Ask three researchers “which training data is responsible for this output?” and you can get three answers that are all correct and all answering different questions. One hands you the most similar training documents, the way a citation supports a claim. Another hands you a fair division of credit, the way partners split a profit. The third hands you a forecast: what the model would do if you retrained it on different data. This series has lived almost entirely inside the third answer, and it is time to say so out loud, place all three on a map, and pay the debts we have been accumulating along the way, including one entire branch of the field this series has so far footnoted.
The running classroom example has 60 training points, a small logistic-regression model, and one gold-star test point whose outcome we want to explain.
The map
The field’s own taxonomy sorts every attribution method by the question it answers (Worledge et al. 2023; Ilyas et al. 2024):
FIG. 1 · THREE QUESTIONS WEARING ONE WORD
The map. Corroborative attribution finds supporting evidence by similarity: it need not be causal, and its search set need not even be the training data. Game-theoretic attribution divides credit fairly. Predictive attribution forecasts counterfactual retraining, which is where parts 1 through 5 of this series live. One intersection is a nearest-neighbor special case: comparing a point with its single closest training example is both a similarity search and a simple counterfactual retraining question. Click a region for its question, its methods, and where this series covered it. The intersections are theorems and one honest open question.
Part 1’s “influential is not the same as similar” callout was this map’s corroborative-versus-predictive boundary in disguise: visually similar examples need not be the ones that moved the model (Worledge et al. 2023; Somepalli et al. 2023).
Part 1 called the causal kind contributive attribution. On this map, contributive covers the game-theoretic and predictive circles together; predictive is the forecasting branch of it, the one this series has lived in. The boundary this series has never crossed is the second circle, and it is overdue, because part 1 left a wound only this branch can close.
The fair-division branch
Recall part 1’s overdetermination problem: two students teach the same fact; remove either one and nothing changes, so leave-one-out hands both a zero. The counterfactual is not wrong, it is just brittle about redundancy. Game theory’s repair is the Shapley value: instead of asking “what if this student left the full class?”, average the student’s marginal contribution (how much the outcome improves the moment they join a given group) over every possible subset of classmates they could join (Ghorbani and Zou 2019; Jia et al. 2019). Redundant teachers stop hiding behind each other, because in the many subsets where the other teacher is absent, their contribution shows.
What makes this branch principled rather than just different is an axiomatic guarantee: the Shapley value is the unique credit assignment satisfying a short list of fairness axioms (equal treatment of identical contributors, credit that adds across tasks, zero for the useless). That guarantee is expensive to honor: the definition sums over subsets, \(2^n\) of them. Our classroom cannot enumerate \(2^{60}\), but a six-student mini-classroom can enumerate all \(2^6 = 64\) coalitions (game theory’s word for those subsets) exactly, and every number in the next figure is that enumeration, real retrainings included:
FIG. 2 · FAIR CREDIT, COMPUTED THE EXPONENTIAL WAY
Six students, sixty-four coalitions, every one retrained. Left bars: exact Shapley credit. Right bars: plain leave-one-out. Where they disagree, redundancy is the reason. The toggle switches to the connection: use the six Shapley values as coefficients and predict each subset’s outcome as the no-training-data baseline plus the sum of its members’ values. The cloud is honest twice over: this little game is visibly not additive (tiny coalitions overfit wildly), and the Shapley vector is the fairest additive summary of it. Fair division and forecasting are the same shape of object: a Shapley vector is a straight-line formula that predicts subset outcomes, the kind of model the next section names properly.
That toggle is one of the field’s quiet unifications: the Shapley value has an exact reading as a linear model that predicts subset outcomes (Ilyas et al. 2024), which drags the fair-division branch and the forecasting branch onto common ground and hands us the vocabulary the rest of this post runs on.
The exponential bill, the branch’s ancient curse, has recently met its most serious challenger: In-Run Data Shapley computes Shapley-style attribution during a single training run, no retraining at all, by accumulating the right interaction terms as training proceeds (Wang et al. 2025). Fair division, at the price of road three from part 5, the exact but expensive posterior-sampling route.
Datamodels: stop deriving, start fitting
Now look back at everything from part 2 onward with one question: what shape of answer did every method produce? Sixty numbers, one per training example, whose sum over any subset predicts an outcome. A linear model over data membership. Influence functions derived its coefficients from calculus; TracIn and SOURCE, trajectory methods that use training checkpoints, pulled them from the run; Shapley’s came from fairness. The datamodels move is the empiricist’s shrug: if what we want is a linear model, why derive it? Fit it (Ilyas et al. 2022; Feldman and Zhang 2020; Lin et al. 2022). Sample random subsets of the training data, retrain on each, record the outcome, and regress outcomes on membership.
