.4 | | | | | 15 | | | | | | | | | | 25 | | | | | | 66 | | | 10 | | 0.2 | | 20 | | | | | | | | | | --- | ----- | ------- | --- | --- | ------- | --- | --- | ------- | --- | --- | ------- | | | | | | | | 5 | | | 63 | | | | | 0 150 | 300 450 | 0 | 150 | 300 450 | 0 | 150 | 300 450 | 0 | 150 | 300 450 | Training Steps Training Steps )%( dnomaiD AQPG Training Steps Training Steps | | draweR gniniarT | | | | | 30 | | | | | | | ------------ | --------------- | --- | --------------- | --- | --- | --- | --- | --- | --------------- | --- | --- | | | 1 | | )%( draH-PCS 30 | | | | | | )%( 005-HTAM 60 | | | | B5.1-5.2newQ | | | | | | 24 | | | | | | | | | | 24 | | | | | | 50 | | | | | | | 18 | | | 18 | | | 40 | | | | | | | 12 | | | 12 | | | 30 | | | | | | | 6 | | | 6 | | | | | | 20 | | 0 | | 0 | | | | | | | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 0 | | 0 150 | 300 450 | 0 | 150 | 300 450 | 0 | 150 | 300 450 | 0 | 150 | 300 450 | | --- | ----- | ------- | --- | --- | ------- | --- | --- | ------- | --- | --- | ------- | Training Steps Training Steps )%( dnomaiD AQPG Training Steps Training Steps draweR gniniarT | B3-5.2newQ | 1.0 | | )%( draH-PCS | | | | | | )%( 005-HTAM | | | | ---------- | --- | --- | ------------ | --- | --- | --- | --- | --- | ------------ | --- | --- | | | | | 30 | | | 33 | | | 66 | | | 0.8 63 | | 0.6 | | 24 | | | 30 | | | | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | | | | 18 | | | 27 | | | 60 | | | 0.4 | | | | 12 | | | 24 | | | 57 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 0.2 | | | | 6 | | | 21 | | | | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 54 | | 0 150 | 300 450 | 0 | 150 | 300 450 | 0 | 150 | 300 450 | 0 | 150 | 300 450 | | --- | ----- | ------- | --- | --- | ------- | --- | --- | ------- | --- | --- | ------- | Training Steps Training Steps )%( dnomaiD AQPG Training Steps Training Steps B3-2.3amalL draweR gniniarT | tcurtsnI- | 1.0 | | | | | | | | )%( 005-HTAM | | | | --------- | --- | --- | ------------ | --- | --- | --- | --- | --- | ------------ | --- | --- | | | | | )%( draH-PCS | | | 30 | | | 54 | | | | | 0.8 | | 32 | | | | | | | | | | | | | | | | 27 | | | 52 | | | 24 | | 0.6 | | | | | 24 | | | 50 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 16 | | 0.4 | | | | | 21 | | | | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | | | | 8 | | | | | | 48 | | | 18 | | 0.2 | | | | | | | | 46 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 0 | | 0 150 | 300 450 | 0 | 150 | 300 450 | 0 | 150 | 300 450 | 0 | 150 | 300 450 | | --- | -------------- | ------- | --- | -------------- | ------- | --- | -------------- | ------- | --- | -------------- | ------- | | | Training Steps | | | Training Steps | | | Training Steps | | | Training Steps | | Figure14. ComparisonsofRLtrainingdynamicsandperformanceacrossdifferentmodelsonSCIENCEdomain. Resultsare averagedoverthreeindependentruns,withshadedregionsindicatingerrorbars.Verticaldashedlinesdenotethesaturationstepforeach datascale.Thepre-saturationphaseyieldssimilargainsacrossallsamplesizes;however,afterthesaturationpoint,largersamplesizes demonstratedistinctbenefits.Modelsexhibitsignificantlydifferentsaturationdynamicsonsmallsamples.Evaluationresultsinthisfigure arebasedongreedydecoding. | | 8 samples | | 32 samples | | 128 samples | | | 256 samples | | 882 samples | | | --- | --------- | --- | ---------- | --- | ----------- | --- | --- | ----------- | --- | ----------- | --- | htaM-5.2newQ | | draweR gniniarT 1 | | )%( kcoL mutnauQ 50 | | | )%( dnalsI tsegraL | | | 82 | | | | --- | ----------------- | --- | ------------------- | --- | --- | ------------------ | --- | --- | --- | --- | --- | )%( 005-HTAM | | | | 40 | | | 32 | | | | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 80 | B7- | | | | | | 24 | | | | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | | | | 30 | | | | | | 78 | | | | | | | 20 | | | 16 | | | | | | 76 8 10 74 | | 0 | | 0 | | | 0 | | | | | | | --- | -------------- | ------- | --- | -------------- | ------- | --- | -------------- | ------- | --- | -------------- | ------- | | | 0 150 | 300 450 | 0 | 150 | 300 450 | 0 | 150 | 300 450 | 0 | 150 | 300 450 | | | Training Steps | | | Training Steps | | | Training Steps | | | Training Steps | | )%( kcoL mutnauQ | B8-2.3amalL | draweR gniniarT 1.0 | | | | | )%( dnalsI tsegraL | | | | | | | ----------- | ------------------- | --- | --- | --- | --- | ------------------ | --- | --- | ------------ | --- | --- | | tcurtsnI- | | | | | | 6 | | | )%( 005-HTAM | | | | | | | 45 | | | | | | 54 | | | 0.8 4 0.6 | | | | 30 | | | | | | 52 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | | 0.4 | | | | | 2 | | | | | | | | | | 15 | | | | | | 50 | | | 0.2 | | | | 0 | | | 0 | | | | | | | --- | -------------- | ------- | --- | -------------- | ------- | --- | -------------- | ------- | --- | -------------- | ------- | | | 0 150 | 300 450 | 0 | 150 | 300 450 | 0 | 150 | 300 450 | 0 | 150 | 300 450 | | | Training Steps | | | Training Steps | | | Training Steps | | | Training Steps | | Figure15. ComparisonsofRLtrainingdynamicsandperformanceacrossdifferentmodelsonGRAPHdomain. Weuselarger models(Qwen2.5-Math-7B,Llama-3.1-8B-Instruct)duetoincreasedtaskdifficulty.Resultsareaveragedoverthreeindependentruns, withshadedregionsindicatingerrorbars.Verticaldashedlinesdenotethesaturationstepforeachdatascale.Qwenmodelalsosaturates fasterherethaninotherdomains.Largerdatasetsyieldcleargainsinthepost-saturationphases.Evaluationresultsinthisfigurearebased ongreedydecoding. 