q103/random_k2/random_1.md

|     |     | 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 | <salman@cs.ucla.edu>, |     | Jingyan | Shen |     |     |     |     |
| ---------- | ------ | --------------------- | --- | ------- | ---- | --- | --- | --- | --- |
downstreamevaluationbenchmarks,bothin-domainheld-outsets
<js15262@nyu.edu>.
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,<t)
|     |     |     |     |     |     |     | whereρ i,t | :=  |     | denotestheprobabilityratio |     |     |     |
| --- | --- | --- | --- | --- | --- | --- | ---------- | --- | --- | -------------------------- | --- | --- | --- |
πθold (oi,t|q,oi,<t)
|     |     |     |     |     |     |     | between | the current | and pre-update |     | sampling | policy | and |
| --- | --- | --- | --- | --- | --- | --- | ------- | ----------- | -------------- | --- | -------- | ------ | --- |
2

LLMReasoningwithWeakSupervision
1.00
0.75
0.50
0.25
0.00
0 150 300 450
draweR
gniniarT
60
45
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)%(
005-HTAM
40
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20
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0 150 300 450
)%(
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20
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0 150 300 450
)%(
draH-PCS
1.00
0.75
0.50
0.25
0.00
0 150 300 450
draweR
gniniarT
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)%(
draH-PCS 24
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)%(
dnomaiD
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45
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0 150 300 450
)%(
005-HTAM
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0.00
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draweR
gniniarT
45
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Training Steps
)%(
kcoL
mutnauQ
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dnalsI
tsegraL
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htaM
ecneicS
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005-HTAM
Qwen2.5-Math-1.5B (7B) Qwen2.5-1.5B Llama3.2-3B-Instruct (8B) N=8 N=min(2048, Nmax)
Figure1.Comparisonoftrainingdynamicsandtestperformance(avg@16metric)acrossmodelfamiliesanddomains.Foreach
domain,weplottrainingreward(column1),in-domainbenchmarkperformance(column2-3)andOODbenchmarkperformance(column
4)overRLstepsfortwodatasetsizes:8(solidlines)andN (dashedlines),whereN isthelargestavailabletrainingsetinthe
max max
domainforthemodel.ForMATHandSCIENCE,N
max
=2048.ForGraph,N
max
=882forQwenmodelandN
max
=256forLlama
model.ColoredverticaldashedlinesmarkthesaturationsteptN foreachrun.Theshadedregionindicatesonestandarddeviationover
sat
independentsampling.Qwenmodelsexhibitextendedpre-saturationphasesandgeneralizefrom8samples,whileLlamamodels
saturaterapidlywithlimitedgains.Correspondingresultsfor7Band8BmodelsonMATHandSCIENCEareprovidedinAppendixC.3.
Aˆ i := ri− st m d( e { a r n i ( } { G i r = i} 1 G i ) = ) 1 ) is the advantage of i-th response 3.RLVRUnderWeakSupervision
calculatedbynormalizingthegroup-levelrewards.Rewards
TounderstandwhenRLVRgeneralizesunderweaksuper-
r ∈{0,1}arebinaryandassignedbyground-truthanswer
i vision,westudythreesettings: scarcedata(§3.1),noisy
verification. TheKLregularizationD (π ||π )isapplied
KL θ ref rewards(§3.2),andself-supervisedproxyrewards(§3.3).
toafixedreferencepolicyπ ,weightedbyascalarcoeffi-
ref Wethenanalyzepolicybehaviortoexplainwhysomemod-
cientβ. Allexperimentsusetheverlframework(Sheng
els succeed and others fail under these conditions (§3.4).
etal.,2024)(hyperparameterdetailsinAppendixB.3).
