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If Influence Functions are the Answer, Then What is the Question?
Juhan Bae, Nathan Ng, Alston Lo, Marzyeh Ghassemi, Roger Grosse
TL;DR
Influence functions are commonly judged by how well they predict leave-one-out retraining, yet that correspondence is often poor in neural networks. This paper decomposes the discrepancy into five components and finds that influence estimates instead often closely approximate the proximal Bregman response function, which supports many motivating applications.
Problem
Influence functions often poorly predict leave-one-out retraining in neural networks, raising questions about the sources of this discrepancy and how to evaluate their usefulness.
Method
The paper decomposes the discrepancy into warm-start, proximity, non-convergence, linearization, and solver components and evaluates them across architectures, datasets, and training conditions.
Results
The first three components are most responsible for the discrepancy, while influence functions closely align with the proximal Bregman response function across a wide variety of tasks.
Takeaways & Limitations
The PBRF is an attractive alternative gold standard because it correlates better with influence functions and supports many use cases such as finding influential or mislabeled examples.
Takeaways & Limitations
Linearization error increases when the downweighting magnitude is large or the PBRF is highly nonlinear, and practical linear solvers introduce additional estimation error and biases.
Abstract
from arXiv · showhide
Influence functions efficiently estimate the effect of removing a single training data point on a model's learned parameters. While influence estimates align well with leave-one-out retraining for linear models, recent works have shown this alignment is often poor in neural networks. In this work, we investigate the specific factors that cause this discrepancy by decomposing it into five separate terms. We study the contributions of each term on a variety of architectures and datasets and how they vary with factors such as network width and training time. While practical influence function estimates may be a poor match to leave-one-out retraining for nonlinear networks, we show they are often a good approximation to a different object we term the proximal Bregman response function (PBRF). Since the PBRF can still be used to answer many of the questions motivating influence functions, such as identifying influential or mislabeled examples, our results suggest that current algorithms for influence function estimation give more informative results than previous error analyses would suggest.
1 Introduction
Influence functions estimate the effect of deleting training examples without retraining, but their agreement with leave-one-out retraining is strong for logistic regression and poor for neural networks. This work decomposes that discrepancy and argues that influence estimates often approximate the PBRF instead.
- Motivation: Influence functions estimate parameter changes from deleting a training example without the computational cost of retraining.They use a second-order Taylor approximation around the learned parameters.
- Motivation: Influence functions align well with leave-one-out retraining for strongly convex objectives such as regularized logistic regression but often fail for neural networks.Prior analyses described neural-network influence estimates as fragile and erroneous.
- Contributions: The paper decomposes the influence–LOO discrepancy into five components: warm-start, proximity, non-convergence, linearization, and solver effects.The components are evaluated across multiple tasks and varied training and architectural conditions.
- Contributions: Across tasks, the warm-start, proximity, and non-convergence components are most responsible for the discrepancy between influence functions and leave-one-out retraining.The study also examines network width and depth, weight decay, training time, damping, and the number of removed examples.
- Contributions: Influence functions are often a poor match to leave-one-out retraining but a much better match to the proximal Bregman response function.The PBRF approximates removing a data point while keeping predictions consistent with the partially trained model.
2 Related Work
Related work places influence functions within instance-based interpretability and evaluates them by comparing their estimates with leave-one-out retraining. Recent empirical work instead reports a fundamental misalignment between those assumed and actual effects in neural networks.
- Instance-based interpretability: Influence functions are instance-based interpretability methods that explain predictions through the training examples used to fit a model.Related methods include TracIn, Representer Point Selection, Grad-Cos, Grad-Dot, and MMD-critic.
- Influence-function extensions: Influence-function research has introduced Fisher-kernel, higher-order, scalable, group-influence, and relative-influence variants.These extensions target different approximation, inference, and formulation needs.
- Evaluation: Influence functions are generally evaluated by correlating their estimates with the effects of leave-one-out retraining from scratch.Recent empirical analyses report fragility and a fundamental misalignment between the assumed and actual effects.
3 Background
The paper formulates deleting or downweighting a training example as a perturbed optimization problem and studies the resulting response of the learned parameters. Influence functions approximate this response locally, but neural-network use introduces computational and convexity challenges.
- Training setup: The model maps inputs x to predictions y=f(θ,x), and training minimizes average loss over the finite dataset, including any regularization in the cost.The loss may be squared error or cross-entropy.
- Downweighting a training example: Downweighting an example z by ϵ defines a perturbed objective whose optimum describes the model response to changing that example’s contribution.When ϵ=1/N, the objective is equivalent to removing z up to a constant factor.
- Response functions: Under strong convexity, the perturbed objective has a unique solution, and influence functions approximate its response at ϵ0=0 with a first-order Taylor expansion.The resulting approximation can estimate parameters and test loss after removing a training example.
