Influence Functions

Inference-time steering offers cheap, fine-grained control over a language model's outputs by estimating a concept's representation in activation space and shifting activations towards it. Existing methods build these representations from activation averages over contrastive datasets. These averages incorporate unrelated concepts and noise, and are dominated by a few tokens, meaning the activation transport encodes token-level rather than thematic concepts. In this work, we steer towards examples that most express a concept thematically, rather than towards an expectation over all. We identify these examples using influence functions, which estimate how much each data point contributes to a model's representation of a concept. Unlike simple model activation similarity, they incorporate the curvature of the model's loss landscape, allowing them to capture concept-relevant relationships beyond superficial token-level similarity. We then propose influence-weighted activation transport, which uses optimal transport to steer activations of non-concept text towards those of concept text, weighting concept examples by their influence scores. We evaluate on toxicity suppression (Jigsaw), object-based concept induction (OneSec) and truthfulness induction (TruthfulQA), outperforming existing activation-transport baselines. We track capability after steering using perplexity and MMLU accuracy, finding that our method improves steering while largely preserving model quality. We further show that influence functions capture concept-relevant information that activation-based methods miss with the two approaches ranking data points significantly differently. Together, these results demonstrate the value of curvature-aware influence information for activation steering.

Influence functions estimate how individual training examples affect the behavior of large language models (LLMs). Analyzing how training data influence different behaviors of an LLM involves repeated influence computation. Reusing stored training gradients reduces the computational cost, but storing full gradients is prohibitively expensive at LLM scale. We study how to compress these gradients while preserving influence estimates for future queries that are unknown at storage time. Through a worst-case analysis, we characterize the optimal fixed-dimensional linear representation and propose eigenbasis-corrected one-bit gradient projection (EOGP) to approximate it at scale. Specifically, EOGP uses EK-FAC to reduce gradient dimensionality, then applies PCA within the retained subspace to learn compression directions from the training gradients. We then apply one-bit quantization to the resulting coordinates, allowing more coordinates to be retained within a fixed storage budget. On GPT-2, EOGP predicts retraining outcomes more accurately than the evaluated compression baselines while using one-sixteenth of their per-example storage. On OLMo 2 SFT models from 1B to 32B parameters, EOGP remains competitive with the baselines allocated over 100 times as much storage per example.

Causal tabular foundation models amortize effect estimation across synthetic mechanisms, but latent-effect supervision rewards posterior shrinkage rather than encoding the repeated-sample response needed in a fixed deployment population. We introduce fluctuation-supervised pretraining (FSP): each synthetic table is labeled by its average treatment effect plus its efficient influence-function fluctuation; deployment remains a frozen forward pass. Along the path $T_{λ,P}=θ(P)+λP_nψ_P$, we prove an endpoint transition: every fixed $λ<1$ retains label ambiguity of order $(1-λ)^2/n$, whereas full fluctuation makes the Gaussian label observable and reduces optimal finite-stratum causal label-prediction risk to order $n^{-2}$. A finite-pretraining bound combines label, network, episode-sampling, and optimization errors; its sampling defect controls fixed-mechanism bias, mean squared error, variance, Gaussian approximation, and, with variance-head accuracy, studentized coverage. Complementary lower bounds separate local $n^{-1}$ ATE risk from the $\log N/M$ excess risk of generic finite-dictionary episode learning. Experiments trace the learned sampling response. Across 24 nonlinear continuous-covariate cells at trained context lengths, continuous-row FSP lowers checkpoint-mean macro RMSE by 7.0% versus S-learner and wins all 12 weak-overlap cells; validation-selected Summary FSP deploys $11.6\times$ faster per table in our warm one-thread benchmark. Under effect shift, matched Raw FSP lowers mean-checkpoint RMSE by 54.2% and teacher defect by 99.0% versus latent-effect supervision, and RMSE by 10.2% versus the released CausalPFN-S checkpoint. Known-effect semisynthesis tests coverage; two randomized-study evaluations show that lower RMSE can coexist with residual attenuation.