We additionally analyze GRPO baseline selection in Ap-
Evaluation. We evaluate reasoning performance using pendixE.
avg@16 accuracy (average pass@1 over 16 independent
Throughout this section, we compare Qwen and Llama
samplesperproblem)withtemperature1.0samplingandre-
modelfamilies.Wetreatthiscomparisonasaproxyforvari-
portpass@kfork ∈{4,8,16}intheAppendix.ForMATH,
ationinpretrainingpriorsratherthananintrinsicpropertyof
weuseMATH-500,AMC,AIME2024,AIME2025,Min-
eitherfamily: Qwen2.5-Mathispretrainedonanadditional
ervaMath,andOlympiadBenchevals. For SCIENCE,we
1Tmath-specifictokens,whileLlama-3.2-Instructisaligned
useGPQA-Diamond,aheld-outSCP-Hardset(Liuetal.,
forgeneralinstruction-following. Thecontrastwereport
2025a) (a subset of SCP problems where both Qwen2.5-
isbetweenmodelswithstrongdomain-alignedpretraining
1.5BandLlama-3.2-3B-Instructachievesolve@16=1pre-
and those without, and §4 confirms this interpretation by
RL),ScienceBench,MMLU-Science,andSuperGPQA.For
showingthatcontinualpre-trainingonmathdatatransforms
GRAPH,weuseheld-outQuantumLockandLargestIsland
Llama’sRLbehaviortoresembleQwen’s.
tasksfromReasoningGym(Stojanovskietal.,2025),fil-
teredsimilarlytosolve@16=1. Foreachdomain,wedes-
3.1.ScarceData
ignatebenchmarksasin-domainorout-of-domain(OOD).
For example, for MATH training, MATH-500 and AMC To understand how data scarcity affects RLVR general-
arein-domain,whileSCP-HardandGPQA-Diamondare ization, we investigate training dynamics across dataset
OOD (full assignments in Appendix Table 2). We report sizesN ∈{8,32,64,512,2048}acrossdiversemodelfam-
representativeresultsinthemaintextandfullresultsinthe iliesanddomains. Unlikepriorworkonsample-efficient
Appendix. RLVR (Wang et al., 2025a; Sun et al., 2025), which se-
lectspecificdatapoints,weusestratifiedrandomsampling
3

LLMReasoningwithWeakSupervision
Table1.Comparisonofsaturationstepst(8),pre-saturationgain∆(8)andpost-saturationresidual∆∗(8) acrossmodelfamilies
sat sat post
andtrainingdomainswhentrainingon8examples.Weadditionallyreportthelarge-smallgapG(n1,8)andG(n1,8) .ForGraph,the
sat,in sat,ood
largestavailablesettingisn =882forQwenmodelandn =256forLlamamodel(markedwith†).Thegreencellsmark∆(8) >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- |     |     |     |     |     |     |     |     |     |     |     |
| ----------------- | --- | --------------------------------- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
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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
<question>
assistant
Figure8.PrompttemplateusedforRLtrainingandevaluationonMATHandGRAPH.Theplaceholder<question>isreplaced
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
<question>
assistant
Figure9.PrompttemplateusedforRLtrainingandevaluationonSCIENCE.Theplaceholder<question>isreplacedwiththe
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 <N:
total
• IteratethroughbinsB fori=1,...,15.
i
• IfB containsunsampledproblems,randomlyselectoneproblemwithoutreplacement,addittothetrainingset,
i
andincrementn .
total
• Terminateimmediatelyifn =N.
total
16

LLMReasoningwithWeakSupervision
Thisapproachensuresthatalldifficultylevelsarerepresentedasuniformlyaspossibleacrossalldatascales.
B.3.ImplementationDetailsofRLTraining
Allexperimentsareimplementedusingtheverlframework(Shengetal.,2024)withitsdefaulthyperparameters:
learning
rate 10−6, KL coefficient β = 0.001, clip ratio ϵ = 0.2 and no entropy regularization. We set group size G = 8 for
computationalefficiency. Forresponsesampling,wefixthesamplingtemperature1.0andamaximumresponselengthof
2048tokensunlessotherwisenoted. Inverl,wesetboththetrainingbatchsizeandmini-batchsizeto64prompts,yielding
exactlyonegradientupdatepertrainingstep. Eachexperimentisrunfor496totalgradientupdates. Asimplerule-based
rewardfunctionisused,assigningreward1tocorrectanswersand0otherwise,withoutincorporatinganyformat-related
signals. ForMATHandSCIENCE,answermatchingandrewardcomputationisimplementedwithMath-Verify3library;for
GRAPH,weusetheinternaltask-specificevaluationprotocolfromReasoningGym. PrompttemplatesaredetailedinFig.8
andFig.9.