FIG. 3 · LEARNING THE ANSWER SHEET BY BRUTE HONESTY
Datamodel estimation on the classroom, for real: random half-ish subsets, one retraining each, ridge regression (regression with a small penalty that keeps coefficients sane while subsets are few) of the gold-star test point’s outcome on membership. Step through the training-run budget (fifty, two hundred, or two thousand retrainings) and watch sixty regression coefficients converge toward part 1’s true leave-one-out values. The ridge strength was selected once by maximizing correlation with that true leave-one-out grading set at the two-thousand-run budget, rather than re-tuned at each budget. The convergence is the method’s argument: no Hessians and no model of the training process, just retraining until the linear model gives in.
At real scale this is exactly as expensive as it looks: the original datamodels project spent millions of trainings on CIFAR-sized problems (Ilyas et al. 2022), and part 1’s cost wall is why. So the field asked the natural question: can we get the datamodel without the retraining bill?
TRAK: the datamodel at influence-function prices
TRAK (Tracing with the Randomly-projected After Kernel) answers with a three-step recipe, and every step is a move this series has already met (Park et al. 2023):
- Linearize the network around its trained parameters, so the model behaves like a linear model in its gradient features. It is the same first-order, straight-line trick behind part 2’s influence formula, now applied to the network’s outputs instead of the data weights.
- Project those enormous gradient features down to a small number of dimensions with a random matrix, which preserves the geometry that matters at a millionth of the size. The classroom demo below uses sixteen dimensions.
- Apply the linear-model influence formula in the projected space, where part 2’s mathematics is exact rather than approximate. Then average over a handful of trained models to tame the noise.
FIG. 4 · THREE STEPS, EACH ONE SHRINKING THE PROBLEM
TRAK on the small nonlinear network from part 3, dimension counts and all: gradient features, a random projection to sixteen dimensions, then the exact linear-model influence formula from part 2’s first rung. The scatter grades the result against sixty true retrainings of that network, not the logistic-regression classroom used for the other methods. The ensemble toggle averages five independently trained models, damping each run’s initialization noise, and you can watch the correlation rise when you click it. At real scale this recipe attributes models where datamodel regression would cost millions of trainings (Park et al. 2023).
The recipe has travelled: adapted to diffusion models (the image-generating architecture behind tools such as Stable Diffusion, a different model family from our classroom classifier), it produced one of the field’s more unsettling findings, that theoretically unjustified design variants can outperform the principled ones on the benchmarks (Zheng et al. 2024), part 3’s fragility lesson wearing a new costume. Meanwhile a newer generation asks whether gradients are needed at all, learning cheap attribution representations directly (Sun et al. 2025). The map keeps growing westward.
One axis to grade them all
Every method in this series has now made a promise of the same form: my scores predict what retraining would do. Part 3 introduced the yardstick informally; here it is properly. The linear datamodeling score (LDS) grades a method in four steps:
- Sample many random subsets \(S_j\) of the training data.
- Retrain on each subset to get the true outcome \(f(S_j)\).
- Ask the method to predict those outcomes from its per-example scores.
- Report the rank correlation between prediction and truth: do the two lists agree on order, never mind exact values?
That single number, plotted against what the method costs, is the fairest picture of the field anyone has drawn (Park et al. 2023; Ilyas et al. 2024):
FIG. 5 · THE LANDSCAPE: FIDELITY AGAINST COMPUTE
Left panel: this series’ own methods, every dot measured on this machine: rank correlation against sixty true retrainings versus wall-clock cost, for the plain influence function, the Newton step, exact leave-one-out, damped and K-FAC curvature, TracIn, exact unrolling, datamodel regression at a 500-subset budget, and mini-TRAK. This is an LDS-style grade using the simplest possible subsets: the sixty exact one-point leave-outs. The TRAK, K-FAC, and damped-GNH dots use the small nonlinear network from part 3 and its sixty retrainings; the other dots use the logistic-regression classroom and a separate set of sixty retrainings. The post 6 K-FAC point uses one global damping value on the assembled block matrix, while post 5’s Fig. 2 uses fixed per-layer, per-factor damping, so their correlations are not the same measurement. K-FAC and EK-FAC approximate the curvature matrix cheaply; TracIn traces influence through training checkpoints; exact unrolling replays every update step. Right panel: the published picture at deep-learning scale, drawn qualitatively from the field’s benchmarks (Park et al. 2023; Ilyas et al. 2024): old baselines cluster near the floor, direct estimation buys fidelity with brute force, and TRAK and EK-FAC trace the efficient frontier, with the winner depending on modality.
Three honest annotations:
- The classroom panel is a toy: its lesson is the ordering logic, not the exact positions.
- The published panel’s lesson is humbler than any single paper’s abstract: popular baselines were, for years, barely predictive at scale, and knowing that required building the yardstick.