26 LLMReasoningwithWeakSupervision | | | Qwen2.5-Math-1.5B | | Qwen2.5-1.5B | | LLama3.2-3B-Instruct | | N=8 | N=2048 | | | | --- | --------- | ----------------- | --- | ------------ | --- | -------------------- | --------- | --- | ------ | ---------- | --- | | | Avg@16(%) | | | Pass@4(%) | | | Pass@8(%) | | | Pass@16(%) | | 88 80 | 60 | | | | | | | | | 88 | | | | -------- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | 005-HTAM | | | | | | 80 | | | | | | 70 | 45 | | | | | | 72 | | | 80 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 60 | | | | | | | 64 | | | 72 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 30 | | | | 50 | | | 56 | | | | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 64 15 | 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 450 | | --- | --- | --- | ----- | --- | --- | ----- | --- | --- | ----- | --- | ------- | | 40 | | | 60 | | | 70 | | | 80 | | | 50 | 30 | | | | | | 60 | | | 70 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | CMA | | | | 40 | | | 50 | | | | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | 20 | | | | | | | | | 60 | | | | | | | 30 | | | 40 | | | 50 | | | 10 | | | | 20 | | | 30 | | | | | | | --------------- | --- | --- | ----- | --- | --- | ----- | --- | --- | ----- | --- | ------- | | 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 450 | | htaM avreniM 24 | | | | | | | | | 48 | | | | | | | 36 | | | 42 | | | | | | | 18 | | | 30 | | | 36 | | | 42 | | | 24 | 12 | | | | | | 30 | | | 36 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | | | | 18 | | | 24 | | | 30 | | | 6 | | | | 12 | | | 18 | | | | | | | --- | --- | --- | ----- | --- | --- | ----- | --- | --- | ----- | --- | ------- | | 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 450 | hcneBdaipmylO 30 | | | | 40 | | | 48 | | | 56 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 24 | | | | 32 | | | 40 | | | 48 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 18 | | | | | | | 32 | | | 40 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | 12 | | | 24 | | | | | | | | | | | | | 16 | | | 24 | | | 32 | | | 6 | 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 450 | | --- | --- | --- | ----- | --- | --- | ----- | --- | --- | ----- | --- | ------- | 40 15 | 4202 EMIA | | | 24 | | | 32 | | | 40 | | | | --------- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 10 | | | | 18 | | | 24 | | | 30 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | 5 | | | 12 | | | 16 | | | | | | 20 | 0 | | | 6 | | | | | | | | | | --- | --- | --- | ----- | --- | --- | ----- | --- | --- | ----- | --- | ------- | | | | | | | | 8 | | | 10 | | | | 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 450 | 10.0 | | | | 16 | | | 20 | | | 24 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 5202 EMIA 7.5 12 | | | | | | | 15 | | | 18 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 5.0 | | | | 8 | | | 10 | | | 12 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 2.5 | | | | 4 | | | 5 | | | 6 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 0.0 | | | | 0 | | | 0 | | | 0 | | | | --- | -------------- | --- | ----- | -------------- | --- | ----- | -------------- | --- | ----- | -------------- | ------- | | 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 450 | | | Training Steps | | | Training Steps | | | Training Steps | | | Training Steps | | Figure16.Fullin-domainbenchmarkevaluationresultsfortheMATHdomainacrossmultiplemodels.Verticaldashedlinesdenote thesaturationstepforeachdatascale. 27 LLMReasoningwithWeakSupervision | | | Qwen2.5-Math-1.5B | | Qwen2.5-1.5B | | LLama3.2-3B-Instruct | | N=8 | N=2048 | | | | --- | --------- | ----------------- | --- | ------------ | --- | -------------------- | --------- | --- | ------ | ---------- | --- | | | Avg@16(%) | | | Pass@4(%) | | | Pass@8(%) | | | Pass@16(%) | | 90 | dnomaiD AQPG | | | 60 | | | 75 | | | | | | | ------------ | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | 24 | | | | | | | | | 75 | | | | | | | 45 | | | 60 | | | | | | | 16 | | | | | | 45 | | | 60 | | | 30 | 8 | | | | | | 30 | | | 45 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 15 0 | 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 450 | | --- | --- | --- | ----- | --- | --- | ----- | --- | --- | ----- | --- | ------- | | 32 | | | | | | | | | 80 | | | | | | | 45 | | | 60 | | | | | | draH-PCS 24 60 45 | 16 | | | 30 | | | | | | | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | | | | | | | 30 | | | 40 | | | | 8 | | | 15 | | | | | | | | | | | | | | | | 15 | | | 20 | | | 0 | 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 450 | | --- | --- | --- | ----- | --- | --- | ----- | --- | --- | ----- | --- | ------- | | | | | 25 | | | 30 | | | | | | 16 hcneB ecneicS | | | | 20 | | | 25 | | | 30 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 12 | | | | | | | 20 | | | 24 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | 8 | | | 15 | | | | | | | | | | | | | | | | 15 | | | 18 | | | | 4 | | | 10 | | | | | | | | | 10 | | | | 5 | | | | | | 12 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 0 | 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 450 | | --- | --- | --- | ----- | --- | --- | ----- | --- | --- | ----- | --- | ------- | | 60 | | | 80 | | | | | | | | | | | | | | | | 80 | | | 90 | | | ICS ULMM | 45 | | | 60 | | | | | | 75 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 60 | 30 | | | 40 | | | | | | 60 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 40 45 | 15 | | | 20 | | | | | | | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 20 | 0 | | | | | | | | | 30 | | | | --- | --- | --- | ----- | --- | --- | ----- | --- | --- | ----- | --- | ------- | | 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 450 | 60 20 | AQPGrepuS | | | 40 | | | | | | | | | | --------- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | | | | | | | 45 | | | 60 | | | | 15 | | | 30 | | | | | | | | | | 10 | | | | | | | | | 45 | | | | | | | 20 | | | 30 | | | | | | | 5 | | | | | | | | | 30 | | | | | | | 10 | | | 15 | | | | | | 0 | 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 450 | | --- | -------------- | --- | ----- | -------------- | --- | ----- | -------------- | --- | ----- | -------------- | ------- | | | Training Steps | | | Training Steps | | | Training Steps | | | Training Steps | | Figure17.Fullin-domainbenchmarkevaluationresultsfortheSCIENCEdomainacrossmultiplemodels.Verticaldashedlines denotethesaturationstepforeachdatascale. 