Training Steps  (log scale)
|     | 10  | 100 | 1k 10k | 100k |     |
| --- | --- | --- | ------ | ---- | --- |
Training loss  (EMA, 50-step window)
1.20
1.10
1.00
ssoL
0.90
0.80
0.70
0.60
|     |     | 0.1 B | 1 B | 10 B 51 B |     |
| --- | --- | ----- | --- | --------- | --- |
Tokens Seen  (log scale)
Figure10.Traininglossduringcontinualpre-trainingofLlama3.2-3Bonapproximately52BtokensofNemotron-CC-Mathdata.
|       | Training Steps |         |       | Training Steps |         |
| ----- | -------------- | ------- | ----- | -------------- | ------- |
| 0 100 | 200 300        | 400 500 | 0 100 | 200 300        | 400 500 |
0.50
|      | Training loss  (EMA, 50-step window) |     |      | Training loss  (EMA, 50-step window) |     |
| ---- | ------------------------------------ | --- | ---- | ------------------------------------ | --- |
| 0.80 |                                      |     | 0.45 |                                      |     |
0.40
0.70
0.35
| ssoL |     |     | ssoL |     |     |
| ---- | --- | --- | ---- | --- | --- |
0.30
0.60
0.25
| 0.50 |     |     | 0.20 |     |     |
| ---- | --- | --- | ---- | --- | --- |
0.15
0.00 B 0.20 B 0.40 B 0.60 B 0.80 B 1.00 B 0.00 B 0.05 B 0.10 B 0.15 B 0.20 B 0.25 B
|     | Tokens Seen            |     |                            | Tokens Seen |     |
| --- | ---------------------- | --- | -------------------------- | ----------- | --- |
|     | (a) CPT + Thinking SFT |     | (b) CPT + Non-Thinking SFT |             |     |
Figure11.Traininglossfor(a)ThinkingSFTand(b)Non-ThinkingSFTon43.5Kmathprompts,initializedfromtheCPTcheckpoint.
B.4.ImplementationDetailsofEvaluation
Weevaluatereasoningperformanceusingavg@16accuracy(averagepass@1over16independentsamplesperproblem)
withtemperature1.0samplingandreportpass@kfork ∈{4,8,16}.
3https://github.com/huggingface/Math-Verify
17

LLMReasoningwithWeakSupervision
Table3.Math-domaintraining(1.5B/3B):in-domainbenchmarks.
Model Metric t(8) MATH-500 AMC-2023 MinervaMath OlympiadBench AIME-2024
sat
∆sat ∆∗
post
Gsat ∆sat ∆∗
post
Gsat ∆sat ∆∗
post
Gsat ∆sat ∆∗
post
Gsat ∆sat ∆∗
post
Gsat
Qwen2.5-Math-1.5B Avg@16 302 29.7 1.5 -2.0 18.7 0.6 -0.1 14.3 1.5 -1.0 13.7 1.0 -0.8 7.3 0.6 0.2
Pass@4 12.3 0.5 -1.2 15.3 -0.1 0.2 14.0 2.0 -1.0 10.4 1.0 0.5 14.8 0.4 2.0
Pass@8 6.2 0.2 -1.0 13.1 -0.1 -1.2 11.1 2.0 -1.4 8.0 1.2 1.0 16.5 0.4 2.4
Pass@16 2.6 0.4 -0.4 10.8 -0.1 -2.8 10.3 1.2 -3.3 5.8 1.8 1.6 16.7 0.5 3.3
Qwen2.5-1.5B Avg@16 170 42.6 0.9 -0.8 15.8 3.3 -1.0 12.6 0.9 -1.3 13.5 1.2 0.2 1.0 0.9 -0.2
Pass@4 25.4 0.6 0.5 16.7 5.2 0.1 16.1 1.1 -1.1 13.9 1.3 0.6 3.4 2.9 0.3
Pass@8 18.6 0.6 1.0 15.0 7.1 2.1 15.7 1.0 -0.3 12.9 1.4 1.2 4.2 4.4 2.2
Pass@16 13.2 0.7 1.2 8.4 11.0 3.6 14.0 1.6 1.5 11.4 1.7 1.6 3.3 6.4 6.7
Llama3.2-3B-Instruct Avg@16 55 10.8 -1.9 -1.4 8.8 -2.1 -0.4 7.4 -1.1 -1.7 6.5 -0.7 0.5 7.1 -0.6 -3.8
Pass@4 23.9 -3.2 -0.6 21.3 -2.7 3.2 7.7 -1.3 -1.2 14.3 -1.1 2.4 0.0 0.0 -4.2
Pass@8 21.7 -3.9 -0.7 20.0 -2.1 6.4 5.6 -0.4 0.1 13.8 -0.9 3.3 0.0 0.4 -3.7
Pass@16 18.8 -3.8 0.2 15.7 -1.1 10.1 1.9 2.2 2.3 11.6 -1.0 4.4 0.0 1.1 -3.3
Table4.Math-domaintraining(1.5B/3B):out-of-domainbenchmarks.
