Intrinsic-Extrinsic Coupling in Learning Dynamics

Qinyou Wang

cs.LGSep 24, 2026

Intrinsic-Extrinsic Coupling in Learning Dynamics

Authors: Qinyou Wang

Abstract

A learner's current observations need not determine its response to further training. We formulate intrinsic-extrinsic coupling through the continuation-conditioned value of a constrained learning-state intervention, with observation-relative fibers describing present agreement. An executable finite-frame classifier-head write protects current logits while repairing specified historical margins under finite-precision acceptance checks. We distinguish local admissibility, continuation-conditioned intervention value, and complete-policy performance. A matched four-cell contrast identifies readout-specific non-additivity between the same intrinsic intervention and alternative external continuations. In a CLINC-derived class-incremental setting, replay changes the write's 32-update contribution from five correct predictions to zero. Nonzero interactions also occur under output distillation, with a RoBERTa backbone, and under optimizer-native SGDW dynamics. Under SGDW, correct-count interactions are negative in all three activated roots at 128 updates, showing that coupling need not imply positive synergy. The mathematical analysis distinguishes feasible local repairs and favorable terminal outputs from training-reachable repair regions. Separate coordination tests show that content controls match or exceed the development gain, while a five-root fresh-test comparison with Fiber present in every arm shows root-dependent rather than uniformly beneficial correct-count effects. On the secondary cross-entropy readout, guided allocation yields lower mean loss than standard replay in all five pairs. Together, these results make intrinsic-extrinsic coupling operational by connecting executable state geometry to continuation-conditioned value, matched interaction identification, and closed-loop coordination, while separating identified coupling from complete-policy performance.

Figures & tables

Comparison	Parent/root design	Identified question and readout scope
Replay interaction	One development root; matched full- or 75%-replay parents	Continuation-conditioned effects; validation with endpoint-dependent panels
Distillation interaction	Development root; first three legal commits on the distillation prefix	Distillation-off/on four cells; $H=32,128$
Backbone extension	Three RoBERTa roots; first legal commit by 671	Replay interaction under a second backbone; $H=32,128$
Native-optimizer comparison	Three new roots, paired AdamW/SGDW; two task phases	Phase activation and conditional replay interaction; $H=32,128$
Update-scale sensitivity	Three distinct roots; fixed anchors and independent rate calibration	Interaction under sampled update-scale matching; associated diagnostics reuse these roots
Allocation attribution	Development root; three control trajectories and archived comparators	Complete-policy attribution on the reused 1,000-item validation endpoint

Table 1: Scientific questions and experimental units. Every nonactivation remains in its prescribed denominator. Multiple anchors and horizons do not create additional independent roots. All comparisons reuse the class partition; only the five-root policy study adds the previously unused test split.

Evaluation task	Fiber without future replay	Replay without anchor Fiber
	$Y_{10}-Y_{00}$	$Y_{01}-Y_{00}$
Task 0 (old)	0	$+47$
Task 1 (old)	$+5$	$+38$
Task 2 (current)	0	$-2$
Total	$+5$	$+83$

Table 2: Conditional effects by task at the 32-update endpoint. Both contrasts use

Y_{00}

as reference. Each task has 200 validation items. These are two edges of the same four-cell experiment, not independent experiments; all five Fiber gains are in label 11.

Future replay	Identity	Execute	Fiber increment	Would either commit?
Off	$-0.7705$	$0.4308$	$1.2013$	Yes
On	$7.1279$	$7.6531$	$0.5252$	No

Table 3: Label-11 endpoint margin after 32 updates. The diagnostic is performed on an endpoint clone; it is not an additional training action.

Anchor	$H$	Eval. step	Items/classes	$V_{F}^{R}$	$V_{F}^{0}$	$I_{F,R}$
511	128	639	800/40	0	$-5$	$+5$
511	288	799	1,000/50	$-3$	$-1$	$-2$
351	128	479	600/30	$-1$	$+5$	$-6$
351	288	639	800/40	$+1$	$+1$	0
351	448	799	1,000/50	0	0	0

Table 4: Continuation-dependent effects at longer horizons. Each row compares the same parent and proposal under two future replay conditions.

H

counts subsequent ordinary updates. Evaluation populations and active classes grow across task ends.

