Learning Dynamics of Continual Learning: A Unified View of Data Attribution, Forgetting, and Plasticity Loss
Yi Ren, Wenlong Deng, Guanzhe Hong, Clare Lyle, Yarin Gal
cs.LG, cs.AI
2026-09-27
Decomposes how a token's update shifts another prediction into two forward-computable channels, unifying data selection, forgetting, and plasticity loss as regimes of one dynamic.
LLMs do not stay frozen after deployment. Feedback, retrieved documents, and agent trajectories keep turning into parameter updates, and every update raises three questions: which experience to learn from, which existing behavior it will disturb, and whether the model can still learn after many more updates. The literature treats these as three separate fields, data attribution, catastrophic forgetting, and plasticity loss, each with its own metrics. The paper's claim is blunt: all three are regimes of one quantity, how learning from one token changes the prediction of another, written Δt(o,u). Earlier learning-dynamics work (Ren & Sutherland, ICLR 2025) studied this interaction at the example level, but example granularity is too coarse, and exact evaluation needs full-parameter Jacobians that cannot be recomputed repeatedly at LLM scale.
The base object is a first-order gradient inner product: after one gradient step on u, the log-probability of o shifts by roughly the inner product of their gradients. The paper opens this black box along three architectural seams.
The result (Proposition 2.1) is two channels. CH1 is direct force alignment scaled by final-hidden similarity: sparse and token-aligned. CH2 projects both forces through the shared readout (go^T ww^T gu) and multiplies by summed per-layer similarities; off-diagonal entries of ww^T let unrelated vocabulary directions interact, so CH2 keeps transmitting when direct overlap is weak.
A byproduct becomes the hinge of the paper. Under residual architecture, every layer receives gradients through the same output-side projection ∇hL log π = w^T g, so ‖w^T g‖²/‖g‖² measures how strongly a learning signal enters the backbone. The geometry that broadens local interaction is the same geometry that gates future learning. Everything is computable from logits, hidden states, and w in one forward pass, no backprop.
Three regimes follow over time. Selection: positive interaction predicts useful experience and ranks candidate data. Interference: negative interaction splits into two diseases. Collision is concentrated conflict from a few high-energy updates (Eu = ‖gu‖²; Proposition 4.1 shows Eu > 1+δ forces a negative component larger than δ on some non-target token). Erosion is coherent accumulation of many weak conflicts: GSM8K chain-of-thought supervision keeps reinforcing "To solve this problem..." continuations, each step small, gradually draining probability from MMLU's A/B/C/D answer tokens. Sustainability: readout transmission RD = E[‖w^T g‖²/‖g‖²], a Rayleigh quotient of ww^T along task directions, decays for future tasks as training reshapes the readout toward current ones; Appendix B.3 proves this under weak task overlap plus a norm constraint.
Cross-lingual selection (English queries against Chinese, French, Korean, Spanish candidates, where direct overlap weakens and CH2 matters):
| Setting | Method | Result | Baseline |
| Cross-lingual MMLU (Qwen2.5-1.5B) | CH1+2 | 0.625 | CH1 0.488; LESS 0.469 |
| Cross-lingual GSM8K | CH1+2 | 0.600 | CH1 0.445; LESS 0.140 |
| Multilingual memory top-1 (Llama3.2-3B) | +CH2 | 1.000 | 0.861; downstream accuracy 0.944 |
| GSM8K 5% budget FT (Qwen2.5-1.5B) | CH1+2 | 0.636 | random 0.610; LESS 0.587 |
In same-language settings with strong overlap, CH1 alone nearly saturates (sentence-transform AUC 1.000) and CH2 adds nothing, matching the geometric prediction.
Forgetting: two diseases, two drugs. Masking updates with Eu above 1.8 or 1.5 (EAFT is the same idea) treats collision at a downstream cost of at most 2.6 points; a threshold of 1.0 masks ordinary semantic tokens too and nearly erases the fine-tuning gain. Energy control improves general-capability retention, more so at larger learning rates, but leaves the erosion metric P(A...D) essentially unchanged. Format separation treats erosion: fine-tuning GSM8K in a response format different from deployment preserves instruction following, redirects interference toward the fine-tuning format, and GSM8K is still learned; swapping formats reverses the pattern. This also explains why on-policy RL forgets less than off-policy SFT: on-policy rollouts almost never produce extreme-energy updates.
Plasticity, over roughly 100M PubMed tokens: RD declines progressively on three held-out OOD tasks (GSM8K, MBPP, Dolly-QA) while in-distribution PubMed stays flat; later checkpoints adapt more slowly on OOD tasks but not in-distribution; restoring the readout to its base value substantially improves downstream validation loss, with resetting the last four layers adding little; and readout degeneration predicts reset gain with Spearman ρ = 0.83 (0.73 on OOD).
Three things practitioners can pick up directly: a forward-only data-selection score (gradient methods such as LESS require backprop; here one forward pass at a fixed checkpoint suffices); a triage procedure for forgetting, since energy control simply does not work on erosion, and knowing that saves wasted effort; and a diagnostic that predicts whether resetting the readout is worth it. The unification is the larger contribution: three subfields become regimes of one dynamic. To be honest about magnitude, per-application gains are incremental, about 2 points over random selection at a 5% budget; the value is mechanism and cost, not leaderboard numbers.
Stated by the authors: the theory is first-order and single-step, while forgetting and plasticity emerge over many steps, so the mechanistic story is a local structure extrapolated along a trajectory and validated empirically; the derivation assumes SGD while all experiments use AdamW, preserving structure but not exact values or rankings; the identity-path and linear-block approximations drop higher-order residual paths; MoE and QKV-level structure are not resolved; signed fidelity fails at cold start and needs a brief warm-up; the readout is a major but not the sole source of lost plasticity, and readout reset is a mechanistic test rather than a deployment recipe, since it discards learned capability.
Two gaps the paper leaves open: format separation requires changing the training response format, and the cost of doing that in a real pipeline is never discussed; and HyperINF actually beats CH1+2 on cross-lingual retrieval (0.694 versus 0.625), a defensible trade given its backward passes and higher complexity, but the forward-only advantage only pays off at large selection scale.