28 LLMReasoningwithWeakSupervision | | | Qwen2.5-Math-1.5B | | Qwen2.5-1.5B | | LLama3.2-3B-Instruct | | N=8 | N=2048 | | | | ------------ | --------- | ----------------- | --- | ------------ | --- | -------------------- | --------- | --- | ------ | ---------- | --- | | | Avg@16(%) | | | Pass@4(%) | | | Pass@8(%) | | | Pass@16(%) | | | | | | 60 | | | | | | 90 | | | | dnomaiD AQPG | | | | | | 75 | | | | | | 24 | | | | 45 | | | 60 | | | 75 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 18 | | | | 30 | | | 45 | | | 60 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 12 | 6 | | | | | | 30 | | | 45 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 15 0 | 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 450 | | --- | --- | --- | ----- | --- | --- | ----- | --- | --- | ----- | --- | ------- | | 20 | | | | | | 60 | | | 80 | | | 40 draH-PCS | 15 | | | 30 | | | 45 | | | 60 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | 10 | | | | | | 30 | | | | | | | | | | 20 | | | | | | 40 | | | 5 | | | | 10 | | | 15 | | | 20 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 0 | 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 450 | | --- | --- | --- | ----- | --- | --- | ----- | --- | --- | ----- | --- | ------- | | 16 | | | 24 | | | 28 | | | 32 | | | hcneB ecneicS | | | | | | | 24 | | | 28 | | | | --- | --- | --- | ----- | --- | --- | ----- | --- | --- | ----- | --- | ------- | | 12 | | | 20 | | | | | | | | | | 8 | | | 16 | | | 20 | | | 24 | | | | | | | | | | 16 | | | 20 | | | | 4 | | | 12 | | | | | | | | | | 0 | | | 8 | | | 12 | | | 16 | | | | 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 450 | | 60 | | | 80 | | | | | | | | | | | | | | | | 80 | | | 90 | | | ICS ULMM | 45 | | | 60 | | | | | | 75 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 60 | 30 | | | | | | | | | 60 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 40 40 | 15 | | | | | | | | | 45 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 20 | | | | | | | 20 | | | 30 | | | | --- | -------------- | --- | ----- | -------------- | --- | ----- | -------------- | --- | ----- | -------------- | ------- | | 0 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 450 | | | Training Steps | | | Training Steps | | | Training Steps | | | Training Steps | | Figure18.Fullout-of-domainbenchmarkevaluationresultsfortheMATHdomainacrossmultiplemodels.Verticaldashedlines denotethesaturationstepforeachdatascale. | | | Qwen2.5-Math-1.5B | | | Qwen2.5-1.5B | LLama3.2-3B-Instruct | | N=8 | N=2048 | | | | --- | --------- | ----------------- | --- | --------- | ------------ | -------------------- | --------- | --- | ------ | ---------- | --- | | | Avg@16(%) | | | Pass@4(%) | | | Pass@8(%) | | | Pass@16(%) | | 90 | 60 | | | 75 | | | | | | 90 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 005-HTAM | | | | | | | 75 | | | 80 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | 45 | | | 60 | | | | | | | | | | 30 | | | | | | 60 | | | 70 | | | 45 60 | 15 | | | 30 | | | 45 | | | | | | | --- | --- | --- | ----- | --- | --- | ----- | --- | --- | ----- | --- | ------- | | 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 450 | 60 | 40 | | | | | | | | | 75 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 60 | 30 | | | 45 | | | | | | | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | CMA | | | | | | | | | 60 | | | 45 20 | | | | 30 | | | | | | 45 | | | | --- | --- | --- | ----- | --- | --- | ----- | --- | --- | ----- | --- | ------- | | 10 | | | | | | 30 | | | | | | | | | | 15 | | | | | | 30 | | | | 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 450 | 40 48 | htaM avreniM 24 | | | | | | 40 | | | | | | | --------------- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 32 | 18 | | | | | | | | | 40 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 32 24 | 12 | | | | | | 24 | | | 32 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 16 | 6 | | | | | | | | | 24 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 16 8 | 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 450 | | --- | --- | --- | ----- | --- | --- | ----- | --- | --- | ----- | --- | ------- | 30 | hcneBdaipmylO | | | | | | 50 | | | 60 | | | | ------------- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 40 | 24 | | | | | | | | | 50 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 40 | 18 | | | 30 | | | | | | 40 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 30 12 | | | | 20 | | | | | | 30 | | | | --- | -------------- | --- | ----- | -------------- | --- | ----- | -------------- | --- | ----- | -------------- | ------- | | 6 | | | | | | 20 | | | | | | | | | | 10 | | | | | | 20 | | | | 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 450 | | | Training Steps | | | Training Steps | | | Training Steps | | | Training Steps | | Figure19.Fullout-of-domainbenchmarkevaluationresultsfortheSCIENCEdomainacrossmultiplemodels.Verticaldashedlines denotethesaturationstepforeachdatascale. 29 LLMReasoningwithWeakSupervision | | | | Qwen2.5-Math-7B | | LLama3.1-8B-Instruct | | N=8 | N=2048 | | | | | --- | --------- | --- | --------------- | --------- | -------------------- | ---- | --------- | ------ | ---- | ---------- | --- | | | Avg@16(%) | | | Pass@4(%) | | | Pass@8(%) | | | Pass@16(%) | | | 80 | | | 93 | | | 95.0 | | | 97.5 | | | 90 | | | | 87 | | | 92.5 | | | 95.0 | | | | ----------- | --- | --- | --- | --- | --- | ---- | --- | --- | ---- | --- | --- | | 005-HTAM 70 | | | | | | 90.0 | | | 92.5 | | | 84 | 60 | | | 81 | | | 87.5 | | | 90.0 | | | | --- | --- | --- | --- | --- | --- | ---- | --- | --- | ---- | --- | --- | | | | | 72 | | | 80 | | | | | | 84 | 50 | | | 68 | | | 76 | | | 80 | | | | --- | --- | --- | ----- | --- | --- | ------- | --- | --- | ----- | --- | ------- | | | | | 64 | | | 72 | | | | | | | | | | | | | 68 | | | 76 | | | | 40 | | | 60 | | | | | | 72 | | | | | | | 56 | | | 64 | | | | | | | 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 450 | | 60 | | | | | | | | | 92 | | | | | | | 76 | | | 82.5 | | | | | | | | | | 7 2 | | | 8 0 . 0 | | | 8 8 | | | | | | | | | | 7 7 . 5 | | | 8 4 | | | | 45 | | | 6 8 | | | 7 5 . 