Continuation/backbone	Root	Anchor	$n_{32}/n_{128}$	$I_{J}(32)$	$I_{J}(128)$
Distillation / BERT	Development	191	400/400	$+3$	$+1$
Distillation / BERT	Development	223	400/600	$+2$	$-4$
Distillation / BERT	Development	255	400/600	$+5$	$-7$
Replay / RoBERTa	1	511	800/800	$+2$	$+5$
Replay / RoBERTa	2	511	800/800	$+1$	$+3$
Replay / RoBERTa	3	575	800/1,000	$-3$	$+2$

Table 5: Interaction beyond replay and the original backbone. Every row is one activated parent, not an independent replication at each horizon.

n_{32}/n_{128}

records the evaluation sizes. The distillation comparison uses one development root; the backbone comparison uses three RoBERTa roots, numbered locally. Complete branch counts and CE contrasts appear in Appendix C .

Root	Optimizer	$I_{J}(32)$	$I_{J}(128)$	$I_{-\mathrm{CE}}(32)$	$I_{-\mathrm{CE}}(128)$
1	AdamW	$-2$	$-6$	$-0.024153$	$-0.048234$
1	SGDW	0	$-1$	$-0.105356$	$-0.118092$
2	AdamW	$-6$	$-8$	$-0.017977$	$-0.043018$
2	SGDW	$-2$	$-2$	$-0.024880$	$-0.029122$
3	AdamW	0	0	$-0.009414$	$-0.009821$
3	SGDW	0	$-3$	$-0.054441$	$-0.067756$

Table 6: Optimizer-native replay interaction at the activated Task-3 parents. All rows use anchor 511 and the same 800-item, 40-class panel at both horizons. Task 2 has six phase nonactivations, excluded from this conditional-interaction table but not from the study denominator. Root numbers are local to this study; the two optimizer conditions are paired within each root.

I_{-\mathrm{CE}}

uses larger-is-better negative CE.

Root	$J_{S}$	$J_{G}$	$J_{M}$	$G-S$	$G-M$	$M-S$
1	1,343	1,344	1,341	$+1$	$+3$	$-2$
2	1,351	1,348	1,350	$-3$	$-2$	$-1$
3	1,323	1,327	1,324	$+4$	$+3$	$+1$
4	1,363	1,360	1,363	$-3$	$-3$	0
5	1,346	1,344	1,347	$-2$	$-3$	$+1$
Mean difference				$-0.6$	$-0.4$	$-0.2$

Table 7: Fresh-test primary outcomes of the five-root policy study. All counts are out of the same 1,500 items.

S

is standard full replay plus period-32 Fiber,

G

is guided reinforcement, and

M

is mask-matched random content. Each row is one paired training root. Both primary contrast vectors are mixed or inconclusive under the frozen directional criterion.

Appendix figures & tables16 assets

Supplementary material from the paper’s appendix.

Appendix

Component	Setting
Model	BERT-base-cased with a growing classifier head
Task stream	Five tasks, ten selected classes each; five epochs per task
Current batch	32, except the eight-item epoch tail
Optimizer	AdamW [ 30 ] ; encoder LR $10^{-5}$ , head LR $10^{-3}$ ; weight decay $5\times 10^{-4}$ ; betas $(0.9,0.999)$ , epsilon $10^{-8}$
Gradient clipping	Global norm 1
Memory/replay	150 reservoir slots; at most 16 replay examples per update

Appendix

Table 8: Base settings for the development and five-root policy comparisons; backbone and optimizer exceptions are specified in Section 4 and Appendix C . The Fiber actuator and its numerical acceptance rule are unchanged across replay and distillation.

Anchor action	Future replay	$J$	Old-task $J$	$m_{11}$	Would commit
Execute	On	578	385	7.6531	No
Identity	On	578	385	7.1279	No
Execute	Off	500	305	0.4308	Yes
Identity	Off	495	300	$-0.7705$	Yes

Appendix

Table 9: Branch readouts underlying Figure 3 and Table 2 , at a 32-update horizon. All branches share the same pre-action parent, proposal specification, and 600-item panel. The last column records a diagnostic on a copy of the endpoint.

Root	Anchor	$H$	$n$	$Y_{00}$	$Y_{01}$	$Y_{10}$	$Y_{11}$	$I_{J}$	$I_{-\mathrm{CE}}$
Dev.	191	32	400	315	346	315	349	$+3$	$+0.015386$
Dev.	191	128	400	305	305	305	306	$+1$	$+0.006555$
Dev.	223	32	400	318	335	322	341	$+2$	$-0.041508$
Dev.	223	128	600	372	458	376	458	$-4$	$-0.014583$
Dev.	255	32	400	299	320	299	325	$+5$	$+0.000954$
Dev.	255	128	600	336	375	344	376	$-7$	$+0.003637$

Appendix

Table 10: Full distillation four-cell readouts. One development root, three anchors, two horizons; panels expand where shown.

