Theoretical and Practical Perspectives on what Influence Functions Do

• Recent studies report low, hyper-parameter-sensitive correlation between IF rankings and Leave-Some-Out Retraining rankings in deep neural networks.Source: §1 · p. 1
• The analysis identifies convexity, numeric stability, training trajectory, and parameter divergence as assumptions that may explain the empirical discrepancy.Source: §1 · pp. 1–2
• Most assumptions can be addressed, but parameter divergence makes IF predictions unreliable as training proceeds and limits their theoretical interpretation.Source: §1 · p. 2
• IF generally predict parameter changes during fine-tuning on a perturbed loss for a limited time, rather than the outcome of general retraining.Source: §1 · p. 2
• A simple intervention using a few gradient steps on influential examples successfully corrects mis-predictions without full retraining.Source: §1 · p. 2

• IF forecast prediction changes by tracing how up- or down-weighting a training example changes model parameters.Source: §2 · p. 2
• Removing or adding a training point is represented as a smooth perturbation of the loss function.Source: §2 · pp. 2–3
• The paper generalizes the perturbation from one scalar weight to a vector ε with Q > 1 for differently reweighting multiple points or datasets.Source: §2 · p. 3
• IF approximate the perturbed final parameters with the first-order expansion θε ≃ θ0 + εT ∇ε|0θε, where the Jacobian captures sensitivity to the perturbation.Source: §2 · p. 3
• Different IF methods mainly differ in how they derive the Jacobian ∇ε|0θε used in this approximation.Source: §2 · p. 3

3.1 Problematic Assumption #1: Convexity can be used to show that θε is a function of ε

• 3.1 Problematic Assumption #1: Convexity can be used to show that θε is a function of ε: strict convexity is unrealistic for neural networks, but a nearby perturbed stationary point can still be modeled near the original one.Source: §3.1 · p. 3

3.2 Problematic Assumption #2: The model Hessian is not degenerate

• 3.2 Problematic Assumption #2: The model Hessian is not degenerate: clustered near-zero eigenvalues make Hessian inversion unstable, while regularization introduces a tuning parameter.Source: §3.2 · p. 3

3.3 Problematic Assumption #3: Training trajectory can be ignored in Hessian-based Influence

• 3.3 Problematic Assumption #3: Training trajectory can be ignored in Hessian-based Influence: HIF gives no guarantee that retraining reaches its predicted stationary point, motivating trajectory-aware TracIn.Source: §3.3 · pp. 3–4

3.4 Problematic Assumption #4: The training trajectory can be modelled additively

• 3.4 Problematic Assumption #4: The training trajectory can be modelled additively: TracIn omits first-order dependencies of later steps on earlier perturbed steps.Source: §3.4 · p. 4

3.5 Problematic Assumption #5: θε can be expanded to first order in ε

• 3.5 Problematic Assumption #5: θε can be expanded to first order in ε: the Jacobian may grow with training time, violating the O(1) condition required by IF.Source: §3.5 · p. 4
• Assumption #5 substantially limits IF predictive power but supports a local interpretation in which influence holds for only a limited number of fine-tuning steps.Source: §3.5 · p. 4

4.1 HIF does not need Assumption #1

• HIF does not require strict convexity when the Hessian at θ0 is non-singular, allowing a locally unique parameter function through the Implicit Function Theorem.Under stationarity, the perturbed solution remains stationary; local minima likewise remain local minima for sufficiently small perturbations.Source: §4.1 · pp. 4–5

4.2 If Assumption #2 is not satisfied, use Arnoldi-based Influence Functions

• When the Hessian is singular, Arnoldi-based Influence Functions restrict parameter variation to the non-degenerate Hessian subspace.The subspace is approximated using eigenvectors associated with the Hessian’s top-k absolute eigenvalues.Source: §4.2 · p. 5

4.3 The training trajectory can be traced to address Assumptions #3–#4

• The trajectory analysis tracks parameters, losses, learning rates, and randomness across training steps to account for batch selection and dropout.The resulting expression includes earlier time-step contributions omitted by prior analysis, so parameter sensitivity can grow faster than linearly.Source: §4.3 · p. 5
• With constant learning rate, parameter-sensitivity growth is empirically super-linear rather than the linear growth suggested by the simpler analysis.This behavior is tied to the requirement that the perturbed and unperturbed parameters remain O(1)-close.Source: §4.3 · p. 6

4.4 Assumption #5 becomes problematic over time

• Theorem 3 bounds parameter divergence using a discrete Gronwall argument, showing that divergence is no longer O(ε) as integrated learning rate increases.The bound assumes bounded trajectories and Lipschitz losses and gradients, but is described as pessimistic because it involves an exponential of integrated learning rate.Source: §4.4 · p. 6
• Influence functions should therefore be evaluated over limited fine-tuning horizons, with deterministic training used to reduce additional divergence from randomness.The paper recommends reporting performance as a function of fine-tuning steps rather than evaluating leave-some-out retraining from scratch.Source: §4.4 · p. 6

5 Illustrating the Theory

• The experiments use deterministic training on SST2 with BERT and CIFAR10 with ResNet, holding batch order and relevant random generators fixed.The evaluation compares influence scores with loss shifts produced by down-sampling individual training points and reports Pearson correlation over time.Source: §5 · p. 6, §5.2 · p. 7

5.1 Illustrating Parameter Divergence (Theorem 3)

• Parameter divergence follows exponential growth in integrated learning rate, with a fast initial phase and a slower later phase across BERT and ResNet.For BERT, the later-phase log-divergence fits have R2 of at least 0.9, with fitted slopes between 30 and 130.Source: §5.1 · pp. 6–7, Figure 1 · p. 7