Data attribution seeks to quantify how individual training examples shape a model's predictions and underpins problems including data valuation, machine unlearning, and model interpretability. Despite having a long line of work, computationally scalable methods often struggle to predict the effect of removing training data in neural networks due to their non-convex nature. To overcome this challenge, metagradient-based methods such as MAGIC (Ilyas and Engstrom, 2025) differentiate each prediction through the entire training run and compute its exact influence with respect to the training data, but require a separate run for every prediction. To reduce this cost, we cast budgeted attribution as estimating a large influence matrix from a small number of measurements. We show that the measurements most appropriate for recovering this matrix differ from those best suited for attribution itself. We then present two algorithms, MAGE and SPELL, suited for reconstruction and attribution respectively, that run on existing metagradient machinery at no extra cost. Empirical studies demonstrate strong performance over existing baselines across training scales and measurement budgets.

The increasing adoption of machine learning in network and distributed security systems has created an urgent need for mechanisms that can selectively and efficiently remove the influence of specific training data to eliminate compromised or adversarial data points from production models. Privacy regulations such as GDPR's \emph{right to be forgotten} also pose similar requirements. However, existing approximate unlearning techniques remain computationally prohibitive for deployment in real-world security systems, as they require repeated expensive Hessian-inverse-vector computations for each data point removal, creating a bottleneck when processing multiple related requests in scenarios such as intrusion detection systems, spam filters, and threat intelligence platforms. Thus, we introduce a computationally efficient unlearning framework that identifies correlated data points in the training set and applies a theoretically derived closed-form parameter update rule, achieving an $82\times$ wall-clock speedup over standard influence function unlearning while preserving model utility with a $10^{-2}$ improvement in accuracy over state-of-the-art baselines. Our method establishes theoretical guarantees and ensures numerical stability through Hessian damping. Our evaluation across seven diverse dataset architecture combinations, including large-scale CIFAR-100 with ResNet-50, demonstrates superior forgetting effectiveness, with membership inference attack success rates of 0.660 and tug-of-war scores of 0.950.

Normative age models are trained to predict chronological age in a nominally healthy cohort. Applied to patients, they deviate, and the gap between predicted and chronological age is read as disease risk. Here, we attribute the disease-related effect size of the age gap directly to individual training samples, rather than using a prediction-level loss as the attribution target. For Cohen's $d$, the resulting closed-form influence functional, validated against leave-one-out retraining, ranks training samples by their effect on held-out case-control separation. Across four diseases and two biomarker modalities in UK Biobank, removing the 10% most influential training samples raises held-out disease-related effect size in every seed. It more than doubles the metabolomic-age effect for type-2 diabetes and raises the brain-age effect for multiple sclerosis by roughly a third. Random removal leaves effect size flat even at 50% removal, confirming the gain comes from which samples are removed, not how many. Flagged subjects carry subclinical cardiometabolic burden that diagnosis-based exclusion misses, on markers the model never sees. For type-2 diabetes, where the method gains most, the marker recovered is HbA1c, the standard measure of blood sugar control. We release pyinfluence, our influence-function package, for reproducibility and reuse.

Training data attribution (TDA) aims to identify training examples that shape model behavior, but its intervention value depends on both which examples are selected and how they are modified. Influence functions (IF) estimate behavioral changes under infinitesimal reweighting, yet IF-selected examples often show limited advantages over random selection under conventional weight-based interventions. This raises the question of whether influential examples lack intervention value or whether reweighting fails to realize their behavioral leverage.We introduce influence-guided response rewriting, which uses IF to identify intervention targets and replaces their responses with behavior-aligned or behavior-opposed supervision while keeping instructions fixed. Across four open-weight LLMs, we compare rewriting and reweighting on the same influence-selected examples using epistemic abstention as our primary testbed. Response rewriting produces stronger, more persistent, and bidirectional behavioral shifts, while reweighting the same examples yields weak and inconsistent effects. Further analyses show that influence-selected examples provide greater rewriting leverage than alternative selectors, with changes remaining concentrated on target-relevant behaviors. The same qualitative contrast extends to safety refusal. These results distinguish the local reweighting effects captured by influence estimates from the broader intervention leverage of the examples they identify, motivating intervention-aware evaluation of TDA methods.