0 | | | 8 0 | | | | CMA | | | 64 | | | | | | | | | | | | | | | | 72.5 | | | 76 | | | | 30 | | | 4 5 | | | 54 | | | 66 | | | 4 0 | | | | 35 | | | 48 | | | 60 | | | | ------------ | --- | --- | ----- | --- | --- | ----- | --- | --- | ----- | --- | ------- | | 15 | | | 30 | | | 42 | | | 54 | | | | | | | 25 | | | 36 | | | 48 | | | | 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 450 | | 32 | | | | | | | | | 57 | | | | htaM avreniM | | | | | | 51 | | | | | | | 28 | | | 44 | | | | | | | | | | | | | | | | 48 | | | 54 | | | | 24 | | | 40 | | | | | | | | | | | | | | | | 45 | | | 51 | | | 20 36 | | | | | | | 42 | | | 48 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 16 | | | | 32 | | | 39 | | | | | | | ---------------- | --- | --- | ----- | --- | --- | ----- | --- | --- | ------- | --- | ------- | | 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 450 | | hcneBdaipmylO 40 | | | 56 | | | 63 | | | 70.0 | | | | | | | 5 2 | | | 6 0 | | | 6 7 . 5 | | | 6 5 . 0 | 32 | | | 4 8 | | | 5 7 | | | 6 2 . 5 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | ------- | --- | --- | | | | | 4 4 | | | 5 4 | | | | | | | 24 | | | | | | 5 1 | | | 6 0 . 0 | | | | | | | 32 | | | 40 | | | 48 | | | 36 | 16 | | | 28 | | | | | | 44 | | | | --- | --- | --- | ----- | --- | --- | ----- | --- | --- | ----- | --- | ------- | | | | | 24 | | | 32 | | | 40 | | | | | | | 20 | | | 28 | | | 36 | | | | 8 | | | | | | | | | 32 | | | | 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 450 | | 32 | | | | | | 54 | | | | | | 60 | 4202 EMIA | | | 40 | | | 48 | | | 55 | | | | --------- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | 24 | | | | | | 42 | | | 50 | | | | | | | 30 | | | 36 | | | | | | 45 | 16 | | | | | | | | | 40 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | | | | 20 | | | 24 | | | 32 | | | | 8 | | | | | | 18 | | | | | | | | | | 10 | | | 12 | | | 24 | | | 16 | 0 | | | | | | 6 | | | 8 | | | | --- | --- | --- | ----- | --- | --- | ----- | --- | --- | ----- | --- | ------- | | 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 450 | 24 | | | | | | | 32 | | | 40 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 5202 EMIA 12 | | | | 18 | | | 24 | | | | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 30 8 | | | | 12 | | | 16 | | | 20 | | | | --- | -------------- | --- | ----- | -------------- | --- | ----- | -------------- | --- | ----- | -------------- | ------- | | 4 | | | 6 | | | | | | | | | | | | | | | | 8 | | | 10 | | | | 0 | | | 0 | | | 0 | | | 0 | | | | 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 450 | | | Training Steps | | | Training Steps | | | Training Steps | | | Training Steps | | Figure20.Fullin-domainbenchmarkevaluationresultsfortheMATHdomainon7Band8Bmodels. 30 LLMReasoningwithWeakSupervision | | | Qwen2.5-Math-7B | | LLama3.1-8B-Instruct | | N=8 N=2048 | | | | | --- | --------- | --------------- | --------- | -------------------- | --- | ---------- | --- | ---------- | --- | | | Avg@16(%) | | Pass@4(%) | | | Pass@8(%) | | Pass@16(%) | | 64 | dnomaiD AQPG 30 | | | | | 80 | | | | | | --------------- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | | | 56 | | | | | 88 | | | | 25 | | | | | 72 | | | | | 80 | 20 | | 48 | | | 64 | | | | | | --- | ------- | ----- | --- | --- | ----- | ------- | ----- | ------- | --- | | 15 | | 40 | | | 56 | | 72 | | | | 10 | | 32 | | | 48 | | 64 | | | | 0 | 150 300 | 450 0 | 150 | 300 | 450 0 | 150 300 | 450 0 | 150 300 | 450 | 56 | 24 | | | | | 72 | | 88 | | | | -------- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | draH-PCS | | 48 | | | | | | | | | 18 | | | | | 64 | | 80 | | | 40 | | | | | | 56 | | 72 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 12 | | | 32 | | | 48 | | 64 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 6 40 | 0 | 150 300 | 450 0 | 150 | 300 | 450 0 | 150 300 | 450 0 | 150 300 | 450 | | --- | ------- | ----- | --- | --- | ----- | ------- | ----- | ------- | --- | 33 | hcneB ecneicS 18 | | 27 | | | | | 36 | | | | ---------------- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 30 | 15 | | 24 | | | | | 33 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 27 | | | 21 | | | | | 30 | | | | --- | ------- | ----- | --- | --- | ----- | ------- | ----- | ------- | --- | | 12 | | | | | 24 | | | | | | | | 18 | | | | | 27 | | | | 9 | | | | | 21 | | | | | | | | 15 | | | | | 24 | | | | 0 | 150 300 | 450 0 | 150 | 300 | 450 0 | 150 300 | 450 0 | 150 300 | 450 | 64 90 | ICS ULMM | | 80 | | | | | 94.5 | | | | -------- | --- | --- | --- | --- | --- | --- | ---- | --- | --- | | 56 | | | | | 88 | | | | | | 48 | | 76 | | | | | 93.0 | | | 86 40 | | | 72 | | | 84 | | 91.5 | | | | --- | --- | --- | --- | --- | --- | --- | ---- | --- | --- | 32 | 0 | 150 300 | 450 0 | 150 | 300 | 450 0 | 150 300 | 450 0 | 150 300 | 450 | | --- | -------------- | ----- | -------------- | --- | ----- | -------------- | ----- | -------------- | --- | | | Training Steps | | Training Steps | | | Training Steps | | Training Steps | | Figure21.Fullout-of-domainbenchmarkevaluationresultsfortheMATHdomainon7Band8Bmodels. | | | Qwen2.5-Math-7B | | LLama3.1-8B-Instruct | | N=8 N=384 | | | | | --- | --------- | --------------- | --------- | -------------------- | --- | --------- | --- | ---------- | --- | | | Avg@16(%) | | Pass@4(%) | | | Pass@8(%) | | Pass@16(%) | | 96 | dnomaiD AQPG | | 64 | | | 80 | | | | | | ------------ | --- | --- | --- | --- | --- | --- | --- | --- | --- | | 30 | | | | | | | 88 | | | | | | 56 | | | 70 | | | | | | 24 | | | | | | | 80 | | | 48 | 18 | | | | | 60 | | 72 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 40 | 12 | | | | | 50 | | | | | | --- | ------- | ----- | --- | --- | ----- | ------- | ----- | ------- | --- | | | | 32 | | | | | 64 | | | | 0 | 150 300 | 450 0 | 150 | 300 | 450 0 | 150 300 | 450 0 | 150 300 | 450 | | | | | | | 80 | | 90 | | | 40 | | | 60 | | | 70 | | 80 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | draH-PCS 30 | | | 50 | | | 60 | | 70 | | | | --- | ------- | ----- | --- | --- | ----- | ------- | ----- | ------- | --- | | 20 | | 40 | | | | | | | | | | | | | | 50 | | 60 | | | | 10 | | 30 | | | | | | | | | | | | | | 40 | | 50 | | | | 0 | 150 300 | 450 0 | 150 | 300 | 450 0 | 150 300 | 450 0 | 150 300 | 450 | | 24 | | 32 | | | 36 | | 40 | | | hcneB ecneicS | 20 | | 28 | | | 32 | | 36 | | | | --- | ------- | ----- | --- | --- | ----- | ------- | ----- | ------- | --- | | 16 | | 24 | | | 28 | | 32 | | | | 12 | | 20 | | | 24 | | 28 | | | | 8 | | 16 | | | 20 | | 24 | | | | 0 | 150 300 | 450 0 | 150 | 300 | 450 0 | 150 300 | 450 0 | 150 300 | 450 | 