cs.LGSep 29, 2026

Behavioral Convergence Without Representational Convergence: Persistent Training-History Dependence in Neural Networks

Authors: Ertuğrul Mutlu

Organizations: Aachen, Germany

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Abstract

Neural networks trained toward the same final objective can reach similar predictive performance while retaining internal representations shaped by earlier training history. We study this effect using controlled sequential-training experiments in which paired convolutional networks start from identical weights, experience reversed task orders, and then receive the same deterministic common-relaxation distribution. Across 20 paired MNIST runs, 16 satisfy a predeclared behavioral-matching criterion, yet their matched representations retain a mean history score of 0.139 (95% bootstrap CI: 0.127-0.153) and approximately 3.1% prediction disagreement. Extending common relaxation to 50,000 optimizer updates does not erase the measured difference: across five paired seeds, the representation-history score remains 0.190 (95% bootstrap CI: 0.161-0.219) at the end of the measured horizon while the mean accuracy gap is only 0.18 percentage points. Fresh linear probes show that, with sufficient labeled data, the two histories retain practically equivalent linearly accessible class information. A same-label rotated-MNIST control reproduces the effect: all five paired seeds reach behavioral matching while retaining a mean representation-history score of 0.162. Finally, a matched-learning-rate ReLU-LeakyReLU control reduces the 50,000-update representation residue by 0.040 on average in all five paired seeds, providing directional evidence that activation-mediated plasticity contributes to the persistence of training-history effects. These results provide protocol-scoped evidence that behavioral convergence need not imply representational convergence and that optimization history can leave measurable internal traces after prolonged common training.

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Figure 1: Common-relaxation design. Paired networks start from the same initialization, receive reversed histories A ⁣→ ⁣BA\!\to\!B and B ⁣→ ⁣AB\!\to\!A , then receive the same deterministic final training stream CC .
Figure 2: Main 20-paired-seed common-relaxation experiment. Behavioral differences contract rapidly, while the declared Conv2/FC1 representation-history score remains clearly above zero at the end of the 10,000-update horizon. Thin curves show individual seeds; the heavy curve and band show the aggregate mean and bootstrap interval.
Figure 3: Long-horizon stress test across five paired seeds. The residue remains substantial through 50,000 common updates. Four of five seeds increase from 10k to 50k, but the sample is too small and variable to support a general growth claim.
Figure 4: Fresh FC1 linear-probe differences across labeled-data budgets. Points show the mean SABC-minus-SBAC full-test accuracy difference; error bars are 95% tt intervals. Dashed horizontal lines mark the ±0.5\pm 0.5 percentage-point practical-equivalence margin used at 500 examples per class.
Examples/classMean diff.95% tt CIPaired ppdzd_{z}
25+1.005[−0.031,2.041][-0.031,2.041]0.0560.454
50+0.580[0.072,1.088][0.072,1.088]0.0270.535
100+0.288[−0.094,0.669][-0.094,0.669]0.1310.353
500+0.090[−0.064,0.244][-0.064,0.244]0.2370.273
Table 1: Primary FC1 linear-probe endpoint (epoch 20) across training-data budgets. Differences are SABC minus SBAC in percentage points.
Figure 5: Matched learning-rate activation control. ΔHrepr\Delta H_{\mathrm{repr}} is LeakyReLU minus ReLU, so negative values indicate less residual history dependence under LeakyReLU. All five seeds are negative at 50,000 updates.
Figure 6: Same-label rotated-MNIST control across five paired seeds. Behavioral gaps contract while a non-zero representation residue remains.
ConditionnnBehavioral criterionHreprH_{\mathrm{repr}}95% bootstrap CI
Main matched endpoints16/20≤0.2\leq 0.2 pp gap, ≥97%\geq 97\% acc.0.139[0.127, 0.153]
Main at 10k20mean gap 0.252 pp0.153[0.144, 0.164]
Long horizon at 50k5mean gap 0.180 pp0.190[0.161, 0.219]
Rotated matched endpoints5/5same criterion0.162[0.141, 0.182]
Leaky minus ReLU at 50k5matched LR=0.04−0.040-0.040[-0.077, -0.018]
Table 2: Summary of the main evidence chain. Bootstrap intervals are mean 95% intervals.
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