Model Metric t(8) GPQADiamond SCP-Hard MMLUSCI ScienceBench
sat
∆sat ∆∗
post
Gsat ∆sat ∆∗
post
Gsat ∆sat ∆∗
post
Gsat ∆sat ∆∗
post
Gsat
Qwen2.5-Math-1.5B Avg@16 302 12.6 1.1 0.9 10.5 2.1 2.4 25.4 1.2 -10.8 4.4 0.2 6.5
Pass@4 33.0 1.7 1.7 22.8 6.0 7.2 35.9 -0.0 -6.7 7.6 0.1 -1.6
Pass@8 41.0 2.5 2.2 25.9 8.6 10.2 31.9 -0.4 -5.4 5.6 0.2 -1.5
Pass@16 39.1 3.8 2.5 26.0 10.6 10.0 22.3 -0.4 -3.4 3.5 1.0 -0.7
Qwen2.5-1.5B Avg@16 170 13.8 1.4 -6.6 7.0 0.3 -0.4 29.4 -0.7 -11.2 6.8 0.6 -0.8
Pass@4 33.0 1.5 -15.5 19.0 2.0 -0.6 47.4 -0.7 -14.6 9.2 0.7 -0.6
Pass@8 37.9 1.0 -17.8 25.3 5.0 0.6 46.4 -0.4 -15.3 9.4 1.0 0.0
Pass@16 31.5 2.3 -16.2 28.0 9.8 2.0 37.4 -0.3 -16.1 9.0 1.2 0.4
LLama3.2-3B-Instruct Avg@16 55 -4.3 1.9 0.8 2.0 0.0 1.5 2.4 -0.4 -1.7 4.8 -0.3 0.1
Pass@4 -2.1 2.8 0.6 5.5 -0.4 3.3 0.1 0.6 0.3 6.8 -0.6 -0.5
Pass@8 0.5 3.0 0.7 8.3 -1.2 3.2 -0.3 0.6 0.2 7.7 -1.0 -0.8
Pass@16 -1.1 6.7 2.1 14.0 -4.3 0.0 -0.3 1.1 0.3 8.2 -1.0 -0.3
B.5.ImplementationDetailsofContinualPre-Training
We continually pre-train Llama3.2-3B on the Nemotron-CC-Math-4plus subset (Mahabadi et al., 2025), comprising
approximately52Btokensofmath-relevantdocumentsfilteredatqualityscore≥4. Trainingisconductedforoneepoch
withamaximumsequencelengthof2,048tokensandabatchsizeof128sequences. WeuseAdamWwithapeaklearning
rateof2×10−5,cosinedecayschedule,5%linearwarmup,weightdecayof0.01,andgradientclippingat1.0.