5.2 Illustrating the fading of influence

• Influence-score predictive power is high for BERT after a few retraining steps but then quickly degrades toward zero.For ResNet, predictive power starts lower and declines monotonically over time.Source: §5.2 · pp. 7–8, Figure 2 · p. 7

6 Using Influential Examples for Error Correction

• The correction procedure retrieves influential examples and applies only a few fine-tuning steps on a perturbed loss.Proponents-correction relabels top proponents, whereas Opponents-tuning fine-tunes on top opponents of the current prediction.Source: §6 · p. 8
• On SST2, both Proponents-correction and Opponents-tuning strongly outperform the random-label baseline.The evaluation allows up to 50 steps and measures both correction success and retention of predictions on held-out examples.Source: §6 · p. 8
• On CIFAR10, Proponents-correction improves success rate by 2% on average and reduces correction steps by 6%, while Opponents-tuning is ineffective.Only 54% of retrieved opponents have the desired label, limiting the opponent-based method in the multiclass setting.Source: §6 · p. 8

7 Limitations

• The trajectory and divergence theorems are derived for stochastic gradient descent, with modifications for other optimizers only sketched.The paper does not pursue those optimizer extensions in detail.Source: §7 · p. 9
• The retention metric does not distinguish semantically similar from unrelated changed predictions, leaving consistency and generalization properties for future work.A thorough study of the correction-retention tradeoff is not included.Source: §7 · p. 9

• Parameter divergence severely limits IF predictive power, making influence fade over training time even with deterministic training.Source: §8 · p. 9
• Most problematic IF assumptions can be addressed, but IF can predict perturbed-loss fine-tuning only for a limited amount of time.Source: §8 · p. 9
• IF remain useful for model debugging because the paper proposes and empirically validates a procedure for correcting model mis-predictions.Source: §8 · p. 9

• A non-singular Hessian at θ0 yields neighborhoods where a Ck function Θ uniquely maps perturbations ε to nearby solutions.Source: A Proof of Theorem 1 · p. 11
• The proof treats the stationarity equation as N constraints in θ and ε, then solves for θ using the Hessian as the Jacobian.Source: A Proof of Theorem 1 · p. 11
• The gradient of Θ at zero is obtained by differentiating the stationarity equation with respect to ε and evaluating at ε = 0.Source: A Proof of Theorem 1 · p. 11

B Proof of Corollary 1

• If θ0 is stationary, the perturbed solutions Θ(ε) remain stationary; if θ0 is a strict local minimum, they remain strict local minima for sufficiently small ε.Source: B Proof of Corollary 1 · p. 11
• For a single-point loss perturbation, the corollary recovers the classical influence result for training-parameter changes caused by down- or up-sampling.Source: B Proof of Corollary 1 · p. 11

D Proof of Theorem 3

• Theorem 3 bounds parameter divergence between perturbed and unperturbed SGD trajectories under bounded-region and Lipschitz assumptions, with sharpness established for the resulting inequality.Source: D Proof of Theorem 3 · p. 12

D.1 Sketching modifications needed in the case of optimizers

• For optimizers, the analysis must track both parameters and optimizer state, adding continuity and Lipschitz conditions on their joint update rules.Source: §D.1 · p. 13
• For BERT, influence-score predictive power on loss shifts degrades quickly over time, and later checkpoints or three TracIn checkpoints do not improve the fading pattern.Source: Figure 4 · p. 14

E Why do we use the Conjugate Residual method?

• Conjugate Residual is preferred for HIF because Conjugate Gradient assumes a positive-definite Hessian, which is generally unsuitable for neural networks.Source: E Why do we use the Conjugate Residual… · p. 14

F Further empirical results on Fading of Influence scores

• Influence-score predictive power faded quickly for BERT, while ResNet’s was never particularly high and declined more slowly.For BERT, later checkpoints showed the same qualitative fading; ResNet’s slower fading was associated with slower parameter divergence and SGD.Source: F Further empirical results on Fading… · p. 14, Figure 5 · p. 15, Figure 6 · p. 15
• Using three TracIn checkpoints did not improve BERT’s predictive performance over one checkpoint, so subsequent experiments used a single checkpoint.The multiple-checkpoint peak was slightly lower than the single-checkpoint peak.Source: F Further empirical results on Fading… · p. 14

G How many steps are needed to correct mis-predictions?

• For BERT, Proponent-correction and Opponent-tuning produced a large decrease in the steps needed to correct mis-predictions.For ResNet, Opponent-tuning was omitted because its success rate was poor, while Proponent-correction yielded smaller but consistent gains as ε varied.Source: G How many steps are needed to correct… · p. 14, Figure 8 · p. 16

F Further empirical results on Fading of Influence scores

• With SGD, BERT’s influence-score predictive power faded more slowly than in the standard BERT experiment.The “Gradient” method in Figure 7 is TracIn.Source: F Further empirical results on Fading… · p. 14, Figure 7 · p. 16

H.1 Training

• Correction experiments used ABIF for computational efficiency, with BERT selected at 6000 steps as the best validation checkpoint and ResNet selected after 10 epochs, when about 150 test points were incorrectly classified.ABIF used 32 projectors obtained with 64 Arnoldi iterations.Source: §H.2 · p. 17

I Broader Impact Statement

• The correction procedure may introduce new errors or overfit spurious artifacts, requiring mitigation of the trade-off between fixing mis-predictions and retaining overall performance.This limitation is especially relevant when influence scores poorly predict loss shifts.Source: I Broader Impact Statement · p. 17