Measuring training data influence consistently across language model pretraining is challenging. It is difficult to select downstream tasks or validation sets representative of a model's general capabilities, and reliance on task performance at intermediate checkpoints complicates comparisons across training. We propose a measure of training data influence that does not require selecting a downstream task or validation set as the attribution target. Specifically, we define an example's influence by how much its gradient update reduces the squared distance to the final parameters of a given pretraining run, and estimate this quantity from intermediate checkpoints without retraining. Applying the method to 18 configurations from the Pythia and PolyPythia suites, we find systematic temporal changes in influential data. Early in training, literature-related data are more strongly aligned with the trajectory toward the final parameters, whereas STEM data become more strongly aligned in later stages. This qualitative crossover is broadly consistent across model configurations. Our results provide a tractable trajectory-level view of how influential data change throughout pretraining, complementing influence analyses defined with respect to specific downstream tasks or validation sets.

We developed a unified covariate-adjusted causal inference framework for estimating the desirability of outcome ranking (DOOR) probability for benefit-risk evaluation in randomized trials and observational studies. The framework expresses the DOOR probability as a bilinear functional of the marginal ordinal outcome distributions under the two treatment strategies, estimates conditional ordinal distributions through sequential risk-set hazards, and derives the efficient influence function (EIF) of the DOOR probability. The point-estimation simulations compared G-computation, normalized inverse probability weighting (IPW), augmented IPW (AIPW), and targeted maximum likelihood estimation (TMLE), with nuisance functions estimated using generalized linear models or Super Learner (SL). TMLE-SL showed the strongest and most consistent point-estimation performance, with AIPW-SL ranking second. EIF-based inference was then evaluated for AIPW-SL and TMLE-SL, with and without cross-fitting, across settings varying in overlap, treatment-effect heterogeneity, and treatment allocation. CVTMLE-SL showed the strongest overall performance across DOOR-scale bias, recovery of the underlying ordinal distributions, standard-error accuracy, and confidence-interval coverage. We illustrate the methodology using data from the multidrug-resistant organism network of the Antibacterial Resistance Leadership Group.

Graphical models are ubiquitous in social and empirical science as they are intuitive and easy to use. These models belong to the broader class of Markov models, defined using solely conditional independence (CI) restrictions. In order to estimate finite-dimensional target parameters in such models efficiently, semi-parametric theory provides a principled framework for constructing regular and asymptotically linear estimators via influence functions (IFs). These estimators are asymptotically normal and root-$n$ consistent. Characterizing the class of all influence functions for a target parameter is crucial for statistically efficient inference in these models. For models that are Markov relative to directed acyclic graphs (DAGs), the orthogonal complement of the tangent space is known, implying that for any target the class of all influence functions can be derived once an influence function is obtained. On the other hand, for Markov models not equivalent to a DAG model -- such as ordinary Markov models associated with undirected graphs, chain graphs, or acyclic directed mixed graphs -- the orthogonal complement has not been characterized, impeding semi-parametric inference in these models. We derive closed form expressions for the orthogonal complement of the tangent space for general Markov models and illustrate our results by characterizing the class of influence functions for the conditional mean parameter in several graphical models.

As constrained learning becomes increasingly common, models are trained under explicit feasibility requirements to enforce fairness, safety, robustness, regulariza- tion, and physics or logic constraints. Understanding how training samples in- fluence the model solution (e.g., learned parameters) is crucial for interpretability and robustness. The classical influence function (IF) estimates sample contribu- tions via local sensitivity analysis, measuring how the solution changes when a specific training sample is perturbed or removed. However, IF becomes unreli- able in constrained settings: data perturbations can reshape both the objective and the feasible region, leading to estimates that violate feasibility. In response, we propose the Directional Influence Function (DIF), a novel estimator that explicitly incorporates these constraints into influence estimation. DIF formulates the opti- mality conditions of constrained learning as a variational inequality (VI) and ana- lyzes how perturbing training data affects this VI. We validate DIF on constrained linear regression and demonstrate that it recovers leave-one-out retraining results, whereas IF and penalty-based IF exhibit significant bias. We further apply DIF to fairness-constrained CNNs, where DIF accurately predicts test loss changes under data removal and aligns closely with actual retraining. Our results establish DIF as an efficient and reliable tool for data attribution in constrained learning.

