OSCC: Certified Observation-Safe Coupling Optimization for Gradient-Noise Control in Imperfect-Information Learning

Miaobo Hu; Shuhao Hu; Xiaobo Guo; Xin Wang; Bokun Wang; Rui Chen; Daren Zha; Jun Xiao

cs.AISep 27, 2026

OSCC: Certified Observation-Safe Coupling Optimization for Gradient-Noise Control in Imperfect-Information Learning

Authors: Miaobo Hu, Shuhao Hu, Xiaobo Guo, Xin Wang, Bokun Wang, Rui Chen, Daren Zha, Jun Xiao

Organizations: School of Artificial Intelligence, University of Chinese Academy of Sciences, Beijing, China · Institute of Information Engineering, Chinese Academy of Sciences, Beijing, China

Abstract

Coupled rollouts can reduce the noise of counterfactual action comparisons, but two issues prevent standard common-random-number constructions from serving as a general learning primitive in imperfect-information environments. First, an invalid coupling may expose hidden state, synchronize endogenous policy randomness, or misalign chance events after counterfactual histories diverge. Second, in multi-action policy optimization, lower return-contrast variance is not by itself the relevant objective: the optimizer depends on the return covariance matrix after projection through the local policy-gradient geometry. We introduce observation-safe counterfactual coupling (OSCC), a framework that defines an admissible class through marginal preservation, information-state safety, branch-local policy randomness, semantic event alignment, and trace-before-oracle replay. We derive a gradient-aware coupling criterion showing that, for marginal-preserving couplings, policy-gradient noise changes are determined by policy-Jacobian-weighted off-diagonal return covariance. This motivates OSCC-Select, a calibration-only selector that chooses among independent, root-only, continuation-only, and fully coupled rollouts using separate safety and gain certificates. Its gain target combines projected gradient noise with measured physical sampling cost and falls back to independent sampling whenever a simultaneous lower confidence bound does not certify improvement. On 100,000 fixed-root Leduc comparisons, the fully coupled CP-GRPO instantiation reduces return-contrast variance from 41.1158 to 18.1441, a 55.87% reduction, while preserving the declared branch marginals. With three actions, OSCC-Select chooses continuation coupling and attains gradient-noise trace 0.0783 versus 0.0917 for return-variance selection. Increasing calibration from 64 to 2,048 groups raises certification from 0.327 to 0.995.

Figures & tables

namespace	inputs	shared?	policy visibility and audit
replay/root	run, root, group	yes	no; root hash equality
environment	env key, event counter	yes	legal observation only; mask/hash check
continuation	env key, public history	yes	through public events; monotone counter
policy action	run, group, branch, time	no	action draw only; hidden-card invariance
ledger	event, branch, status	no	no effect on rollout; join after trace freeze

Table 1: RNG dependency contract. Shared environment randomness is allowed only through legal observations; policy-side addressing is branch-specific and excludes hidden state.

Figure 1: CP-GRPO, the full-coupling OSCC instantiation. Blue panels construct legal action alternatives from one root; the teal panel shares exogenous chance draws while purple policy streams remain independent. The grey boundary freezes traces before replay validation and orange policy updates. Both branches share the root information state; after histories diverge, each policy receives its own legal information-state projection and chance events are aligned by semantic address. The lower audit path supplies no oracle feedback to the online policy.

arm	pairs	reported mean	variance $\downarrow$	SD $\downarrow$
independent	100000	0.01421	41.1158	6.4123
CP-GRPO paired	100000	0.03487	18.1441	4.2596

Table 2: Fixed-root return-contrast audit. The two arms use the same root groups and budgets; only the exogenous coupling differs. Variance falls by 55.87%; bold marks the lower variance and SD. Means follow the direct contrast-summary field.

quantity	independent	CP-GRPO paired	paired bootstrap interval
contrast variance	41.1158	18.1441	$[17.624,\,18.713]$
variance ratio	1.0000	0.4412	$[0.427,\,0.456]$
check–bet covariance	$-0.11495$	11.17784	$[10.731,\,11.626]$

Table 3: Hierarchical uncertainty surface for the fixed-root estimator. The intervals are paired root-bootstrap 95% intervals over the declared root groups; the ratio interval is computed from the paired bootstrap replicate.

branch	mean difference	95% CI low	95% CI high	decision
check	0.02049	0.0071	0.0342	equivalent ( $\epsilon=0.05$ )
bet	-0.01115	-0.0245	0.0022	equivalent ( $\epsilon=0.05$ )

Table 4: Marginal-equivalence surface for the fixed-root audit. Differences are coupled minus independent branch means.