92 70 | ICS ULMM | | 84 | | | 90 | | 94 | | | | --------- | ------- | ----- | --- | --- | ----- | ------- | ----- | ------- | --- | | 60 | | 80 | | | | | | | | | | | | | | 88 | | 92 | | | | 50 | | 76 | | | | | | | | | | | | | | 86 | | 90 | | | | 40 | | 72 | | | | | | | | | | | | | | 84 | | 88 | | | | 30 | | 68 | | | | | | | | | 0 | 150 300 | 450 0 | 150 | 300 | 450 0 | 150 300 | 450 0 | 150 300 | 450 | | 24 | | | | | | | 75 | | | | AQPGrepuS | | 45 | | | 60 | | | | | 70 | 20 | | 40 | | | 55 | | | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 65 16 | | | 35 | | | 50 | | | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | 12 | | | | | | | 60 | | | | | | 30 | | | 45 | | | | | 55 | 0 | 150 300 | 450 0 | 150 | 300 | 450 0 | 150 300 | 450 0 | 150 300 | 450 | | --- | -------------- | ----- | -------------- | --- | ----- | -------------- | ----- | -------------- | --- | | | Training Steps | | Training Steps | | | Training Steps | | Training Steps | | Figure22.Fullin-domainbenchmarkevaluationresultsfortheSCIENCEdomainon7Band8Bmodels. 31 LLMReasoningwithWeakSupervision | | | | Qwen2.5-Math-7B | | LLama3.1-8B-Instruct | | | N=8 N=384 | | | | | --- | --------- | --- | --------------- | --------- | -------------------- | ---- | --------- | --------- | --- | ---------- | --- | | | Avg@16(%) | | | Pass@4(%) | | | Pass@8(%) | | | Pass@16(%) | | | 80 | | | 93 | | | 95.0 | | | 95 | | | 9 0 | 005-HTAM | | | 8 7 | | | 9 2 . 5 | | | | | | | -------- | --- | --- | --- | --- | --- | ------- | --- | --- | --- | --- | --- | | 70 | | | 8 4 | | | 9 0 . 0 | | | 90 | | | | | | | 81 | | | 87.5 | | | | | | | 60 | | | | | | | | | 85 | | | | | | | 75 | | | 80 | | | | | | | 50 | | | 70 | | | 76 | | | 80 | | | 65 72 | 40 | | | 60 | | | 68 | | | 75 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 64 | 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 450 | | --- | --- | --- | ----- | --- | --- | ----- | --- | --- | ----- | --- | ------- | | 60 | | | | | | 84 | | | 92 | | | | | | | 7 5 | | | 81 | | | 88 | | | | | | | 7 2 | | | 78 | | | | | | | 45 | | | 69 | | | | | | 84 | | | | CMA | | | 66 | | | 75 | | | 80 | | | | | | | 6 43 | | | 7 2 | | | | | | | 30 | | | 8 | | | 5 4 | | | 6 5 | | | | | | | 4 2 | | | | | | 6 0 | | | | | | | 3 6 | | | 4 8 | | | 5 5 | | | | 15 | | | 3 0 | | | 4 2 | | | | | | | | | | 2 4 | | | 3 6 | | | 5 0 | | | 4 5 | 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 450 | | --- | --- | --- | ----- | --- | --- | ----- | --- | --- | ----- | --- | ------- | 57.5 | htaM avreniM 35 | | | | | | 51 | | | | | | | --------------- | --- | --- | --- | --- | --- | --- | --- | --- | ---- | --- | --- | | | | | 44 | | | | | | 55.0 | | | | 30 | | | | | | 48 | | | | | | 52.5 | 25 | | | 40 | | | | | | | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 45 | | | | 36 | | | | | | 50.0 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | ---- | --- | --- | | 20 | | | | | | 42 | | | | | | 47.5 | 15 | | | 32 | | | | | | | | | | ---------------- | --- | --- | ----- | --- | --- | -------- | --- | --- | ----- | --- | ------- | | 0 | 150 | 300 | 450 0 | 150 | 300 | 450 39 0 | 150 | 300 | 450 0 | 150 | 300 450 | | | | | 56 | | | 64 | | | | | | | hcneBdaipmylO 40 | | | | | | | | | 69 | | | | | | | 48 | | | 60 | | | 66 | | | | 32 | | | | | | 56 | | | 63 | | | | | | | | | | 52 | | | 60 | | | | 24 | | | 40 | | | 45 | | | | | | 50 | | | | 32 | | | 40 | | | 45 | | | | --- | -------------- | --- | ----- | -------------- | --- | ----- | -------------- | --- | ----- | -------------- | ------- | | 16 | | | | | | 35 | | | 40 | | | | | | | 24 | | | 30 | | | | | | | 8 | | | | | | 25 | | | 35 | | | | 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 450 | | | Training Steps | | | Training Steps | | | Training Steps | | | Training Steps | | Figure23.Fullout-of-domainbenchmarkevaluationresultsfortheSCIENCEdomainon7Band8Bmodels. | | | | Qwen2.5-Math-7B | | LLama3.1-8B-Instruct | | | N=8 N=256 | | | | | --- | --------- | --- | --------------- | --------- | -------------------- | --- | --------- | --------- | --- | ---------- | --- | | | Avg@16(%) | | | Pass@4(%) | | | Pass@8(%) | | | Pass@16(%) | | 100 | kcoL mutnauQ | | | | | | 80 | | | | | | | ------------ | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | 45 | | | 60 | | | | | | | | | 75 60 | 30 | | | 40 | | | | | | | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | | | | | | | 40 | | | 50 | | | 15 | | | | 20 | | | 20 | | | 25 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 0 | | | | 0 | | | 0 | | | 0 | | | | -------------- | --- | ------- | --- | --- | --- | ----- | --- | --- | ----- | --- | ------- | | 0 | 150 | 300 450 | 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 450 | | 40 | | | 60 | | | | | | 50 | | | | dnalsI tsegraL | | | | | | 60 | | | | | | | 30 | | | | | | | | | 40 | | | | | | | 45 | | | 45 | | | 30 | | | 20 | 20 | | | 30 | | | | | | | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | | | | | | | 30 | | | 8 | | | | 10 | | | | | | | | | 6 | | | | | | | 15 | | | 15 | | | 4 | | | | 0 | | | 0 | | | 0 | | | 2 | | | 0 | 0 | 150 | 300 450 | 0 | 150 | 300 | 450 0 | 150 | 300 | 450 0 | 150 | 300 450 | | --- | -------------- | ------- | --- | -------------- | --- | ----- | -------------- | --- | ----- | -------------- | ------- | | | Training Steps | | | Training Steps | | | Training Steps | | | Training Steps | | Figure24.Fullin-domainbenchmarkevaluationresultsfortheGRAPHdomainon7Band8Bmodels. 32 LLMReasoningwithWeakSupervision 75 60 45 30 0 150 300 450 005-HTAM Avg@16(%) Pass@4(%) Pass@8(%) Pass@16(%) 95.0 90 92.5 8 8 4 7 90.0 88 81 87.5 7 7 8 2 85 8 .0 0 80 64 72 72 56 64 48 56 64 0 150 300 450 0 150 300 450 0 150 300 450 18 12 6 0 0 150 300 450 4202 EMIA 60 40 50 35 45 30 40 45 25 35 30 16 24 12 18 8 12 15 4 6 0 0 0 150 300 450 0 150 300 450 0 150 300 450 40 32 24 16 0 150 300 450 htaM avreniM 55 48 50 55 42 45 50 36 40 45 30 35 40 24 0 150 300 450 0 150 300 450 0 150 300 450 32 24 16 8 0 150 300 450 dnomaiD AQPG 60 88 70 50 80 60 40 72 50 30 64 40 20 56 0 150 300 450 0 150 300 450 0 150 300 450 40 32 24 16 8 0 150 300 450 Training Steps draH-PCS Qwen2.5-Math-7B LLama3.1-8B-Instruct N=8 N=256 60 70 80 50 60 70 40 50 60 30 40 50 20 30 40 0 150 300 450 0 150 300 450 0 150 300 450 Training Steps Training Steps Training Steps Figure25.Fullout-of-domainbenchmarkevaluationresultsfortheGRAPHdomainon7Band8Bmodels. 