We revisit dataset distillation from an outcome-centric perspective. Rather than aligning process surrogates (per-step gradients or training trajectories), Influence Matching (Inf-Match) aligns the final outcome of training: it learns a compact synthetic set whose effect on the converged parameters matches that of the full dataset. Concretely, we introduce a fully differentiable, sample-level influence estimator that quantifies parameter shifts from adding or removing data, without time-consuming inverse-Hessian products or convexity assumptions. The estimator runs in linear time by unrolling the optimization dynamics and applying a first-order Taylor approximation. We then learn the synthetic set by minimizing the mismatch between its influence and that of the real dataset, yielding outcome alignment rather than heuristic process imitation. Inf-Match delivers the best accuracy across standard classification benchmarks. For instance, on Tiny-ImageNet (IPC=10), Inf-Match attains 31.5%, a +4.7% improvement over NCFM. Beyond classification, Inf-Match scales to vision-language distillation on Flickr30K, outperforming strong process-matching baselines. For instance, with 200 to 1000 synthetic samples, our method achieved a leading impressive average on image/text retrieval tasks, higher than NCFM by 2.5%. The code will be released via https://github.com/hrtan/infmatch.

The impact of a given training point on a statistical model is classically measured through its leave-one-out influence, which quantifies the effect of its removal from the training set on the model accuracy. While the statistics of leave-one-out influences are well understood in the low-dimensional, large sample limit $n\to \infty, d=O(1)$, they become more intricate in high dimensions, as the influence of a given sample develops non-trivial dependencies on all other training samples. For convex M-estimation under Gaussian design, in the high-dimensional limit $n\asymp d$, we show that the distribution of the influences across the training set converges to a limiting measure which we sharply characterize. Building on these results, we provide evidence that influential samples tend to lie close to the decision boundary, thereby making contact with a standard data selection heuristic in active learning.

One way to understand LLM behavior is to trace its output back to the training data. Two types of measures are commonly used for output tracing: data-similarity and data-influence. The former is cheaper while the latter is believed to be more accurate. Even though many works have compared them for ground-truth tasks, no such comparisons exist for output tracing. Here, we fill this gap and precisely quantify the commonalities and differences between the two measures. We do this by first ranking the training documents according to each measure and then computing the overlap between the two rankings. Our main finding is that the two rankings agree significantly, but there is an asymmetry between them: The top documents of data-similarity are assigned more consistent ranks by data-influence than the other way around. This result is valid across a range of experiments involving OLMo2-1B, Qwen3-1.7B, LlaMa3.2-1B, Gemma3-1B, and GPT2. We exploit the asymmetry to obtain a favorable cost-accuracy trade-off by using the costly data-influence to refine the results of data-similarity.

Optimizing the training data distribution for Supervised Fine-Tuning (SFT) dictates the capability of Large Language Models (LLMs). While existing data curation methods excel at accelerating training under constrained budgets, they are less suited to elevating the capability upper bound. The challenge here is no longer to identify a smaller subset that preserves performance, but to refine the data distribution toward instances most capable of improving the final model. To address this problem, we explore instance-level data attribution using Influence Functions (IF). We identify that standard IF formulations struggle in this setting due to two structural limitations: a proximity gap caused by off-policy validation targets, and a severe bias towards gradient norm. We propose DRIFT (Data Refinement via On-Policy Influence Functions for Supervised Fine-Tuning). Instead of relying on external reference data, DRIFT utilizes the model's on-policy rollouts as validation targets, which empirically minimizes the parameter proximity gap and better aligns with the local neighborhood assumption of IF. It further applies signed weighting based on trajectory correctness and debiases influence scores against the gradient hacking issue, allowing a small set of validation queries to act as reliable anchors for attributing the full dataset. Experiments on 7B-parameter instruction and reasoning models show that DRIFT consistently raises the performance ceiling on both, outperforming existing data curation baselines.