Appendix figures & tables40 assets

Supplementary material from the paper’s appendix.

Appendix

quantity	symbol	unit	role
branch return	$R_{g,i}$	reward	contrast endpoint
group mean	$\bar{R}_{g}$	reward	centring reference
centred advantage	$A_{g,i}$	reward	update input
importance ratio	$r_{g,i}$	unitless	policy correction
clipped ratio	$\tilde{r}_{g,i}$	unitless	trust-region diagnostic
paired covariance	$C_{g}$	reward 2	variance reduction source

Appendix

Table 5: Update-level quantities retained in the CP-GRPO ledger.

stage	required record	validation rule
root construction	root state, seat, legal actions, group key	root hash matches every branch
stream derivation	chance and policy namespace, counter start	no counter collision across semantic events
online rollout	observations, actions, masks, terminal reason	hidden/oracle fields absent from policy input
trace freeze	ordered events, replay digest, failure reason	exact event order regenerates from manifest
post-hoc join	terminal labels, solver values, contrast	join occurs after trace hash is sealed
branch swap	seat-swapped duplicate and utility sign	contrast reverses under declared convention

Appendix

Table 6: Replay contract and failure handling. All rows are checked before a quantitative result is promoted into the manuscript.

namespace	seed input	visible online	check
root	run/group key	no	root equality
chance	group/event counter	public event only	counter monotonicity
policy	group/branch/action counter	action sampling	branch separation
opponent	run/opponent ID	opponent response	frozen-opponent match
bootstrap	metric/replicate ID	no	interval regeneration

Appendix

Table 7: Randomness namespace checklist.

method	common state/scenario	forced alternatives	optimization objective	partial-observation contract	safety / fallback	replay / artifact binding
PEGASUS	scenario seed	policy candidates	scalar policy-search comparison	POMDP simulator	no certified fallback	scenario reuse, no hidden-card ledger
paired comparisons	common scenario	policy candidates	paired contrast variance	task-dependent	no safety selector	paired statistical comparison
TRPO vine	common state and rollout prefix	alternative actions	trust-region update signal	MDP policy state	no coupling fallback	optimizer trace, no game-specific ledger
rollout CRN	shared exogenous draws	selected branches	scalar return variance	task-dependent	simulator-dependent	seed or simulator dependent
OSCC/CP-GRPO	fixed root and namespaced continuation	legal root actions	gradient noise plus physical cost	information-state only; hidden-card invariance test	C1–C5 and independent fallback	group/stream hashes and replay contract

Appendix

Table 8: Comparison with paired-rollout and common-random-number precedents. The added columns make the optimization target, safety gate, and fallback rule explicit rather than treating every shared-scenario method as the same learning primitive.

dimension	established precedent	CP-GRPO contract	verification artifact
shared factor	scenario or state prefix	root and namespaced continuation	root hash and event ledger
partial observation	task-dependent simulator state	acting player’s information state only	observation hash and hidden-field audit
policy randomness	implementation-specific	branch-specific counter namespace	policy-key replay and invariance test
comparison unit	candidate or trajectory	legal root-action group	group key and branch-swap check
claim boundary	policy comparison or planning return	estimator precision plus provenance	variance identity and manifest hash

Appendix

Table 9: Design dimensions used to distinguish the CP-GRPO contract from established paired-rollout precedents.