B.6.ImplementationDetailsofSFT
ForSFT,wetrainforthreeepochswithabatchsizeof16andamaximumsequencelengthof8192tokens. Wetunethe
learningrateforeachmodelwithinthe1×10−5,5×10−5]andreportresultsforthebest-performingsetting. Forthe
subsequentRLphase,weevaluateperformanceacrosstrainingsamplesizesN ∈ {8,2048}. Allotherhyperparameters
follow the configurations established in Section B.3, with the maximum response length extended to 8192 tokens to
accommodatelong-formreasoningtraces.
C.DataScaleEffect
C.1.AdditionalExperimentalResultsfromSmalltoLargeDataScale
Figs. 13, 14, and 15 present domain-specific training dynamics and generalization performance across sample sizes
N ∈ {8,32,64,512,2048}. Each figure tracks the training reward, two in-distribution benchmarks, and one OOD
benchmark,aslistedinTable2.
IntheMATHdomain,Llamamodelsexhibitrapidsaturationinsmall-sampleregimesandrelyheavilyondatascale. In
contrast,Qwenmodelsyieldcomparableperformanceacrossvaryingsamplesizes,characterizedbyextendedsaturation
periods. Specifically,themath-specializedQwen2.5-Math-1.5Bsustainsapre-saturationphasefor330gradientstepson8
18

LLMReasoningwithWeakSupervision
Table5.Science-domaintraining(1.5B/3B):in-domainbenchmarks.
t(8)
Model Metric ScienceBench SCP-Hard GPQA-Diamond MMLUSCI SuperGPQA
sat
∆sat ∆∗ Gsat ∆sat ∆∗ Gsat ∆sat ∆∗ Gsat ∆sat ∆∗ Gsat ∆sat ∆∗ Gsat
|     |     | post | post |     | post |     | post | post |
| --- | --- | ---- | ---- | --- | ---- | --- | ---- | ---- |
Qwen2.5-Math-1.5B Avg@16 268 11.1 0.2 0.0 14.5 1.1 0.8 16.9 1.6 1.3 19.1 1.0 2.0 6.6 0.3 1.4
Pass@4 7.5 -0.1 -0.5 35.0 -1.3 -1.3 38.0 2.3 1.4 34.7 1.1 -0.4 18.1 0.7 2.8
Pass@8 6.0 -0.2 -0.7 43.5 -2.7 -1.6 44.5 1.7 0.5 31.4 0.9 0.3 24.7 0.4 2.8
Pass@16 4.6 0.2 -0.7 44.0 -1.2 -2.0 41.1 0.2 0.5 22.3 0.5 -0.0 29.4 -0.1 1.3
Qwen2.5-1.5B Avg@16 161 6.8 0.5 -0.0 6.4 0.2 0.6 13.3 1.7 2.9 25.4 4.4 8.7 7.9 0.9 2.4
Pass@4 9.7 0.4 0.0 20.4 -1.5 -0.5 33.5 1.9 5.1 54.9 2.0 7.3 21.1 1.5 5.8
Pass@8 9.9 0.3 0.3 31.1 -1.7 -2.3 39.7 1.5 5.4 62.5 0.1 4.7 28.5 2.0 7.4
Pass@16 9.4 0.3 0.7 40.0 1.7 -2.0 32.0 3.0 5.6 60.5 0.2 2.6 32.8 3.1 8.8
Llama3.2-3B-Instruct Avg@16 61 2.6 0.5 0.7 1.8 1.7 5.9 11.9 3.0 4.3 10.6 0.8 2.3 5.2 2.2 3.7
Pass@4 3.5 0.4 0.6 5.1 2.9 11.1 24.0 4.8 3.8 8.7 1.4 0.7 8.5 3.9 5.6
Pass@8 4.1 0.3 0.3 8.2 1.9 11.9 25.7 4.2 1.5 6.7 1.8 -0.2 10.6 3.9 4.5
Pass@16 4.8 0.1 -0.3 16.0 -4.3 6.0 22.3 2.3 -1.1 2.6 2.6 -0.5 11.7 3.4 2.2
Table6.Science-domaintraining(1.5B/3B):out-of-domainbenchmarks.