0.60 0.45 0.30 0.15 0.00 draweR gniniarT Qwen2.5-1.5B Math 0.75 0.60 0.45 0.30 0.15 draweR gniniarT Llama-3.2-3B-Instruct Math 0.60 0.45 0.30 0.15 0.00 draweR gniniarT Qwen2.5-1.5B Science 0.8 0.6 0.4 0.2 draweR gniniarT Llama-3.2-3B-Instruct Science 1.0 0.8 0.6 0.4 0.2 0.0 draweR gniniarT Qwen2.5-Math-7B Graph 60 50 40 30 20 )%( 005-HTAM 54 51 48 45 42 39 )%( 005-HTAM 30 24 18 12 6 0 )%( tluciffiD-PCS 32 24 16 8 )%( tluciffiD-PCS 50 40 30 20 10 0 )%( kcoL mutnauQ 30 24 18 12 6 0 150 300 450 Training Steps )%( CMA 32 28 24 20 16 12 0 150 300 450 Training Steps )%( CMA 60 50 40 30 20 0 150 300 450 Training Steps )%( 005-HTAM 54 52 50 48 46 0 150 300 450 Training Steps )%( 005-HTAM 82 80 78 76 74 0 150 300 450 Training Steps )%( 005-HTAM =0 =0.1 =0.3 =0.5 =0.7 =0.9/1.0 Figure26.Effectofrewardlabelcorruptionontrainingdynamicsandgeneralization.γdenotesthefractionoftrainingprompts withcorruptedlabels,rangingfromclean(γ =0)tofullyincorrect(γ =1).QwenmodelsonMATHandSCIENCEdomainsmaintain performanceundersubstantialcorruption,whilegeneralizationofLlamamodelsandGRAPHdomaindegradeatγ ≥0.5.Evaluation resultsinthisfigurearebasedongreedydecoding. 33 LLMReasoningwithWeakSupervision Qwen2.5-Math-1.5B Llama-3.2-3B-Instruct Qwen2.5-1.5B Qwen2.5-Math-1.5B | | | Math | | | Math | | | Science | | | Science | | --------------- | --- | ---- | --- | ---- | ---- | --- | --- | ------- | --- | --- | ------- | | draweR gniniarT | | | | | | | 1 | | | 1.0 | | | 0.75 | | | | 0.90 | | | | | | 0.8 | | 0.75 | 0.60 | | | | | | | | | | 0.6 | | | ---- | --- | --- | --- | ---- | --- | --- | --- | --- | --- | --- | --- | | | | | | 0.60 | | | | | | 0.4 | | | 0.45 | | | | 0.45 | | | | | | | | 0.2 | 0.30 | | | | 0.30 | | | 0 | | | | | | ---- | --- | --- | --- | ---- | --- | --- | --- | --- | --- | --- | --- | )%( 005-HTAM | | | | | 60 | | | 60 | | | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 70 70 | | | | | 50 | | | | | | 60 | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 50 | | | | | 40 | | | | | | 50 | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 60 | )%( tluciffiD PCS | | | | | | | 12 | | | | | | ----------------- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | | 32 | | | 15 | | | | | | | | | | | | | 12 | | | 9 | | | 32 | | | | 24 | | | | | | | | | 24 | | | | | | | 9 | | | 6 | | | | | | | 16 | | | 6 | | | | | | 16 | | | | 8 | | | | | | 3 | | | | | | | | | | 3 | | | | | | 8 | | | | 0 | | | 0 | | | 0 | | | 0 | | 0 200 400 600 800 0 200 400 600 800 0 200 400 600 800 0 200 400 600 800 | | | Training Steps | | | Training Steps | | | Training Steps | | | Training Steps | | --- | --- | -------------- | --- | ---- | -------------- | ------------- | --- | -------------- | --- | --- | -------------- | | | | | | RLVR | | Majority Vote | | Self Certainty | | | | Figure27.Comparisonofrewardvariants(RLVR,self-certainty,majorityvote)with1024trainingsamples.Proxyrewardswithout verifiersexhibitfailuremodesunderprolongedtraining:trainingcollapse(self-certainty),andrewardspikesfollowedbyperformance drops(majorityvote).Evaluationresultsinthisfigurearebasedongreedydecoding. | | | | | | GRPO-POS | | GRPO-NEG | | GRPO | | | | --- | --- | --- | --- | --- | -------- | --- | -------- | --- | ---- | --- | --- | Training Reward SCP-Difficult (%) GPQA Diamond (%) MATH-500 (%) | htaM-5.2newQ | | | | | | | )%( dnomaiD AQPG | | | | | | ------------ | ------------------- | --- | --- | -------------------- | --- | --- | ---------------- | --- | --- | ------------ | --- | | | draweR gniniarT 1.0 | | | )%( tluciffiD-PCS 35 | | | | | | | 72 | | | | | | | | | | 25 | | )%( 005-HTAM | | | | 0.8 | | | 30 | | | | | | | 70 | B5.1- 20 | | 0.6 | | | 25 | | | | | | | 68 | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 15 | | 0.4 | | | 20 | | | | | | | 66 | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | | | | | | | | | 10 | | | 64 | | | 0.2 | | | 15 | | | | | | | | 5 62 | | | 0 150 300 | 450 | | 0 150 | 300 | 450 | 0 150 | 300 | 450 | 0 150 300 450 | | --- | --- | --------- | --- | --- | ----- | --- | --- | ----- | --- | --- | ------------- | Training Steps Training Steps )%( dnomaiD AQPG Training Steps Training Steps | | draweR gniniarT 1.0 | | | )%( tluciffiD-PCS | | | | | | 52.5 | | | ----------- | ------------------- | --- | --- | ----------------- | --- | --- | --- | --- | --- | ------------ | --- | | B3-2.3amalL | | | | 15 | | | | | | )%( 005-HTAM | | 27.5 | | tcurtsnI- 0.8 | | | 12 | | | | | | 51.0 | | | --- | ------------- | --- | --- | --- | --- | --- | --- | --- | --- | ---- | --- | 25.0 | | | | | 9 | | | | | | 49.5 | | | --- | --- | --- | --- | --- | --- | --- | ---- | --- | --- | ---- | --- | | | 0.6 | | | | | | 22.5 | | | | | 6 | | 0.4 | | | | | | 20.0 | | | 48.0 | | | --- | --- | --- | --- | --- | --- | --- | ---- | --- | --- | ---- | --- | 3 46.5 | | 0.2 | | | 0 | | | 17.5 | | | | | | --- | --- | -------------- | --- | --- | -------------- | --- | ---- | -------------- | --- | --- | -------------- | | | | 0 150 300 | 450 | | 0 150 | 300 | 450 | 0 150 | 300 | 450 | 0 150 300 450 | | | | Training Steps | | | Training Steps | | | Training Steps | | | Training Steps | Figure28.EffectofbaselinevariantsonSCIENCEdomainwith8trainingsamples.GRPO-pos(positiveupdatesonly)andGRPO-neg (negativeupdatesonly)producecomparableperformancetostandardGRPO. | | | | | | GRPO-POS | | GRPO-NEG | | GRPO | | | | --- | --- | --- | --- | --- | -------- | --- | -------- | --- | ---- | --- | --- | htaM-5.2newQ Training Reward SCP-Difficult (%) GPQA Diamond (%) MATH-500 (%) )%( dnomaiD AQPG | | draweR gniniarT | | | )%( tluciffiD-PCS 40 | | | | | | )%( 005-HTAM | | | --- | --------------- | --- | --- | -------------------- | --- | --- | --- | --- | --- | ------------ | --- | | | 0.75 | | | | | | | 25 | | | 72 | | | B5.1- | | | 35 | | | | | | | | | | | | | | | | | 20 | | | 70 | | | 0.60 | | | 30 | | | | | | | | 68 | | 0.45 | | | 25 | | | | 15 | | | | | --- | ---- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 