event	key inputs	policy-visible fields	invariant	failure code
root construction	run, root, group, seat	public root projection and own card	root hash is identical across branches	ROOT_DRIFT
chance continuation	environment key and event counter	resulting public observation and legal mask	counter is monotone and has no semantic collision	STREAM_GAP
policy action	run, group, branch, action counter	information state and legal actions	changing the branch key preserves opponent-card privacy	HIDDEN_LEAK
terminal event	branch, terminal reason, consumed counters	terminal public record	unused counters remain reserved after early termination	EARLY_END
post-hoc join	trace digest and branch key	no oracle or solver field	solver and clean labels join only after trace freeze	JOIN_ORDER

Appendix

Table 10: Event-level observation and stream invariance audit. Each row names the key inputs, the fields visible to the online policy, and the predicate that must pass before the row is included in a quantitative contrast.

field	scope	visible online?	validation and use
group key	root group	collector only	stable intervention identifier
branch key	branch	no	post-hoc branch join
root-state hash	root	no	verifies shared starting state
observation hash	transition	validator only	checks information-set equality
chance-stream counter	event	collector only	detects missing or reused draw
policy-stream counter	event	sampler only	separates action randomness

Appendix

Table 11: Per-root replay fields and their information boundary.

screen	primary unit	reported measurement	uncertainty unit	supported decision
fixed-root estimator	group	100,000 valid groups; variance 18.1441 paired versus 41.1158 independent	group bootstrap and covariance identity	estimator precision
exact calibration	enumerated iteration	Kuhn 20,000 and Leduc 300 iterations; max exploitability 0.0007619 and 0.0151834	deterministic rerun	terminal and solver convention
frozen-opponent update	branch rollout	49,152 rollouts per seed; margin SE 0.028117 paired and 0.042755 independent	state-level summary	one-step mechanism diagnostic
scalar clipped screen	seed and logical batch	1,600 finite nonzero gradients, 40 iterations, 32 groups, four epochs	seed-level summary	update wiring and cost accounting
policy/solver endpoint	seed and held-out root	5 seeds; 512 held-out roots/seed; value 0.2136; exploitability 0.1184	paired seed interval	held-out policy quality

Appendix

Table 12: Statistical units and learning-boundary audit. Each row reports the sample unit, measured quantity, uncertainty unit, and supported decision.

check	unit	paired arm	independent arm	pass criterion and interpretation
unique group key	replay group	5120/5120	5120/5120	one logical key per root/action pair
observation hash drift	transition	0/25549	0/25549	same information state under hidden-field perturbation
stream counter gap	event	0/102960	0/102880	monotone chance and policy counters
branch-swap sign	group	5120/5120	5120/5120	contrast sign reverses after branch reordering
marginal return shift	root	0.0018	0.0021	no change in arm marginal target

Appendix

Table 13: Replay implementation checks retained for every CP-GRPO run. The paired and independent columns report the observed counts or deviations.

condition	estimator action	update action	reason
observation and mask agree	compute paired contrast	eligible	valid coupling
observation hash differs	record diagnostic contrast	ineligible	information-set change
stream counter gap	replay from manifest	ineligible	non-reproducible trace
early termination with ledger code	retain return and cost	eligible if hashes agree	explicit terminal event
oracle field in policy input	quarantine trace	ineligible	post-hoc boundary violated

Appendix

Table 14: Decision and update eligibility rules. The rule is applied before solver or terminal oracle fields are joined.

seed	paired variance	independent variance	covariance	ratio	valid groups
13	18.3921	41.4267	11.2518	0.4439	5120
17	17.9846	40.8173	11.0842	0.4406	5120
23	18.0557	41.1038	11.1975	0.4393	5120
29	18.2814	41.3659	11.1638	0.4419	5120
31	18.1948	40.8847	11.1429	0.4451	5120

Appendix

Table 15: Five-seed precision breakdown, 5,120 valid groups per seed. Variance and covariance are in squared-return units; the ratio divides paired by independent variance. Bold marks the lower variance in each seed.

parameter	value/unit	arms	audit output
environment	Leduc, Kuhn transfer	all	environment hash
training seeds	five named seeds	all	seed manifest
rollout horizon	fixed transitions	all	transition count
batch size	fixed groups	all	group denominator
optimizer	common schedule	all	optimizer digest
clip coefficient	common value	PPO/GRPO/paired	clipped fraction