t(8)
| Model | Metric | MATH-500 |     | AMC |     | OlympiadBench |     | MinervaMath |
| ----- | ------ | -------- | --- | --- | --- | ------------- | --- | ----------- |
sat
|     |     | ∆∗        |           | ∆∗   |           | ∆∗   |      | ∆∗             |
| --- | --- | --------- | --------- | ---- | --------- | ---- | ---- | -------------- |
|     |     | ∆sat post | Gsat ∆sat | post | Gsat ∆sat | post | Gsat | ∆sat post Gsat |
Qwen2.5-Math-1.5B Avg@16 268 25.3 0.8 1.1 14.7 1.5 0.1 11.4 0.9 0.5 12.7 1.1 0.6
|     | Pass@4  | 10.2 0.5 | 0.7 12.8 | 0.6  | 0.5 9.7  | 0.5  | -1.1 | 12.5 1.3 0.7 |
| --- | ------- | -------- | -------- | ---- | -------- | ---- | ---- | ------------ |
|     | Pass@8  | 4.5 0.8  | 0.5 10.8 | 0.1  | 0.1 8.1  | 0.0  | -1.6 | 9.4 1.8 0.7  |
|     | Pass@16 | 1.2 1.3  | 0.4 9.4  | -0.0 | -1.3 7.4 | -0.3 | -2.1 | 7.7 1.7 0.7  |
Qwen2.5-1.5B Avg@16 161 32.3 2.1 1.2 14.5 0.7 -0.2 10.7 1.5 1.6 10.6 1.9 0.7
|     | Pass@4 | 30.9 1.1 | 0.9 21.1 | 1.9 | 2.0 16.9 | 1.4 | 2.1 | 18.3 1.4 0.7 |
| --- | ------ | -------- | -------- | --- | -------- | --- | --- | ------------ |
|     | Pass@8 | 24.5 0.6 | 0.1 20.4 | 3.0 | 5.3 16.9 | 1.0 | 1.8 | 20.4 0.5 0.1 |
Pass@16 19.0 0.2 -0.6 15.7 4.1 11.2 14.1 1.1 1.4 21.3 0.3 -1.5
Llama3.2-3B-Instruct Avg@16 61 7.3 2.2 0.6 4.9 2.7 1.5 5.8 1.2 -0.1 5.9 1.2 1.2
|     | Pass@4  | 4.8 2.4 | 2.3 6.6 | 3.9 | 1.5 8.4 | 2.2 | 0.3 | 7.0 1.5 1.8 |
| --- | ------- | ------- | ------- | --- | ------- | --- | --- | ----------- |
|     | Pass@8  | 3.2 2.4 | 3.2 6.0 | 4.5 | 0.2 8.3 | 2.6 | 0.4 | 5.9 2.2 2.0 |
|     | Pass@16 | 1.0 3.0 | 3.6 4.9 | 5.1 | 0.0 7.7 | 2.7 | 0.4 | 4.3 2.4 1.5 |
samples,drivingcontinuousimprovementsonin-domainbenchmarks.
IntheSCIENCEdomain,thepre-saturationphaseyieldssimilargainsacrossallsamplesizes;however,afterthesaturation
point,largersamplesizesdemonstratedistinctbenefits. SimilartoMATHdomain,modelsexhibitsignificantlydifferent
saturationdynamicsonsmallsamples.
IntheGRAPHdomain,wecomparetwolargermodels,Qwen2.5-Math-7BandLlama3.1-8B-Instruct. TheQwenmodelalso
saturatesfasterherethaninotherdomains,implyingthatthelackofdomain-specificpre-trainingacceleratessaturationin
small-sampleregimes.
C.2.FullEvaluationResults
Inthissection,wewillreportthefullevaluationresultswithallbenchmarksandpass@k(k ∈{1,4,8,16}metrics. Fig.16,
Fig. 17, Fig. 18, Fig. 19, Fig. 24, and Fig. 25 include in-domain and out-of-domain evaluation results across multiple
benchmarksinMATH,SCIENCEandGRAPHdomains.