In robot imitation learning, influence functions provide a principled approach to quantify each demonstration's effect on robot task outcomes, yet scaling them to billion-parameter Vision-Language-Action (VLA) models is limited by computational and multitask bottlenecks. To this end, we propose ATHENA, an influence function framework tailored for multitask VLA data curation at a billion-parameter scale. Concretely, it leverages the Kronecker structure of linear-layer gradients to reduce projection cost, and approximates dense Hessian inversion with a rank-r Random Truncated Approximation, achieving about a 313.4x speedup in influence computation. Furthermore, ATHENA formulates global and local interactive influence to balance data curation across 50 jointly trained tasks. Extensive evaluations on RoboTwin 2.0 and real-robot deployment, covering 9.34 and 6.90 hours of demonstrations, respectively, show that ATHENA matches or exceeds full-data joint fine-tuning using only 50% of demonstrations in simulation and 66.7% of data across six real-robot tasks. Overall, ATHENA demonstrates its effectiveness for data curation in billion-parameter multitask VLA fine-tuning.

With the growth of LLMs' (Large Language Models) capabilities, there has been an increasing push to curate high quality datasets by filtering samples in the training data. In general, Data Attribution (DA) methods aim to estimate how individual samples in a training dataset can precondition a model to generate certain outputs. As an example, one might be interested in which samples in the data could be the source of toxic behavior after training the LLM. Many methods quantify this conditioning through the paradigm of influence functions. While methods of this family are effective in its function, they lack the necessary processing speed and storage compactness to be practically implemented on large datasets. We propose a method, Influcoder, as a quick and cost-effective approach to influence-based Data Attribution at scale.

As Text-to-Video (T2V) generation models continue to evolve, the complexity of video evaluation necessitates a fine-grained assessment across various axes. To address this, recent works have focused on developing Multidimensional Video Reward Models (MVRMs), which decompose the evaluation process to better align with the multifaceted nature of human visual perception. However, training effective MVRMs is fundamentally challenged by the complex nature of video data. In this work, we identify a critical phenomenon termed Dimensional Heterogeneity: the reliability of a training sample can vary substantially across evaluation dimensions, meaning that a sample may provide reliable supervision for one objective while inducing high supervision risk for another. Consequently, prevailing data-centric methods that filter based on global scalar metrics are ill-posed for T2V tasks. To address this, we propose a disentangled influence framework that that efficiently estimates dimension-specific supervision risk. Leveraging this framework, we introduce two dimension-disentangled refinement strategies: Dimension-Disentangled Pruning, which removes extreme high-risk samples, and Dimension-Disentangled Reweighting, which softly down-weights high-risk supervision. Extensive experiments demonstrate that our disentangled strategies significantly outperform global filtering baselines, yielding reward models with superior alignment to ground truth.

As LLMs continue to scale up, improving training efficiency heavily relies on effective data utilization. Data selection mitigates this issue by allocating the limited training budget to high-value examples that optimally facilitate the model's target behavior. Most existing approaches define target behavior via a set of target examples and score candidate training data based on their estimated influence on these samples. However, such methods uniformly treat all target examples as equally important, ignoring the varying relevance of individual examples to model optimization. Specifically, target examples that align closely with the model's inherent behavior deliver stronger supervisory signals, whereas discrepant examples yield only weak and ineffective local guidance. We propose PRISM, a Preference-aware Influence function based Data Selection Method. It leverages model preference to assign weights to target examples and builds a preference-aware target direction. PRISM evaluates candidate training samples according to their influence on this direction, and prioritizes data budget allocation to samples that effectively drive the model to match expected target behavior. Theoretical analysis verifies that weighted preference construction generates a superior first-order gradient direction for boosting target preference, compared with uniform aggregation strategies. Extensive experiments covering diverse model architectures and parameter scales demonstrate that PRISM achieves better performance in efficient fine-tuning and safety-aligned supervised fine-tuning rectification. The results validate that accurate characterization of target behavior serves as the core of cost-effective data selection.