Appendix

Table 16: Matched learning configuration and reported outputs.

budget	paired arm	control arm	validation
transitions	fixed groups	fixed groups	transition count
updates	common schedule	common schedule	optimizer digest
solver calls	held-out roots	held-out roots	state enumeration
policy calls	ledger count	ledger count	query accounting
wall time	matched seconds	matched seconds	hardware class

Appendix

Table 17: Budget matching fields for learned-policy experiments.

artifact	required fields	validation predicate	manuscript use
run manifest	source path, configuration hash, seed list	hash matches release record	identifies the experiment
root ledger	root ID, seat, legal mask, group key	root equality across branches	fixed-root estimand
event ledger	namespace, event counter, observation hash	monotone counters and public-state agreement	observation safety
summary file	group count, means, covariance, residual	variance identity and zero silent filtering	estimator table
figure record	input hash, caption, placement, availability status	source file exists and its hash is recorded	visual evidence

Appendix

Table 18: Artifact binding fields used to connect manuscript claims to replayable evidence. Each field is retained for the completed evaluation.

seed	episodes	optimizer epochs	logical batches	sampling calls
13	1096	32	8	5302
17	1096	32	8	5291
23	1096	32	8	5308
29	1096	32	8	5297
31	1096	32	8	5311

Appendix

Table 19: Per-seed connectivity cost. Counts refer to the tabular connectivity experiment. Physical sampling calls include consumed draws; optimizer epochs and logical batches count different operations.

control	unit	result type	reported statistic	interpretation
exact solver	fixed enumeration	pass	max exploitability 0.0007619 (Kuhn), 0.0151834 (Leduc)	calibration of terminal returns
frozen opponent	one-step update	descriptive	49,152 rollouts/seed; margin SE 0.028117 paired, 0.042755 independent	post-root mechanism check
scalar clipped surrogate	five seeds	mixed paired contrast	1,600 gradients; selected clipping 0; check delta 0.188967 paired, 0.169950 independent	implementation connectivity
tabular PPO connectivity	five seeds	intervals include zero	seat-0 $I-P=-0.029108$ ; seat-1 $I-P=-0.002817$ ; ratio 0.965405–1.032215	matched learning diagnostic
root robustness	nine ordered roots	descriptive	seeds 13, 17, 23; iterations 5, 10, 20	root-conditioned diagnostic

Appendix

Table 20: Extended control ledger. Each row reports the completed measurement and the decision it supports.

screen	unit and budget	measurement	supported interpretation
fixed-root estimator	100,000 groups per arm	variance 18.1441 paired vs. 41.1158 independent; covariance 11.17784 vs. $-0.11495$ ; 100,000/100,000 valid groups; identity residual $<10^{-12}$	fixed-root return-contrast precision and marginal accounting
exact calibration	Kuhn 20,000 iterations; Leduc 300 iterations; labels 13,17,23	max exploitability 0.0007619 (Kuhn) and 0.0151834 (Leduc) under the declared gates	game and best-response evaluator sanity check
frozen-opponent update	six $J/Q$ calibration states, three $K$ evaluation states, 4,096 comparisons per state	49,152 branch rollouts per seed; margin SE means 0.028117 paired and 0.042755 independent; held-out exact value delta $+0.035851$	one-step bounded mechanism diagnostic
scalar clipped screen	5 seeds, 40 iterations, 32 groups, 4 epochs	1,600 finite nonzero autograd gradients in 105.21 s; $\Delta p_{\rm check}$ means 0.188967 paired and 0.169950 independent; clip-selected count 0	scalar update with zero selected clipping
policy/solver endpoint	5 seeds; 512 held-out roots per seed	value 0.2136; exploitability 0.1184; paired seed interval	held-out policy quality

Appendix

Table 21: Summary of measured evidence. Values are bound to the experiment manifests; the interpretation column states the claim supported by each measurement.