- Influence estimation: Exact inverse-Hessian vector products are infeasible for large networks, so practical influence estimation uses truncated conjugate gradients or LiSSA.These methods avoid explicitly storing and inverting the Hessian.
- Influence estimation: Neural networks often violate the strong-convexity assumption, so damping and Gauss–Newton approximations are used to obtain an invertible influence-estimation system.The Gauss–Newton Hessian is positive semidefinite when the loss is convex in network outputs, while λ>0 ensures invertibility.
4 Understanding the Discrepancy between Influence Function and LOO Retraining in Neural Networks
The paper decomposes influence-function misalignment with LOO retraining into five gaps and finds the first three dominate because they reflect different retraining questions. Influence estimates instead closely approximate the PBRF, whose proximal objective preserves predictions while modeling data-point removal.
- Decomposition: The misalignment comprises warm-start, proximity, non-convergence, linearization, and solver gaps, which together bound parameter or output differences.The first three are termed gaps because they compare different influence-related solutions rather than purely algorithmic errors.
- 4.1 Warm-Start Gap: In overparameterized neural networks, initialization-dependent optima create a warm-start gap between retraining from the current solution and retraining from scratch.For quadratic costs, gradient descent converges to the optimum closest to its initialization, so cold-start and warm-start optima need not coincide.
- 4.2 Proximity Gap: Damping adds an L2 proximity penalty, making influence functions approximate a proximal warm-start objective rather than ordinary warm-start retraining.Early stopping can have a similar effect to L2 regularization, so the proximal response can represent limited optimization steps from the warm-start solution.
- 4.3 Non-Convergence Gap: With non-converged parameters, LOO retraining partly reflects additional training, whereas influence estimation isolates the effect of removing the example.The resulting discrepancy is the non-convergence gap, and extended-training effects are described as a nuisance for understanding influence.
- PBRF: The PBRF combines prediction-preserving Bregman divergence, a negative loss term for the removed example, and a proximity term.It replaces the original objective with one that keeps predictions near those of the partially trained model while modeling deletion.
- 4.4–4.5 Approximation Errors: Linearized PBRF optimization equals damped influence estimation at possibly non-converged parameters, while linearization and solver errors remain separate approximations.Linearization error grows with larger downweighting magnitude or greater PBRF nonlinearity, and practical solvers introduce additional iHVP error.
5 PBRF: The Question Influence Functions are Really Answering
The PBRF reframes influence functions as approximating prediction-preserving data removal rather than exact LOO retraining. This reframing supports practical evaluation and use of influence estimates without modifying typical training regimes.
- PBRF approximates removing a data point while keeping predictions consistent with those of the partially trained model.
- PBRF differs from influence estimates only through linearization and solver errors, while warm-start, proximity, and non-convergence gaps separate it from LOO retraining.
- PBRF can serve as a gold-standard ground truth for evaluating influence-function approximations without added regularization or modified training regimes.
- Influence functions remain useful for identifying influential or mislabeled examples because these applications typically do not require exact LOO retraining.
6 Experiments
Experiments decompose influence–LOO misalignment across models, datasets, and training factors. Across tasks, warm-start, proximity, and non-convergence gaps dominate, while influence estimates align strongly with PBRF estimates.
- Influence Misalignment Decomposition: Across binary classification, regression, image reconstruction, image classification, and language modeling, the first three gaps dominate misalignment over linearization and solver errors.
- Model Comparisons: Logistic regression exhibits uniformly low misalignment because its objective is strongly convex and the model is trained to convergence.
- Model Comparisons: MLPs and autoencoders show larger warm-start, proximity, and non-convergence gaps than linearization and solver errors, while influence estimates accurately predict PBRF.
- Influence Misalignment Decomposition: Influence estimates show high Pearson and Spearman correlations with PBRF test-loss estimates but align poorly with cold-start and warm-start retraining.
- Factors Affecting Misalignment: Increasing network width decreases linearization error, whereas depth shows no strong relationship with component contributions.
- Factors Affecting Misalignment: Increasing weight decay decreases overall discrepancy, while higher damping reduces solver and linearization errors but increases the proximity gap.
- Factors Affecting Misalignment: Removing more training examples sharply increases linearization error and also increases the warm-start gap.
7 Conclusion
The paper decomposes the discrepancy between influence functions and leave-one-out retraining into five components, finding the first three most responsible across architectures and datasets. It introduces PBRF as a better-matching object that can support influence-function use cases despite not necessarily aligning with leave-one-out retraining.
- The discrepancy comprises warm-start, proximity, non-convergence, linearization, and solver gaps.