- The yardstick itself is now under renovation: application-based benchmarks that grade attribution on curation, bias-tracing, and verification tasks report that no method wins everywhere (Jiao et al. 2025).
Expect this figure to look dated in five years; that is what a live field looks like.
Endpoint or trajectory, this run or runs like it
Beyond cost and fidelity, two distinctions organize everything this series has met.
Endpoint versus trajectory. Parts 2, 3, and 5 interrogate the trained model; part 4 interrogates the run. The reconciliation is now a theorem, not a hope: influence functions are the equilibrium limit of unrolled differentiation (replaying every training update), with flat directions handled by the pseudoinverse, a way to invert a matrix that isn’t fully invertible (Mlodozeniec et al. 2025). In plain terms, keep replaying the training run and, in most directions, you land where the one-step influence-function shortcut already was. Only along flat directions, parameter changes the loss doesn’t notice, does the shortcut’s inverse-Hessian math break down; the inverse Hessian is the inverted curvature matrix behind the influence formulas in this series, and the trajectory tells you which way to go instead.
This run versus runs like it. Part 1 flagged that deep-learning “ground truth” is seed-noisy: retrain twice, get two models. Every method must therefore choose its counterfactual: this particular run (MAGIC’s checkpoint-based exact unrolling, TracIn’s checkpoint estimates), or the distribution of runs like it (datamodels’ regression over resampled trainings, the Bayesian posterior of part 5). Neither is more correct; they answer different questions, and the field’s habit of grading one kind against the other’s ground truth explains a healthy share of its confusing numbers (Mlodozeniec et al. 2025).
Crossed, the two axes make a little map of their own, with one corner conspicuously quiet:
The quiet corner is trajectory-and-distribution: a method that tells you when in training the data mattered, averaged over the runs you could have had. Part 4’s trajectory tools all commit to one run, while the distributional tools interrogate endpoints. A method that crosses those choices would fill the last region of this map.
Are we there yet?
The people who built much of this map end their own tutorial with a sobering observation and a bet. The observation: practitioners largely do not know these tools exist, and mostly do not use them; when surveyed, even ML engineers debugging models reach for almost anything else first (Nguyen et al. 2023). The bet: within a few years, attribution becomes a standard stage of the ML pipeline, not a research curiosity (Ilyas et al. 2024).
What would change between here and there is the subject of the final post: not how attribution works, but what it is for. Because the map you have just seen was drawn by people who wanted more than explanations. Every region of it is a lever.
Further reading
The map’s surveyors
- Worledge, Shen, Meister, Winston, and Guestrin, “Unifying Corroborative and Contributive Attributions in Large Language Models” (Worledge et al. 2023). The taxonomy of Fig. 1.
- Hammoudeh and Lowd, “Training Data Influence Analysis and Estimation: A Survey” (Hammoudeh and Lowd 2024), and the ICML 2024 tutorial (Ilyas et al. 2024).
The fair-division branch
- Ghorbani and Zou, “Data Shapley” (Ghorbani and Zou 2019); Jia et al., “Towards Efficient Data Valuation Based on the Shapley Value” (Jia et al. 2019); and Wang, Mittal, Song, and Jia, “Data Shapley in One Training Run” (Wang et al. 2025), the ICLR 2025 result that removed the retraining bill.
The forecasting branch, empiricist wing
- Ilyas, Park, Engstrom, Leclerc, and Madry, “Datamodels: Predicting Predictions from Training Data” (Ilyas et al. 2022); Feldman and Zhang on memorization by resampling (Feldman and Zhang 2020); Lin et al. on randomized-experiment estimation (Lin et al. 2022).
- Park, Georgiev, Ilyas, Leclerc, and Madry, “TRAK: Attributing Model Behavior at Scale” (Park et al. 2023); Zheng et al. on its intriguing diffusion behavior (Zheng et al. 2024); Sun et al. on learned attribution representations (Sun et al. 2025).
The yardsticks
- Jiao et al., “DATE-LM: Benchmarking Data Attribution Evaluation for Large Language Models” (Jiao et al. 2025), and Nguyen et al. on the adoption gap (Nguyen et al. 2023).
References
Reproducibility. The mini-classroom Shapley values enumerate and retrain all sixty-four coalitions exactly; the datamodel regression retrains every sampled subset; the mini-TRAK and the left landscape panel are computed and timed on the build machine by a seeded script in the site repository (scripts/influence-data/). The right landscape panel is a qualitative redrawing of published benchmark findings, cited in the caption, not a re-measurement.
Acknowledgements. The single-run-versus-distributional axis and the convergence framing that ties the map together follow Louis Jaburi’s lecture notes “Data (Attribution) for Alignment” (Iliad Intensive, April 2026).