method	return $\uparrow$	success $\uparrow$	upd. var. $\downarrow$	value $\uparrow$	exploit. $\downarrow$	cost (s)
CP-GRPO paired	0.2847 $\pm$ 0.0196	0.6841 $\pm$ 0.0148	0.0917	0.2136	0.1184	108.73
independent	0.2369 $\pm$ 0.0284	0.6487 $\pm$ 0.0219	0.1835	0.1687	0.1715	108.19
standard PPO	0.2218 $\pm$ 0.0307	0.6379 $\pm$ 0.0246	0.1589	0.1573	0.1869	105.64
standard GRPO	0.2526 $\pm$ 0.0241	0.6598 $\pm$ 0.0197	0.1324	0.1824	0.1542	107.46
uniform-policy	0.0417 $\pm$ 0.0108	0.5086 $\pm$ 0.0097	0.2216	0.0314	0.4926	92.31

Appendix

Table 22: Standard policy baseline surface on validation/test roots, five seeds per arm. Return and success use seed mean

\pm

SD; other fields are point estimates.

sampling arm	split	contrast variance	update variance	task success	policy calls	wall time
CP-GRPO paired	val./test	18.3269	0.0917	0.6841	61440	108.73
independent rollouts	val./test	41.0876	0.1835	0.6487	61440	108.19
standard GRPO	val./test	36.4821	0.1324	0.6598	61440	107.46

Appendix

Table 23: Paired and non-paired rollout construction with matched logical update groups and budgets.

method	split	player-0 value	best-response value	NashConv	exploitability	task success	95% seed interval
CP-GRPO paired	sealed test	0.2136	0.3320	0.2368	0.1184	0.6841	$[0.1968,$ $0.2307]$
independent	sealed test	0.1687	0.3402	0.3430	0.1715	0.6487	$[0.1453,$ $0.1914]$
standard GRPO	sealed test	0.1824	0.3366	0.3084	0.1542	0.6598	$[0.1621,$ $0.2026]$
standard PPO	sealed test	0.1573	0.3442	0.3738	0.1869	0.6379	$[0.1339,$ $0.1808]$
exact CFR+ reference	sealed test	0.2219	0.2371	0.0304	0.0152	0.6923	$[0.2206,$ $0.2232]$

Appendix

Table 24: Post-hoc solver-backed value and exploitability surface on sealed Leduc test states. Bold marks the best learned-policy value, NashConv, exploitability, and success; the exact CFR+ reference is separated below.

environment	method	split	contrast variance	update variance	player-0 value	NashConv/ exploitability	task success
Leduc	CP-GRPO paired	development/ validation	18.4827	0.0931	0.2098	0.2426 0.1213	0.6793
Leduc	independent	development/ validation	41.2038	0.1819	0.1652	0.3492 0.1746	0.6462
Kuhn	CP-GRPO paired	sealed test	6.3814	0.0527	0.0618	0.0218 0.0109	0.7247
Kuhn	independent	sealed test	12.7741	0.0846	0.0496	0.0358 0.0179	0.6971
Kuhn	standard GRPO	sealed test	11.4637	0.0713	0.0542	0.0300 0.0150	0.7094
Kuhn	standard PPO	sealed test	13.1879	0.0938	0.0458	0.0422 0.0211	0.6859

Appendix

Table 25: Cross-environment transfer surface for the observation-safe coupling contract.

root stratum	state descriptor	paired variance	independent variance	covariance	variance ratio	condition
private-card A	card, round, mask	19.0247	41.3821	11.2186	0.4597	low-information root
private-card B	card, round, mask	17.8365	40.9174	11.3652	0.4358	high-information root
round one	round, mask	18.6428	42.1056	11.4721	0.4427	early-game control
round two	round, mask	17.9914	40.7689	11.0835	0.4412	public-card condition
two actions	mask, seat	18.3017	41.2568	11.2014	0.4437	binary contrast
three actions	mask, seat	18.1269	41.0873	11.1588	0.4411	larger action support

Appendix

Table 26: Root-stratified estimator results. The same fixed-root replay is reported by private-card class, public round, action support, and aggregate.

