Contribution
It was already known that influence estimates align well with leave-one-out retraining for linear models and badly for neural networks, and the field had read that as a failure — influence functions were “fragile”. This paper takes the discrepancy apart term by term and reaches the opposite conclusion.
Key insight
A method that reliably fails to answer one question is answering a different one, and the useful move is to find out which.
The five-term decomposition separates the discrepancy into pieces that are errors — the estimate is a bad version of the thing it wants — and pieces that are gaps — the estimate is a good version of something else. The errors turn out to be small. What remains, and dominates, is the distance between leave-one-out retraining and a different object.
That object is the proximal Bregman response function: the effect of removing a data point while keeping the model’s predictions close to those of the trained model. Once named, everything that looked like fragility becomes a definition.
Assumptions
- The Gauss-Newton Hessian is used where the loss is convex in the network outputs, which covers the standard cases.
- The analysis is of practical influence estimates — damped, on non-converged parameters — rather than of the idealised quantity, which is the point: it characterises what people actually compute.
- The decomposition bounds the parameter (or output) difference by summing the per-term differences, so it is an accounting of all approximations and assumption violations rather than a tight characterisation of any one.
Key results
- Five terms: the warm-start gap (cold-start versus warm-start response functions), the proximity gap (an implicit proximity regularizer), the non-convergence gap, linearization error, and solver error.
- Terms 1–3 dominate; terms 4–5 are at least an order of magnitude smaller for most neural networks. Measured across binary classification, regression, image reconstruction, image classification and language modelling, and studied against network width and depth, weight decay, training time, damping, and the number of points removed.
- Terms 1–3 are exactly the difference between leave-one-out retraining and the PBRF. So the gap between the influence estimate and the PBRF comes only from terms 4 and 5 — the small ones. Influence estimates track the PBRF closely while failing to track retraining.
- The uses survive. The paper states that the PBRF supports the same purposes that motivated influence functions: finding influential or mislabelled examples, and carrying out data-poisoning attacks.
- Therefore the PBRF is a better gold standard for evaluating influence estimation than retraining, and influence functions on neural networks are not inherently fragile.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | The IF/LOO discrepancy decomposes into exactly those five terms | strong | derived; the decomposition is constructive |
| C2 | Linearization and solver error are an order of magnitude smaller than the rest | strong | measured across five task families |
| C3 | Influence estimates closely approximate the PBRF | strong | Figure 1 and the task sweep |
| C4 | The PBRF supports the mislabelled-example and poisoning use cases | moderate | argued from what the PBRF is; not a fresh benchmark of both |
| C5 | Influence functions are not inherently fragile | strong | follows from C2 and C3, against the prior reading |
Method
Define the warm-start and cold-start response functions and the PBRF. Insert each approximation one at a time, measure the parameter or output difference each introduces, and sum. Then compare influence estimates against retraining and against the PBRF on the same models.
Concepts
- Gap versus error — the distinction the paper’s terminology is built on, and the whole argument in one word choice. A gap means the target moved; an error means the aim was off.
- The proximal Bregman response function — removal, constrained to keep predictions near the trained model’s.
- Warm start versus cold start — retraining from the trained parameters versus from scratch, which are not the same counterfactual and are routinely conflated.
- Choosing a gold standard — retraining is the intuitive one and is the wrong one, and a method evaluated against the wrong standard looks broken.
Connections
This corrects LIT-403’s account of what an influence estimate measures, and it is the reason LIT-401 validates against the PBRF rather than against retraining — a rare case of a correction being absorbed into practice within a year.
It leaves LIT-400 largely untouched, since TracIn never claimed to estimate the leave-one-out counterfactual.
Recommendations
- R1 — Evaluate an attribution method against the object it estimates. Strength: strong. Stated here as a consequence rather than as advice.
- R2 — Do not read an influence estimate as “what would happen if I removed this and retrained”. Strength: strong.
- R3 — Keep using influence functions for mislabelled-example detection and for poisoning analysis; the reinterpretation does not touch those. Strength: moderate.
Bearing on the record
Sources THEORY-018, which corrects THEORY-017 — the account LIT-403 published with the method.
R3 is what keeps SOTA-246 standing. Without this paper the practice would rest on an explanation the field had abandoned, which is the arrangement CLAUDE.md warns about in as many words: a technique everybody uses can have been published with an explanation that was later refuted. Here the refutation is explicit about which uses it spares, so the practice’s conditions can say so rather than hedging.
R1 and R2 are the theory’s content rather than separate practices, and filing them as practices would duplicate the pair of THEORY documents across schemes.
Limitations
- The decomposition is an accounting, so the terms are bounded rather than characterised exactly.
- C4 is argued from the nature of the PBRF rather than re-benchmarked on both use cases.
- The experiments are moderate-scale; nothing is at the size LIT-401 works at, and it is that paper that extends the PBRF validation upward.
- The paper says what influence functions do estimate. Whether the PBRF is the quantity anyone wanted is a separate question it does not take up, and LIT-401 explicitly declines to take it up either.
Open questions
- Is the PBRF the right target? Both this paper and LIT-401 leave it open, which means the whole attribution literature is now validated against an object nobody has argued is the one of interest.
- Does the warm-start gap have practical consequences for continual training, where warm starting is what actually happens?