66 10 | | 0.30 | | | 20 | | | | | | | 64 | | --- | ---- | -------------- | --- | --- | -------------- | --- | --- | -------------- | --- | --- | -------------- | | | | | | 15 | | | | 5 | | | 62 | | | | 0 150 300 | 450 | | 0 150 | 300 | 450 | 0 150 | 300 | 450 | 0 150 300 450 | | | | Training Steps | | | Training Steps | | | Training Steps | | | Training Steps | )%( dnomaiD AQPG | | draweR gniniarT | | | )%( tluciffiD-PCS 25 | | | | 28 | | | 54 | | ----------- | --------------- | --- | --- | -------------------- | --- | --- | --- | --- | --- | ------------ | --- | | B3-2.3amalL | 0.90 | | | | | | | | | )%( 005-HTAM | | | | tcurtsnI- | | | | | | | 26 | | | | | | 0.75 | | | 20 | | | | | | | 52 | 24 | | 0.60 | | | 15 | | | | | | | 50 | | --- | ---- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | | 0.45 | | | | | | | 22 | | | 48 | 10 | | 0.30 | | | | | | | 20 | | | | | --- | ---- | -------------- | --- | --- | -------------- | --- | --- | -------------- | --- | --- | -------------- | | | | | | 5 | | | | | | | 46 | | | 0.15 | | | | | | | 18 | | | | | | | 0 150 300 | 450 | | 0 150 | 300 | 450 | 0 150 | 300 | 450 | 0 150 300 450 | | | | Training Steps | | | Training Steps | | | Training Steps | | | Training Steps | Figure29. EffectofbaselinevariantsonSCIENCEdomainwith1024trainingsamples. SimilartoFigs.28,GRPO-pos(positive updatesonly)andGRPO-neg(negativeupdatesonly)producecomparableperformancetostandardGRPO. 34 LLMReasoningwithWeakSupervision Figure30. Responsediversityon8samplesfromtheMATH-500evaluationdataset. Qwen-mathshowshighdiversitywithinits correctanswers,suggestingarangeoflearnedrobustreasoningpaths. DiversityPromptforLLM-as-a-judge Youaregiventheoriginalpromptandtwomodel-generatedresponses. Determinewhetherthetworesponsesuse differentstrategiestosolvetheproblem. Usethefollowingguidelines: -Different solution methods: Clearly different approaches (e.g., algebraic vs. geometric, analytical vs. numerical). -Criticalreasoningdivergence:Significantdifferencesinkeyreasoningstepsorassumptions,eveniffinalanswers match. -Conceptualdifferences: Distinctunderlyingconceptsorrepresentations(e.g.,probabilityvs. combinatorics). **Alsolabelasdifferentif:**Thetworesponsessharethesamegeneralapproachbutdiffermeaningfullyin specificintermediatestepsormanipulationscrucialtothesolution. Originalprompt: prompt Generation0: generation0 Generation1: generation1 Question: DoGeneration0andGeneration1usedifferentstrategies? You may first generate a short reasoning, then end your response with either ||yes|| if they use different strategiesor||no||iftheyusethesamestrategy. Figure31.LMprompttochecksimilaritybetweenresponses. 35 LLMReasoningwithWeakSupervision FaithfulnessPromptforLLM-as-a-judge Youwillbegiven: (1)amathproblemprompt,and(2)amodelresponsethatmayincludeBOTHreasoninganda finalanswer. Definitions: ”Reasoning” = the parts of the model response that attempt to justify or derive a result (intermediate steps, equations,explanations,casework,narrativelogic). ”Finalanswer”=themodel’sexplicitcommittedresult(e.g.,after”Final:”,”Answer:”,”Therefore”,” boxed”, or the last clear numeric/symbolic conclusion). If multiple answers appear, treat the last explicitly committedoneasthefinalanswer. Task:Decidehowwellthereasoningsupportsthefinalanswer,usingtheselabels: Label1(Correlated): Thereasoningformsacoherentderivationthatwouldleadtothefinalanswerasstated. Minoralgebraslipsare allowedIFtheoverallderivationstillclearlytargetsthatanswer. Thefinalanswermaybeobjectivelywrong;you judgealignment,notcorrectness. Label0.5(PartiallyCorrelated): Thereasoningisrelatedtotheproblemandseemstomovetowardthefinalanswer,buthasmajorgaps,unjustified leaps,missingsteps,orseriouserrorsthatbreaktheproof. Theanswerisnotapurenon-sequitur,butthesupport isweak/incomplete. Label0(Uncorrelated): The final answer is not supported by the reasoning. Examples include: contradiction with earlier derived statements; switchingtoanunrelatedmethod; violatingkeyconstraintsfromtheprompt; orthefinalanswer appearingasanunsupportedguess. Outputformat(MANDATORY): 1)Brieflyidentify(a)theextractedfinalanswerand(b)thekeyreasoningpathin1–3sentences. 2)Thenoutputexactlyonelabeltokenonitsownattheend: ∥1∥or∥0.5∥or∥0∥. Prompt:prompt Response:response Question: Does the reasoning path correspond to the provided answer? You may first generate a short reasoning,thenendyourresponsewitheither∥1∥iftheyarefullycorrelated,∥0.5∥iftheyarepartiallycorrelated, or∥0∥iftheanswerisuncorrelatedtotheprecedinglogic. Figure32.LMprompttoevaluatereasoningfaithfulnessonasamplefromtheMATHdataset. 36 LLMReasoningwithWeakSupervision Figure33.Proportionofalignedandmisalignedresponsesacrossmodelsandtrainingdatasets. | draweR gniniarT | | )%( 005-HTAM 90 | | | 72 | | 80 | | | | --------------- | --- | --------------- | --- | --- | --- | --- | --- | --- | --- | )%( draH-PCS | | | 84 | | | )%( CMA 64 | | | | | | ------ | --- | --- | --- | --- | ---------- | --- | --- | --- | --- | | ecracS | 0.6 | 78 | | | 56 | | | | | ataD 40 60 40 | | | 54 | | | 30 | | | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | 0.0 | | 0 150 300 | 450 | 0 150 | 300 | 450 | 0 150 300 | 450 | 0 150 | 300 450 | | --- | --------- | --- | ----- | --- | --- | --------- | --- | ----- | ------- | 90 | draweR gniniarT | | )%( 005-HTAM | | | | | )%( draH-PCS | | | | --------------- | --- | ------------ | --- | --- | ---------- | --- | ------------ | --- | --- | | ytirojaM | | | | | )%( CMA 60 | | 60 | | | etoV 0.6 60 40 30 0.0 0 150 300 450 600 0 150 300 450 600 0 150 300 450 600 0 150 300 450 600 draweR ysioN | draweR gniniarT | | )%( 005-HTAM | | | | | | | | | --------------- | --- | ------------ | --- | --- | ---------- | --- | ------------ | --- | --- | | | 0.6 | | | | | | )%( draH-PCS | | | | )7.0= | | | | | )%( CMA 60 | | | | | | | | 75 | | | | | 60 | | | 0.3 ( 40 30 50 | | 0 150 300 | 450 | 0 150 | 300 | 450 | 0 150 300 | 450 | 0 150 | 300 450 | | --- | -------------- | --- | -------------- | --- | --- | -------------- | --- | -------------- | ------- | | | Training Steps | | Training Steps | | | Training Steps | | Training Steps | | Base + Thinking SFT Base + Non-Thinking SFT CPT + Thinking SFT CPT + Non-Thinking SFT Instruct Figure34.Evaluationresultsofpass@16metricacrossmodelswithdifferentpre-RLinterventiononweaksupervision. 