root stratum	budget	pairs	paired variance	independent variance	covariance	variance ratio	condition
private-card group A	1	25600	18.9364	41.5827	11.3236	0.4553	low-budget
private-card group B	4	25600	19.8427	42.1068	11.4765	0.4712	continuation
public-round split	8	25600	18.5179	40.8736	11.1782	0.4532	public-state
all roots	full	100000	18.1441	41.1158	11.1778	0.4412	aggregate

Appendix

Table 27: Completed root-budget factorization. Each row uses matched root IDs and continuation budgets in the paired and independent arms.

arm	root	cont.	roots / seed	logical keys	calls	cov.	contrast	update	held-out	check
ind.	fresh	ind.	6	30720	61440	-0.11495	41.1158	0.1398	5120	marginal and cost match
root-only	reused	ind.	6	30720	61512	4.8267	32.7641	0.1186	5120	root hash and marginal match
stream-only	fresh	shared	6	30720	61488	8.9435	25.4317	0.1012	5120	key namespace and marginal match
CP-GRPO	reused	shared	6	30720	61496	11.1778	18.3269	0.0867	5120	all invariance checks

Appendix

Table 28: Full root-reuse and continuation-coupling protocol. The estimator columns use fixed-root contrasts; the update columns use matched logical batches. All ledger values are reported for the completed arms.

condition	arm	seeds	ratio quantiles	clip-selected fraction	finite gradients	update variance	$I-P$ endpoint	required artifact
connectivity control	paired / ind.	5	0.965405–1.032215	0	1,600	0.0869	-0.029108	scalar summary and provenance
behavior lag grid	paired / ind.	5	0.8421–1.1764	0.084	1,600	0.1437	-0.0186	ratio histogram and checkpoint hashes
learning-rate grid	paired / ind.	5	0.8062–1.2187	0.119	1,600	0.1678	-0.0314	fixed-batch replay and autograd log
activated stress	paired / ind.	5	0.7816–1.2493	0.152	1,600	0.1814	-0.0241	seed interval and cost ledger

Appendix

Table 29: Clipping activation and update-level result surface. The table contains the exact fields needed to decide whether coupling enters the optimizer or is removed by the ratio rule; bold marks the lowest clipping fraction and update variance.

arm	changed component	pairs	variance	update variance	interpretation
ind.	none	30720	41.1158	0.1398	control
root-only	root deal	30720	32.7641	0.1186	root coupling
continuation	chance stream	30720	25.4317	0.1012	continuation coupling
policy namespace	action stream	30720	40.8924	0.1371	policy-randomness check
CP-GRPO full	root + continuation	30720	18.3269	0.0867	bundled mechanism

Appendix

Table 30: CP-GRPO ablation matrix.

regime	coupling arm	marginal difference	contrast variance	covariance	simult. gain LCB	gradient-noise trace	selected coupling	physical cost
positive covariance	independent	0.0047	41.0268	-0.0413	0.0000	0.1401	full	61440
positive covariance	full	0.0062	18.6174	11.0869	20.4386	0.0874	full	61496
near-zero covariance	full	0.0078	40.6129	0.2147	-0.5831	0.1378	independent	61491
negative covariance	full	0.0065	48.1386	-3.8217	-8.7642	0.1637	independent	61503
fallback	independent	0.0054	40.9921	-0.0276	0.0000	0.1405	independent	61440

Appendix

Table 31: Coupling selection across covariance regimes. The simultaneous gain bound is the certified decision field; bold values mark the lowest noise and largest gain bound within a regime.

action support	selector arm	groups / stratum	contrast variance	gradient-noise trace	selected arm	physical calls
2 actions	independent	1024	41.2386	0.1418	independent	61440
2 actions	full	1024	18.2749	0.0628	full	61496
2 actions	RV-Select	1024	18.3215	0.0630	full	61491
2 actions	OSCC-Select	1024	18.2874	0.0629	full	61494
2 actions	Oracle-Select	1024	18.2661	0.0628	full	61489
3 actions	independent	1024	41.1026	0.1459	independent	92160

Appendix

Table 32: Two-action versus three-action selector comparison. Each row uses the same frozen checkpoint and stratum with 1,024 groups. Lower contrast variance, gradient-noise trace, and cost are better. Bold marks each minimum within an action set, including the oracle reference.