Discussions on pass@k. Despite prior work (Yue et al., 2025) discussing divergent behavior between pass@1 and
pass@k for k > 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 <i
i=1
whereo denotespreviouslygeneratedtokensandU istheuniformdistributionoverthevocabulary. Highervalues
<i
indicategreatermodelconfidence. Foreachprompt,wesample8responsesandusetheself-certaintyscoresdirectlyas
rewardstocomputeadvantages. AllotherRLhyperparametersfollowSectionB.3.
Results. Fig.27showsfullresultsofself-supervisedproxyrewardsacrossmodel-domainpairs. ExceptforQwen2.5-Math-
1.5B,allothermodelsexhibitfailuremodesunderprolongedtraining. ForQwen2.5-1.5BonSCIENCE,bothproxyrewards
collapse: majority voting shows a sharp reward spike followed by performance degradation, while self-certainty leads
tocompletetrainingcollapse. Similarly,Llama-3.2-3B-InstructonMATHshowsdegradedperformancewithbothproxy
rewardsdespiteincreasingtrainingrewards.OnlyQwen2.5-Math-1.5BonMATHmaintainsstableperformancewithmajority
voting,thoughself-certaintystillcollapsesafterapproximately200steps. Theseresultsdemonstratethatself-supervised
proxyrewardsarebrittleandmodel-dependent,withonlymath-specializedmodelsshowingpartialrobustness.
D.3.RewardHackingExampleUnderMajorityVote
Table8showstworolloutsfromQwen2.5-3Btrainedon SCIENCE withmajorityvoterewardsattrainingstep846. In
bothcases,themodelproducesplausibleintermediatereasoningbutconvergestothesamefinalanswer 0 ,regardlessof
theproblemcontent. Themajorityvoterewardis1.0becauseallrolloutsagreeonthisanswer—thepolicyhaslearned
toproduceidenticaloutputstomaximizeconsensus,constitutingrewardhacking. Thecorrectanswers(68.4gandτ /k,
0
respectively)appearinthereasoningtracesbutareoverriddeninthefinalanswer.
Table8.TworolloutsfromQwen2.5-3BonSCIENCEatstep846undermajorityvotereward.Bothproducecoherentreasoningtowardthe
correctanswerbutoutput 0 asthefinalanswer,achievingmajorityvoterewardof1.0.
Rollout1:Sucrosesolutionproblem Rollout2:Momentofinertiaproblem
Prompt:Preparea0.0348molefractionsolutionofsucroseusing Prompt: A wheel with moment of inertia I is acted upon by
100gofwater. torqueτ ,resistedbyτ =−kω.Findthemaximumspeed.
0 f
Reasoning(excerpt):“Massofsucroserequired=0.2moles× Reasoning(excerpt):“ω = τ0”
max k
342g/mole=68.4g”
Finalanswer: 0 Finalanswer: 0
Majorityvotereward:1.0 Majorityvotereward:1.0
E.BaselineEffect
WeanalyzehowthechoiceofrewardbaselineinfluencesgeneralizationinGRPO.StandardGRPOusesthewithin-group
meanreward(µ= 1 (cid:80)G r )asthebaseline. Byreplacingµwithaconstantbaselineb∈{0,1},weisolatethedirection
G i=1 i
of the policy update: b = 0 retains only positive reinforcement from correct samples (GRPO-POS), equivalent to the
REINFORCEalgorithm,whileb=1retainsonlynegativereinforcementfromincorrectsamples(GRPO-NEG),which
(Zhuetal.,2025)studiedinMATHdomain. Weremovethelengthpenaltyterm 1 inGRPOforthisexperiment. Based
|o|
onthepolicygradienttheory,wheresubtractinganaction-independentbaselinedoesnotchangetheexpectedgradientbut
reducesvariance,withalargebatchthesetwomethodsshouldyieldsimilarlearningbehavior(Williams,1992).