37 LLMReasoningwithWeakSupervision | | avg@16 | | pass@16 | | avg@16 | | pass@16 | | | --- | ------ | --- | ------- | --- | ------ | --- | ------- | --- | )%( hcneB ecneicS )%( 4202 EMIA 30 8 | | | 20 | | 10 | | | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | ecracS ataD 15 0 0 150 300 450 0 0 150 300 450 0 0 150 300 450 0 150 300 450 )%( hcneB ecneicS )%( 4202 EMIA ytirojaM | 8 | | 20 | | 10 | | 20 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | etoV | 0 | | 0 | | 0 | | 0 | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | 0 150 300 450 600 0 150 300 450 600 0 150 300 450 600 0 150 300 450 600 | draweR ysioN | | | | )%( hcneB ecneicS | | | | | | ------------ | --- | --- | --- | ----------------- | --- | --- | --- | --- | )%( 4202 EMIA | )7.0= | | | | | | 30 | | | | ----- | --- | --- | --- | --- | --- | --- | --- | --- | | 8 | | 25 | | 10 | | | | | ( 15 | 0 | | 0 | | 0 | | | | | | --- | -------------- | --- | -------------- | --- | -------------- | --- | -------------- | --- | | 0 | 150 300 | 450 | 0 150 300 | 450 | 0 150 300 | 450 | 0 150 300 | 450 | | | Training Steps | | Training Steps | | Training Steps | | Training Steps | | Base + Thinking SFT Base + Non-Thinking SFT CPT + Thinking SFT CPT + Non-Thinking SFT Instruct Figure35. EvaluationresultsonAIME2024andScienceBenchacrossmodelswithdifferentpre-RLinterventiononweak supervision. 38 LLMReasoningwithWeakSupervision DiversityJudgeExampleOutput Prompt: Aboxcontains5whiteballsand5blackballs. Idrawthemoutofthebox,oneatatime. Whatisthe probabilitythatallofmydrawsalternatecolors,startingandendingwiththesamecolor? ResponseA: To determine the probability that all of the ball draws alternate colors, starting and end- ingwiththesamecolor,weneedtoconsiderthefollowing: 1. **Totalnumberofwaystoarrangetheballs**: Thereare5whiteballsand5blackballs,sothetotalnumberof waystoarrangethese10ballsisgivenbythebinomialcoefficient (cid:0)10(cid:1) ,whichrepresentsthenumberofwaysto 5 choose5positionsoutof10forthewhiteballs(theremaining5positionswillbefortheblackballs). Thisvalue iscalculatedas: (cid:18) (cid:19) 10 10! = =252 5 5!5! 2. **Numberoffavorableoutcomes**: Forthedrawingstoalternatecolorsstartingandendingwiththesame color,thereareonlytwopossiblesequences: -Startingwithawhiteballandalternating(WBWBWBWBW B)-Startingwithablackballandalternating(BWBWBWBWBW) Sothereareonly2favorableoutcomes. 3. **Probability calculation**: Theprobability istheratioof thenumberof favorableoutcomestothe total numberofoutcomes. Therefore,theprobabilityP is: Numberoffavorableoutcomes 2 1 P = = = Totalnumberofoutcomes 252 126 1 Sothefinalansweris . 126 ResponseB: Tosolvetheproblem,weneedtodeterminetheprobabilitythatasequenceof10balls(5white and5black)isarrangedsuchthattheyalternatecolors. Thereareonlytwopossiblesequencesthatmeetthis condition: ”WBWBWBWBWB”and”BWBWBWBWBW”. Thetotalnumberofwaystoarrange5whiteand5 blackballsisgivenbythebinomialcoefficient (cid:0)10(cid:1) ,whichrepresentsthenumberofdistinctpermutationsof5 5 whiteand5blackballs. Thisiscalculatedas: (cid:18) (cid:19) 10 10! = =252 5 5!5! Sincethereareonly2favorableoutcomes(thetwoalternatingsequencesmentionedabove),theprobabilityisthe ratioofthenumberoffavorableoutcomestothetotalnumberofoutcomes: 2 1 Probability= = 252 126 1 So,theprobabilitythatallofmydrawsalternatecolors,startingandendingwiththesamecolor,is . 126 JudgeOutput: Both Generation 0 and Generation 1 use the same strategy to solve the problem. They both identifythattherearetwopossiblesequencesthatsatisfytheconditionofalternatingcolorsandstartingand endingwiththesamecolor: ”WBWBWBWBWB”and”BWBWBWBWBW”. Theybothcalculatethetotal numberofwaystoarrangethe10ballsusingthebinomialcoefficient (cid:0)10(cid:1) ,whichaccountsforthepermutations 5 of5whiteand5blackballs. Finally, theybothcomputetheprobabilitybytakingtheratioofthenumberof favorableoutcomes(2)tothetotalnumberofoutcomes(252),resultinginthesamefinalanswerof 1 . 126 Therearenosignificantdifferencesintheirreasoningsteps, assumptions, orintermediatecalculations. Both responsesfollowasimilarlogicalflowandusethesamemathematicalconceptstoarriveatthesolution. JudgeResult: ∥no∥ Figure36.QualitativeExampleofDiversityAnalysis 39 LLMReasoningwithWeakSupervision FaithfulnessJudgeExampleOutput Prompt: Findallrealpairs(a,b)thatsolvethesystemofequation a2+b2 =25, 3(a+b)−ab=15. Response: Tosolvethesystemofequations: (cid:40) a2+b2 =25, 3(a+b)−ab=15, wecanusethefollowingreasoningandalgebraicmanipulations. First,recalltheidentity: a2+b2 =(a+b)2−2ab. Let’sdenotes=a+bandp=ab. So,thefirstequationbecomes: s2−2p=25.Thesecondequationremains: 3s−p=15.Wecansolvethesetwoequationssimultaneously. First,weexpresspfromthesecondequation: p=3s−15.Next,wesubstitutepintothefirstequation: s2−2(3s−15)=25.Simplifytheequation: s2−6s+30=25 =⇒ s2−6s+5=0. √ Thisisaquadraticequationins. Wecansolveforsusingthequadraticformulas= −b± b2−4ac,wherea=1, 2a b=−6,andc=5: √ √ 6± 36−20 6± 16 6±4 s= = = . 2 2 2 So,thesolutionsforsare: s=5 or s=1.Now,wefindthecorrespondingpvaluesforeachs: 1. Ifs=5, thenp=3(5)−15=0. 2. Ifs=1,thenp=3(1)−15=−12. Wenowneedtofindthepairs(a,b)thatsatisfy theequationsa+b=sandab=p. Thesearetherootsofthequadraticequations: 1. Fors=5andp=0,the quadraticequationis: t2−5t=0 =⇒ t(t−5)=0. So,thesolutionsaret=0andt=5. Thus,thepairs(a,b)are(0,5)and(5,0). 2. Fors=1andp=−12,the quadraticequationis: t2−t−12=0. Wesolvethisquadraticequationusingthequadraticformula: √ 1± 1+48 1±7 t= = . 2 2 So,thesolutionsaret=4andt=−3. Thus,thepairs(a,b)are(4,−3)and(−3,4). Therefore,thecomplete setofsolutionsis: (0,5),(5,0),(4,−3),(−3,4) . JudgeOutput: Finalanswergiven: (0,5),(5,0),(4,−3),(−3,4).Reasoningpath: Introducessumss=a+b andproductp=ab,rewritesequationsintoasysteminsandp,solvestogets=5or1,findscorrespondingp, thensolvesquadratict2−st+p=0toobtainallorderedpairsconsistentwitheach(s,p),yieldingthelisted solutions. Thederivationiscompleteandconsistentlyleadstothestatedpairs. JudgeResult: ∥1∥ Figure37.QualitativeExampleofFaithfulnessAnalysis 40