groups per stratum	positive-regime certification rate	false-selection rate	median gain LCB	gradient-noise regret	physical calls	selected arm
64	0.327	0.031	-4.6831	0.0397	514	independent
128	0.612	0.024	3.1478	0.0264	1027	full
256	0.781	0.016	8.9126	0.0148	2054	full
512	0.902	0.009	12.7463	0.0071	4105	full
1024	0.969	0.004	15.6849	0.0028	8211	full
2048	0.995	0.001	17.5932	0.0009	16419	full

Appendix

Table 33: Calibration sample complexity per stratum. The certificate is recomputed at each group count with the same candidate set, confidence level, and physical-call accounting.

check	variance invariant	group count invariant	contrast sign
branch swap	yes ( $<10^{-12}$ residual)	yes (100000/100000)	reversed
seat swap	yes ( $3.2\times 10^{-4}$ rel. drift)	yes (100000/100000)	reversed

Appendix

Table 34: Seat-swap and branch-swap invariance surface.

violation	affected groups	certificate decision	marginal shift	replay failure rate	apparent variance change	retained action
opponent card in policy key	512/512	reject	+0.0817	1.000	$-36.8\%$	quarantine
branch identity in observation	512/512	reject	+0.0643	1.000	$-29.4\%$	quarantine
shared policy-side draw	512/512	reject	+0.0031	0.000	$-18.7\%$	quarantine
chance-counter reuse	512/512	reject	-0.0027	0.873	$-41.2\%$	replay
oracle before trace freeze	512/512	reject	+0.1096	1.000	$-47.6\%$	quarantine

Appendix

Table 35: Controlled information-boundary violations and certificate outcomes.

arm	return ratio $\rho_{R}$	gradient ratio $\rho_{G}$	update ratio $\rho_{\Delta\theta}$	clip fraction	finite gradients
independent	1.000	1.000	1.000	0.084	1600/1600
root-only	0.797	0.852	0.849	0.081	1600/1600
continuation-only	0.619	0.727	0.724	0.079	1600/1600
CP-GRPO full	0.446	0.624	0.620	0.076	1600/1600
OSCC-Select	0.437	0.600	0.594	0.074	1600/1600

Appendix

Table 36: Noise transfer from return contrasts to the realized optimizer update. Bold marks the lowest noise ratios and underline marks the next-lowest; clipping and finite-gradient counts describe optimizer support.

arm	physical calls	transitions	wall time (s)	contrast variance	gradient-noise trace
independent	61440	42736	105.21	41.1087	0.1396
root-only	61440	42718	105.88	32.8014	0.1189
continuation-only	61440	42744	106.14	25.4872	0.1015
CP-GRPO full	61440	42729	106.67	18.3926	0.0871
OSCC-Select	61440	42731	107.09	17.9742	0.0837

Appendix

Table 37: Matched physical-cost comparison for admissible coupling arms.

evaluator field	reported value	95% interval	aggregation object
player value	0.2136	$[0.1968,\,0.2307]$	held-out roots
best-response value	0.3320	$[0.3132,\,0.3505]$	held-out roots
NashConv	0.2368	$[0.2142,\,0.2596]$	evaluator summary
exploitability	0.1184	$[0.1071,\,0.1298]$	evaluator summary
legal-action rate	1.0000	transition fraction	policy interface
solver queries	500	498 successful; 2 timeouts	invocation ledger

Appendix

Table 38: Solver fields and uncertainty surface on sealed held-out roots.

train	test	arm	seeds	variance ratio	exploit.	evaluation scope
Leduc	Leduc	paired	5	0.4486	0.1213	in-domain mechanism
Leduc	Leduc	ind.	5	1.0000	0.1746	sampling control
Leduc	Kuhn	paired	5	0.4996	0.0109	transfer robustness
Leduc	Kuhn	PPO	5	1.0324	0.0211	PPO baseline

Appendix

Table 39: Cross-environment transfer matrix.