- Across varied architectures and datasets, the warm-start, proximity, and non-convergence components are most responsible for the discrepancy.
- The paper introduces PBRF to better capture influence-function behavior in neural networks.
- PBRF is more easily calculated and correlates better with influence functions than leave-one-out retraining.
- Although PBRF may not align with leave-one-out retraining, it remains applicable to many motivating influence-function use cases.
B.2 Proximal Response Function Derivation
With damping, influence estimates are equivalent to first-order Taylor approximations of a proximal response function, corresponding to leave-one-out retraining with a proximity term added.
- Damping adds λI to the influence-function Hessian system and makes it equivalent to a first-order Taylor approximation of the proximal response function.
- The damped influence estimate approximates leave-one-out retraining with a proximity term added to the downweighted objective.
B.3 Proximal Bregman Response Function Derivation
The proximal Bregman response function handles potentially non-converged parameters by penalizing prediction mismatch to the current model. This makes the current parameters optimal at zero downweighting and reframes damped influence estimates as PBRF approximations.
- For non-converged parameters, warm-start retraining can reflect training longer rather than the effect of removing a training example.
- The proximal Bregman objective replaces the cost function with a penalty for mismatch to predictions from the current parameters.
- The Bregman divergence is non-negative and zero at the current parameters, making those parameters optimal at zero downweighting.
- Influence functions computed at non-converged parameters with damping approximate the PBRF rather than leave-one-out retraining.
B.4 Linearized Proximal Bregman Response Function Derivation
The linearized PBRF is equivalent to influence estimation using a Gauss–Newton Hessian and damping. Efficient implementations approximate the inverse-Hessian vector product with conjugate gradients or LiSSA, trading exactness for tractability.
- The linearized PBRF equals influence estimation with a Gauss–Newton Hessian approximation and damping λ > 0.
- Convexity of the loss in network outputs makes the Gauss–Newton Hessian positive semidefinite, while damping guarantees invertibility.
- Influence functions require inverse-Hessian vector products that are intractable to compute exactly for large networks.
- Conjugate gradients iteratively approximate inverse-Hessian vector products, with exact solutions guaranteed after d iterations but truncated iterations used in practice.
- LiSSA approximates inverse-Hessian vector products with a truncated Neumann series that becomes exact as recursion depth T approaches infinity.
E Experimental Details
The experiments decompose influence-function discrepancies through staged retraining and influence estimation across diverse models, datasets, and training configurations. Across these evaluations, the solver error is smaller than the other decomposition terms, while influence estimates align more closely with PBRF than with LOO retraining.
- Experimental pipeline: The experimental pipeline compares cold, warm, proximal, proximal Bregman, linearized proximal Bregman, and LiSSA-based influence optima.These stages isolate warm-start, proximity, non-convergence, linearization, and solver discrepancies.
- Architectures and datasets: The study evaluates binary classification, regression, image reconstruction, image classification, and language modeling across logistic regression, MLPs, autoencoders, CNNs, and Transformers.The experiments use UCI, MNIST, FashionMNIST, CIFAR10, and Penn Treebank settings.
- Solver error: The solver error remains significantly lower than the other decomposition terms, including when influence functions use the Hessian rather than the Gauss–Newton approximation.The warm-start, proximity, non-convergence, and linearization terms are unchanged by how influence estimates are computed.
- Influence versus retraining: Influence estimates align closely with PBRF values while failing to accurately predict cold-start retraining effects for both regression and classification datasets.This pattern holds with both Gauss–Newton and Hessian-based influence approximations.
F.2 Two-Stage LOO Retraining: An Alternative Method for PBRF computation
The paper evaluates two-stage LOO retraining as an alternative way to approximate influence-function behavior and tests PBRF and influence scores for detecting mislabeled examples. Two-stage retraining correlates better with influence estimates than warm-start retraining, while PBRF and influence scores efficiently identify corrupted examples.
- Two-stage LOO retraining: Two-stage LOO retraining performs separate full-dataset and removed-example retrainings to contrast their training-time effects.The resulting parameter-space difference is intended to approximate the effect of removing a data point.
- Two-stage LOO retraining: Two-stage LOO retraining correlates worse with influence estimates than PBRF but significantly better than warm-start retraining.It requires retraining the network twice and is described as a better reflection of what influence functions compute in neural networks.
- Practical use: PBRF can be used instead of exact LOO retraining for tasks such as identifying influential or mislabeled examples.The paper motivates this substitution because these use cases typically do not require exact LOO retraining.
- Mislabeled-example detection: Using PBRF self-influence scores, examining 20% of training examples detects over 80% of mislabeled examples in corrupted MNIST.Influence functions similarly support efficient detection because their scores closely align with PBRF, and both outperform random inspection.