Federated learning systems must support data deletion requests to comply with privacy regulations, yet retraining from scratch after each deletion is computationally prohibitive. We present HF-KCU, a method that removes a client's contribution by approximating the influence function through conjugate gradient iterations in Krylov subspaces, reducing complexity from O(d^3) to O(kd) where k<<d.A causal weighting mechanism ensures that only clients holding the deleted data receive parameter updates, preventing spurious changes to unaffected clients. Our method is designed to handle bounded adversarial perturbations to the Hessian and gradient, providing graceful degradation under realistic threat models. We validate HF-KCU across convolutional (ResNet-18, SimpleCNN) and transformer (ViT-Lite) architectures on CIFAR-10, MNIST, and Fashion-MNIST. On CIFAR-10 under Dirichlet (alpha=0.5) partitioning, HF-KCU achieves 47.75 times speedup over retraining while maintaining test accuracy within 0.60% of the rational baseline(71.16 vs 71.76 %). Membership inference attacks on the forget set yield success rates of 0.499 matching the retrained model and confirming effective privacy restoration. We provide convergence guarantees showing that the Krylov approximation error decreases as O((k ^1/2-1)/(k^1/2+1)) where k is the Hessian condition number. The causal weighting mechanism ensures surgical updates, where only clients holding deleted data are modified, preserving model quality for unaffected participants and avoiding the instability of gradient-based approaches in asynchronous federated settings. This design provides interpretability as each update is directly traceable to the influence of the deleted data. The method's efficiency and precision make it suitable for production federated systems where deletion requests arrive asynchronously and computational budgets are constrained.

In recent years, the black-box nature of deep learning models has limited their application in high-stakes domains such as medical diagnosis and finance, where interpretability is essential. To address this, we propose a novel approach using influence functions to enhance interpretability in NLP models at both the sample and concept levels. Experiments on CEBaB and Yelp datasets show that influence functions effectively identify the most impactful training samples, both helpful and harmful, on model predictions. By adjusting the labels and weights of these samples, we demonstrate that model performance can be restored to baseline levels without retraining, confirming the value of influence functions for efficient data debugging. Furthermore, our concept-level analysis identifies key concepts within Concept Bottleneck Models (CBM) that significantly affect predictions. Modifying these concepts alters model behavior observably, providing clear insights into the decision process.

Influence functions approximate how removing a training example changes a quantity of interest, called the target function, such as a held-out loss. To estimate the influence of a group of examples, the standard practice is to sum the individual influences of its members. However, this sum does not capture how examples jointly affect the target: a pair of examples may be redundant or complementary, but the sum cannot distinguish these cases. We propose an interaction-aware influence function that characterizes how interactions between examples influence the target. By expanding the target to second order around the trained parameters, we obtain an estimator that augments the standard sum with a pairwise interaction term that captures the alignment between two examples' effects on the target. We empirically evaluate our estimator in two settings. First, on six dataset-model pairs spanning logistic regression, MLPs, and ResNet-9, our estimator tracks leave-group-out retraining substantially better than first-order influence across all settings. Second, when used as a greedy selection rule for instruction-tuning data on Llama-3.1-8B, it beats prior influence-based and representation-similarity baselines on five of seven downstream tasks, in a regime where standard influence-based selection underperforms random selection.

A critical step for reliable large language models (LLMs) use in healthcare is to attribute predictions to their training data, akin to a medical case study. This requires token-level precision: pinpointing not just which training examples influence a decision, but which tokens within them are responsible. While influence functions offer a principled framework for this, prior work is restricted to autoregressive settings and relies on an implicit assumption of token independence, rendering their identified influences unreliable. We introduce a flexible framework that infers token-level influence through a latent mediation approach for general prediction tasks. Our method attaches sparse autoencoders to any layer of a pretrained LLM to learn a basis of approximately independent latent features. Unlike prior methods where influence decomposes additively across tokens, influence computed over latent features is inherently non-decomposable. To address this, we introduce a novel method using Jacobian-vector products. Token-level influence is obtained by propagating latent attributions back to the input space via token activation patterns. We scale our approach using efficient inverse-Hessian approximations. Experiments on medical benchmarks show our approach identifies sparse, interpretable sets of tokens that jointly influence predictions. Our framework enhances trust and enables model auditing, generalizing to high-stakes domain requiring transparent and accountable decisions.