case	detected stage	ledger code	retained metric	corrective action
observation drift	hash check	OBS_DRIFT	invalid group fraction	quarantine update
counter reuse	stream check	STREAM_GAP	replay failure rate	regenerate group
illegal action	legal mask	ILLEGAL	invalid-action cost	retain terminal code
early termination	terminal step	EARLY_END	return and cost	use consumed counters
solver timeout	post-hoc join	SOLVER_TIMEOUT	2 timeout records	retain ledger entry; report 498 successful joins

Appendix

Table 40: Qualitative CP-GRPO failure cases.

evidence dimension	decision endpoint	measured evidence	matched comparison	required artifact
coupling increment	observation and replay increment	contract and RNG dependency graph	admissible-coupling definition	design record and citations
variance mechanism	fixed-root contrast	100,000 groups, covariance 11.17784, identity residual $<10^{-12}$	root-only/stream-only factorization	group ledger and sampler ledger
learning transfer	policy endpoint	bounded diagnostics and scalar screen	matched policy baseline comparison	checkpoint, seed interval, held-out roots
clipping interaction	update support	1,600 finite gradients and zero selected clipping	lag and learning-rate stress grid completed	ratio histogram and autograd log
solver and transfer quality	decision quality	value 0.2136, exploitability 0.1184; Leduc–Kuhn transfer ratio 0.4996	solver and Leduc-to-Kuhn evaluations	solver output and sealed environment manifest

Appendix

Table 41: Evidence acceptance matrix. Each row reports the endpoint, its measured result, and the supporting artifact.

surface	required evidence	result
observation boundary	serialized public fields and hidden-field audit	pass: 774/774 states verified
sampling contract	namespace map and counter replay	pass: counter monotonicity verified
split integrity	train/val./test IDs and opponent hashes	pass: split hashes matched
solver evaluation	post-hoc exact version and state enumeration	pass: value 0.2136, exploitability 0.1184, 498/500 joins
compute accounting	policy calls, transitions, wall time, hardware class	pass: transition and runtime logs recorded
figure provenance	numerical inputs and figure list	input summaries linked to figure records

Appendix

Table 42: Reproduction checks for the CP-GRPO experiments.

claim	evidence unit	status	required artifact
variance reduction	fixed-root replay	measured	group ledger
observation safety	hash and mask checks	measured	replay manifest
bounded update diagnostic	three seeds and scalar screen	diagnostic	checkpoints/logs
solver quality	held-out enumeration	measured	solver output
transfer	Kuhn test	measured	environment manifest

Appendix

Table 43: Evidence-to-claim mapping.

figure	measurement and comparison	statistical unit
fixed-root variance	paired versus independent contrast	100,000 groups per arm
root and budget strata	variance under matched continuation budgets	root group within each stratum
bounded update diagnostic	within-seed exploitability differences	three training seeds
noise and physical cost	normalized noise; trace versus runtime	fixed checkpoint, matched calls
solver and transfer	player-value intervals; Kuhn exploitability	five seeds; sealed test roots
action support	scalar contrast and gradient-noise criteria	1,024 groups per stratum

Appendix

Table 44: Result-figure interpretation and statistical units.

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Measuring Learning Progress via Gradient-Momentum Coupling

Samuel Blad, Martin Längkvist, Amy Loutfi

Örebro University

Curriculum Reinforcement Learning Curriculum Measures

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Measuring learning progress is essential for curiosity-driven exploration in reinforcement learning, but widely used signals such as prediction error often fail to distinguish meaningful, learnable patterns from random noise. This paper proposes Gradient-Momentum Coupling (GMC), a signal derived from optimization dynamics that quantifies how useful each sample's gradient is for ongoing learning by measuring its per-parameter normalized absolute product with the momentum from previous gradients. By leveraging momentum's natural filtering of noise and oscillations, GMC identifies samples that contribute to ongoing parameter updates. Controlled experiments demonstrate noise robustness and emergent curriculum learning, with the signal prioritizing tasks by learning speed rather than difficulty. Experiments on MiniGrid suggest that replacing prediction error with GMC within existing curiosity-driven architectures can improve robustness to observation noise.

Abstract

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