Root	Anchor	$H$	$n$	$Y_{00}$	$Y_{01}$	$Y_{10}$	$Y_{11}$	$I_{J}$	$I_{-\mathrm{CE}}$
1	511	32	800	689	752	688	753	$+2$	$+0.013963$
1	511	128	800	660	752	655	752	$+5$	$+0.026789$
2	511	32	800	689	761	688	761	$+1$	$+0.043752$
2	511	128	800	667	759	665	760	$+3$	$+0.046859$
3	575	32	800	752	769	753	767	$-3$	$+0.033377$
3	575	128	1,000	820	954	816	952	$+2$	$+0.027539$

Appendix

Table 11: Full RoBERTa readouts. The three root numbers are local to the RoBERTa comparison. A positive interaction need not mean a positive Fiber effect with replay.

Root	Optimizer	Anchor	$H$	$n$	$Y_{00}$	$Y_{01}$	$Y_{10}$	$Y_{11}$	$I_{J}$	$I_{-\mathrm{CE}}$
1	AdamW	511	32	800	693	747	697	749	$-2$	$-0.024153$
1	AdamW	511	128	800	663	753	667	751	$-6$	$-0.048234$
1	SGDW	511	32	800	704	735	710	741	$0$	$-0.105356$
1	SGDW	511	128	800	673	739	675	740	$-1$	$-0.118092$
2	AdamW	511	32	800	694	755	702	757	$-6$	$-0.017977$
2	AdamW	511	128	800	651	754	659	754	$-8$	$-0.043018$

Appendix

Table 12: Full optimizer-native readouts for the six activated Task-3 parents. Both horizons use the same 800-item panel. Root numbers refer only to this study; optimizer conditions are paired within a root.

Root	Anchor	$H$	$n$	$Y_{00}$	$Y_{01}$	$Y_{10}$	$Y_{11}$	$I_{J}$	$I_{-\mathrm{CE}}$
1	351	32	600	436	574	435	573	$0$	$+0.008095$
1	351	128	600	413	564	414	564	$-1$	$+0.003807$
1	511	32	800	599	755	602	754	$-4$	$+0.023977$
1	511	128	800	564	759	563	761	$+3$	$+0.021861$
2	351	32	600	437	571	437	571	$0$	$+0.000479$
2	351	128	600	412	578	412	579	$+1$	$+0.000080$

Appendix

Table 13: Complete update-scale sensitivity outcomes at the four activated parents. Root 3 has two nonactivations and no corresponding outcome rows. The sign change for Root 1 at anchor 511 uses a common panel across horizons.

Replay retained (%)	Without Fiber $J$	With Fiber $J$	Difference
100	950	950	0
75	948	941	$-7$
50	942	938	$-4$
25	934	933	$-1$
12.5	929	926	$-3$
0	424	507	$+83$

Appendix

Table 14: Terminal outcomes under different retained replay fractions. Each row has its own training history, so comparisons across rows do not isolate the effect of continuation at a fixed state. These are descriptive results, not a jointly randomized dose-response estimate.

Policy	159	319	479	639	799
Full replay + Fiber	196	393	573	751	950
Replay withdrawal	196	393	495	754	851
Replay deprioritization	196	393	569	748	948
Fiber-aligned reinforcement	196	393	574	750	953
Mask-matched random	196	393	573	751	953
Permuted signal	196	393	574	752	955

Appendix

Table 15: Correct counts at task ends. The evaluation sizes are respectively 200, 400, 600, 800, and 1,000. A terminal improvement need not imply non-degradation at all earlier endpoints.

Policy	Terminal $J$	Replay events	Replay examples	Retained writes
Full replay + Fiber	950	799	12,784	3
Replay withdrawal	851	511	8,176	10
Replay deprioritization	948	799	12,784	10
Fiber-aligned reinforcement	953	799	12,784	3
Mask-matched random	953	799	12,784	3
Permuted signal	955	799	12,784	3

Appendix

Table 16: External workload and retained state writes for Figure 4 . All rows except withdrawal use 799 replay events and 12,784 replay examples. The five reallocation policies make 96 or 112 replacements as described in Section 6 . The shadow policy accepts three proposals but retains none; the last column counts retained writes, not accepted proposals or total sensing cost.

Root	$J_{S}$	$J_{G}$	$J_{M}$	$\mathrm{CE}_{S}$	$\mathrm{CE}_{G}$	$\mathrm{CE}_{M}$
1	1343	1344	1341	0.4558487	0.4449773	0.4508730
2	1351	1348	1350	0.4481850	0.4477481	0.4501625
3	1323	1327	1324	0.4612834	0.4606896	0.4618068
4	1363	1360	1363	0.3936347	0.3930567	0.3919950
5	1346	1344	1347	0.4147207	0.4141283	0.4158862

Appendix

Table 17: All terminal fresh-test counts and mean CE. The 1,500-item panel is common across the 15 trajectories. CE is secondary; lower is better.