Figs.28and29presentthetrainingresultsontheSCIENCEdomainfor8and1024samples,respectively. Inbothregimes,
GRPO-POSandGRPO-NEGachievecomparablePass@1performancetostandardGRPO,exhibitingsimilarsaturation
andgeneralizationbehaviors. Wenotethatthiscontrastswithrecentfindingsby(Zhuetal.,2025),whichhighlightthe
superiorityofGRPO-NEG. However,theirimprovementswereprimarilyobservedinPass@kmetricsratherthanPass@1
andevaluatedonMATHdomain. Beyondthesemetricdifferences,it’sworthstudyingwhetherimplementationartifactsmay
alsoinfluenceobservations. Forinstance,clippingtermsintheGRPOformulationcanintroducebiases(Shaoetal.,2025;
Chenetal.,2025a). Whileourstrictlyon-policysetupmitigatessuchclippingeffects,weleaveacomprehensiveanalysisof
theseaffectstofuturework.
21

LLMReasoningwithWeakSupervision
F.DiversityandFaithfulness
Table9.Inter-rateragreementbetweenLLMjudgesmeasuredusingCohen’sKappa.
|     | JudgePair                                    |     |     |     | Cohen’sKappa |     |     |
| --- | -------------------------------------------- | --- | --- | --- | ------------ | --- | --- |
|     | OpenAIo3vs.GPT-OSS-20B(OpenAI,2025b)         |     |     |     | 0.752        |     |     |
|     | OpenAIo3vs.Gemini3Flash(GoogleDeepMind,2025) |     |     |     | 0.649        |     |     |
F.1.Quantificationofgenerationdiversity
Toquantifythegenerationdiversityofamodelonagivenprompt,wegenerateanumberofresponses,y ,...,y andcluster
|     |     |     |     |     |     | 1   | N   |
| --- | --- | --- | --- | --- | --- | --- | --- |
thembasedontheirreasoningsimilarity. Basingouranalysisonthemethodusedby(Lietal.,2025),todeterminereasoning
similaritybetweentwooutputsy ,y ,wedefineafunctions(y ,y ) ∈ {0,1}suchthats(y ,y ) = 1ify ,y aresimilar
|     |     | i j |     | i j |     | i j | i j |
| --- | --- | --- | --- | --- | --- | --- | --- |
and0otherwise. Toevaluates(·,·),wepromptGPT-4o(Hurstetal.,2024)asadiversityjudgetodeterminewhetherthe
reasoningproducedbyanytworesponsesfollowsadifferentreasoningpathusingthepromptspecifiedinFig.31.
Weformsemanticclustersbyiteratingthroughresponsesandcomparingthemtoarepresentativeresponsefromeachexisting
cluster,creatinganewclusteriftheresponseisdissimilartoeachrepresentative. Thisisperformedundertheassumption
oftransitivityofsimilarity. Wecreateclusters{C ,...C }whereC ={y ,...,y }suchthats(y ,y )=1 ∀y ,y ∈C .
|     |     |     | 1 K | i 1 | ni  | i j | i j i |
| --- | --- | --- | --- | --- | --- | --- | ----- |
WethendefinethediversityscoresusingtheShannonDiversityIndex(Shannon,1948)asfollows.
Foragivenprompt,letN bethetotalnumberofresponses,n bethenumberofresponsesinclusterC ,andK bethe
|     |     |     |     | i   |     | i   |     |
| --- | --- | --- | --- | --- | --- | --- | --- |
n
| numberofclusters. | Letp i = | i DefinetheShannonentropy |     |     |     |     |     |
| ----------------- | -------- | ------------------------- | --- | --- | --- | --- | --- |
N
K
(cid:88)
|     |     |     | H(p)=− | p logp |     |     |     |
| --- | --- | --- | ------ | ------ | --- | --- | --- |
i i
i=1
andtheeffectivenumberofclusters
(cid:0) (cid:1)
|     |     |     | N eff =exp | H(p) . |     |     |     |
| --- | --- | --- | ---------- | ------ | --- | --- | --- |
Wethendefinethediversityscore
N −1
|     |     |     | Div (x)= | eff . |     |     | (2) |
| --- | --- | --- | -------- | ----- | --- | --- | --- |
π
K−1
whenK >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:
<think>
Okay,soIneedtofindthelimit...
Substitutingx=0givestheindeterminateform 0,so...
0
Rationalizingthedenominator,weget...
</think>
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
<think>and</think>,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