Learning Dynamics of Continual Learning: A Unified View of Data Attribution, Forgetting, and Plasticity Loss

Learning Dynamics of Continual Learning. A Unified View of Data Attribution, Forgetting, and Plasticity Loss. By Yi Ren and four others, from OATML, the University of Oxford, and two other institutions. Published on the twenty-seventh of September. Its code repository is worked out: about one hundred and thirty source files, with a readme.

Modern language models are likely to be updated throughout their lifetime, rather than trained once and frozen. Every update then sits inside a recurring cycle. Decide which experience to learn from. Understand what the update changes. Remain able to learn from whatever comes next. Those three steps line up with three familiar names: data attribution, forgetting, and plasticity loss. The claim here is that all three are governed by the same evolving interaction between updates and behaviour. Choosing data, losing old abilities, and losing the ability to learn new ones are, on this view, three faces of one process, seen at different moments and over different spans of time.

The foundation is a decomposition, at the level of tokens and layers, of how learning from one token changes another prediction. It separates three ingredients: the softmax force, the shared readout geometry, and the residual connections. Pulling them apart exposes two channels of interaction, and yields an approximation that can be computed in a forward pass.

Followed through time, that interaction gives a unified picture of continual adaptation. Positive interaction identifies useful experience. Negative interaction shows up in one of two forms: concentrated collision, or accumulated erosion. And over longer horizons, updates reshape the shared geometry that carries future learning signals, so those signals are transmitted less well.

The predictions are then put to work in three ways. They lead to effective data selection. They lead to controls for interference that target a specific mechanism. And they lead to a diagnostic, based on the readout, of how learnable the model will be in future. When that diagnostic degrades, it predicts the benefit of restoring the readout.

The authors report the account holding across models and across training regimes. The decomposition is offered as an approximation rather than an exact identity. And the unification is careful about its own reach: data attribution, forgetting, and plasticity loss are kept as distinct regimes of the same evolving learning dynamics, not merged into one phenomenon.

Its reach comes from a single local quantity answering two questions at once: what an update changes now, and how learning today changes what can be learned tomorrow. Help, harm, and a slow loss of learnability are not three mechanisms needing three theories. They are the same interaction, positive, negative, or accumulated over time, and the decomposition is what lets one see which is happening.

Learning Dynamics of Continual Learning. A Unified View of Data Attribution, Forgetting, and Plasticity Loss. By Yi Ren and four others, from OATML, the University of Oxford, and two other institutions. Published on the twenty-seventh of September. This is the long version: the whole paper rather than the abstract.

A language model that keeps learning after deployment faces the same three questions with every update. Which experience should it learn from? What existing behaviour will that update change? And after many updates, will it still be able to learn from whatever comes next? These are usually studied as three separate fields: data attribution, forgetting, and plasticity loss. The claim here is that they are one object seen at three moments. Learning from one token changes the model's confidence in another prediction. Before an update, that change tells you whether the experience helps. During learning, it tells you what gets damaged. Over long horizons, the updates reshape the very geometry that carries the change, and that decides what can still be learned.

Start with a single gradient step on one token, the updating token, and ask how the log probability of some other token, the observing token, moves. To first order, the answer is the inner product of the two tokens' gradients, scaled by the learning rate. The authors then open the output side. The gradient of the log probability with respect to the logits is the one-hot target minus the model's predicted distribution. They call this signed vector the token force. It pushes up the target and pushes down whatever the model currently believes instead. Everything that follows is about how the force of one token reaches the prediction of another.

Exact evaluation needs full-parameter Jacobians, impractical to repeat at scale. The goal is a version computed from a forward pass alone.

Splitting the network into a backbone and a shared linear readout separates the interaction into two parts. Under two further approximations, linearising each residual block locally and keeping only the direct identity path through the residual stream, both parts become forward quantities. The result has two channels. Channel one is direct: the dot product of the two forces in vocabulary space, weighted by how similar the final hidden states are. Token forces are concentrated on a few vocabulary directions, so this channel is sparse and token-aligned. Channel two routes both forces through the readout, so it compares the forces after projection into hidden space. Because the readout couples different vocabulary entries off the diagonal, this channel lets influence pass even when the forces share no tokens. It is then summed over every layer by hidden-state similarity, plus a term counting shared context tokens.

Two consequences follow. Negative interaction comes in two kinds. Collision is a few updates with strongly conflicting forces, and the paper proves that extreme token energy implies a strong negative force on some non-target token, and the reverse. Erosion is many weak conflicts pointing the same way, so no single update looks dangerous. Second, every backbone layer receives its learning signal through the readout projection. The readout is a bottleneck. The authors define readout transmission: how much of a task's force survives projection into hidden space, divided by the force's own size. If training bends the readout away from directions a future task needs, that task's signal arrives weakened, however strong its force.

For selection, the direct channel nearly saturates attribution when tasks share output tokens. Its real test is cross-lingual: matching English problems to equivalents in Chinese, French, Korean and Spanish, where vocabulary overlap collapses. On the smaller Qwen model, adding channel two lifts retrieval from zero point four eight eight to zero point six two five on MMLU. Training on a five percent selected subset of GSM8K reaches zero point six three six, against zero point six one zero for random choice and zero point five eight seven for the gradient-based LESS.

The forgetting experiments directly test the thesis, because they show the two mechanisms answering to different controls. Off-policy fine-tuning produces extreme-energy updates that on-policy rollouts rarely do, which the authors offer as the reason reinforcement learning forgets less. Masking updates with energy above one point eight or one point five costs at most two point six points on GSM8K. But energy control leaves erosion untouched: the total probability on the answer letters barely moves. Erosion instead follows the response format. Fine-tuning in a format different from the deployed one largely preserves the deployed behaviour, and swapping formats reverses the pattern. For plasticity, transmission falls on held-out tasks across long PubMed training but not on PubMed itself, and later checkpoints adapt more slowly on exactly those tasks. Resetting the readout recovers much of it, and the size of the drop predicts the gain from reset with a rank correlation of zero point eight three.

The paper names its limits in an appendix. The most important, in the authors' words, is that the theory is local in time. It describes one update, while forgetting and plasticity loss emerge across many, so those mechanisms are extrapolations validated empirically, not derived from a multi-step theory. The derivation assumes plain gradient descent, and the evolving state of adaptive optimizers is left to future work. There is a quieter catch in the approximation itself. At a cold start, channel two gets magnitudes right but flips many signs, and with all layers summed, sign agreement falls to chance. A brief warm-up of about a hundred updates repairs it, so signed attribution needs one. The readout is called important but not the only source of lost plasticity, and resetting it is a diagnostic, not a deployment method. The authors present the results as mechanistic patterns rather than universal laws.

The idea to keep is that every update does two jobs. It changes what the model predicts now, through a direct channel and a diffuse one. And it changes the geometry through which every later update must pass. Data selection, collision, erosion and lost plasticity all fall out of following one token's force through the readout and the residual stream. Learning today shapes what can be learned tomorrow.