Root	$J_{S}$	$J_{G}$	$J_{M}$	$\mathrm{CE}_{S}$	$\mathrm{CE}_{G}$	$\mathrm{CE}_{M}$
1	915	915	917	0.3796341	0.3671043	0.3744089
2	919	918	922	0.3118337	0.3138216	0.3133275
3	914	916	913	0.3712733	0.3682319	0.3737126
4	928	927	930	0.2735706	0.2711383	0.2722694
5	918	917	921	0.3362293	0.3340249	0.3362681

Appendix

Table 18: Terminal reused-validation readouts on 1,000 items. These do not override the fresh-test primary verdict.

Root	Arm	Commit steps	Writes	Redirected rows	Duplicate unchanged
1	S	511, 543, 671	3	0	0
1	G	511, 671	2	59	2
1	M	511, 543, 671	3	86	5
2	S	511	1	0	0
2	G	511	1	32	0
2	M	511	1	32	0

Appendix

Table 19: Endogenous actions and actual reallocation workload. Each arm has 20 consultations. “Duplicate unchanged” counts active steps on which a duplicate-UID native candidate is left intact under the specified duplicate-candidate rule; the ordinary update is still executed. Redirect counts need not match across divergent whole trajectories.

Readout	LwF-only	LwF + Fiber	Difference
Task 0 correct	40	62	$+22$
Task 1 correct	16	41	$+25$
Task 2 correct	65	58	$-7$
Task 3 correct	96	75	$-21$
Task 4 correct	197	197	0
Total correct	414	433	$+19$

Appendix

Table 20: A complete-policy distillation outcome with task-level trade-offs. Each task has 200 terminal evaluation items. Teacher-forward counts are equal, but complete cost and historical-information use are not. This is a two-policy comparison, not a four-cell distillation-interaction identification.

External continuation	Accepted repair	Saturated identity	Rejected identity
Full replay	3	17	0
Fiber-aligned reinforcement	3	17	0
LwF-style distillation	15	0	5

Appendix

Table 21: Observed decisions under the same 20-point consultation schedule. Consultations occur at steps 191, 223,

\ldots

, 799. The replay trajectories share decision times, not identical hidden states. Every LwF rejection is due to the ideal-write cap.

Output segment	Minimum $J$	Maximum $J$	Breakpoints
Random–permuted	953	955	16
Random–shadow	952	954	31
Permuted–shadow	952	955	23

Appendix

Table 22: Exact saved-logit segment certificates on 1,000 items. The endpoints are three measured attribution controls. The counts apply to output mixtures, not parameter or replay-policy mixtures. Breakpoints include the segment endpoints.

Native-event diagnostic: coupled-decay SGD
Root	Anchor	$H$	$n$	$Y_{00}$	$Y_{01}$	$Y_{10}$	$Y_{11}$	$I_{J}$	$I_{-\mathrm{CE}}$
1	191	32	400	380	387	380	387	$0$	$-0.000007$
1	191	128	400	366	393	366	393	$0$	$-0.000023$
2	191	32	400	381	389	380	389	$+1$	$+0.000593$
2	191	128	400	357	392	357	392	$0$	$-0.001189$
3	191	32	400	382	390	382	390	$0$	$-0.000472$

Appendix

Table 23: Optimizer diagnostic readouts. The native-event, coupled-decay design has three activated roots. The fixed-anchor, linked-rate design has two activated cells and four nonactivations. Both reuse the sensitivity-study roots; they are not additional independent roots for the native-optimizer comparison.

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Modern language models are likely to be updated throughout their lifetime rather than trained once and frozen. Each update therefore participates in a recurring cycle: decide which experience to learn from, understand what that update changes, and remain capable of learning from what comes next. We show that these challenges are governed by the same evolving update--behavior interaction. We derive a token- and layer-wise decomposition of how learning from one token changes another prediction. By separating the softmax force, shared readout geometry, and residual connections, it exposes two interaction channels and yields a forward-computable approximation. Following this interaction through time reveals a unified picture of continual adaptation. Positive interaction identifies useful experience; negative interaction produces either concentrated collision or accumulated erosion; over longer horizons, updates reshape the shared geometry mediating future learning signals, reducing their transmission. These predictions lead to effective data selection, mechanism-specific controls for interference, and a readout-based diagnostic of future learnability whose degradation predicts the benefit of restoring the readout. Across models and training regimes, the same local interaction thus explains both what an update changes now and how learning today changes what can be learned tomorrow. This view connects data attribution, forgetting, and plasticity loss as distinct regimes of the same evolving learning dynamics.

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