In this paper, we present a dual representation of the influence functions, whose computational complexity scales with dataset size rather than model size. Both analytically and experimentally, we show that this representation can be an efficient alternative to the original influence functions for estimating changes in parameters, model outputs and loss due to data point removal, when model size is large relative to dataset size, or when evaluating the original influence functions in parameter space is infeasible. The dual representation, however, is limited to linearizable models, which are models whose behavior can be approximated by their linearizations throughout training, and requires materializing a matrix, whose size grows with the product of model output dimension and dataset size.

We introduce the Geometric Observability Index (GOI), a per-feature sensitivity measure for pose estimation on SE(3). For a Gauss-Newton curvature matrix $H=E[J^\top WJ]$ and a Riemannian metric $G$ on the Lie algebra, the index is the $G$-norm of the influence a single measurement exerts on the estimated pose: $\mathrm{GOI}(z)=\|\mathcal{A}_{OO}^{-1}P_O\,\varphi(z)\|_G$, where $ψ(z)=J^\top Wr(z)$ is the score, $\varphi=G^{-1}ψ$ its gradient representative, $\mathcal{A}=G^{-1}H$ the curvature operator (self-adjoint in the $G$-inner product), $O=\mathrm{range}(\mathcal{A})$ the observable subspace, and $\mathcal{A}_{OO}$ its restriction. This single object (i) equals the norm of the M-estimator influence function, (ii) is governed by the Fisher information, which coincides with the curvature, (iii) exposes weak observability through the smallest eigenvalue $λ_{\min}$, which (iv) also governs finite-sample stability. Operationally the theory cuts both ways. The index is the exact per-measurement attribution: it predicts the true leave-one-out pose shift with log-correlation $r=1.00$. But we also prove that the influence standardized by its inlier null covariance collapses exactly to the classical chi-square residual statistic: residual gating is the leverage-corrected influence test, explaining its robustness from first principles, while raw-influence gating conflates a measurement's information with its harm and over-rejects high-leverage inliers in weakly observable geometry. Experiments on synthetic problems, five TUM RGB-D dynamic sequences, and two KITTI odometry sequences confirm the picture: the two criteria coincide under well-conditioned geometry, and raw-influence gating degrades significantly at $\mathrm{cond}(H)\approx 10^4$, as the leverage analysis predicts for noise-dominated weak directions. All quantitative claims are validated; code is released.

Federated learning (FL) is a machine learning paradigm where clients with different behaviors and preferences can learn collaboratively without compromising data privacy. Typical FL methods often rely on fixed weighting for parameter aggregation, thereby neglecting the mutual influence among clients. In practice, clients with similar preferences or backgrounds may provide more useful knowledge to each other, which can be leveraged to improve local performance. However, how to quantify such cross-client influence and how to exploit it for personalized aggregation remain underexplored. To address this gap, we propose an influence-oriented Federated learning framework which quantitatively measures Client-level and Class-level Influence to realize adaptive parameter aggregation for each client (FedC^2I for short). Our core idea is to explicitly model the inter-client influence within an FL system via the well-crafted influence vector and influence matrix. Specifically, FedC^2I incorporate influence vectors to quantify client-level influence, enables clients to selectively acquire knowledge from others, and guides the aggregation of feature representation layers. Meanwhile, the influence matrix captures class-level influence in a more fine-grained manner to achieve personalized classifier aggregation. We evaluate the performance of FedC^2I against existing federated learning methods under non-IID settings, and the results demonstrate the superiority of our method in terms of effectiveness, robustness, and interpretability.