Influence Functions

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.

Understanding whether attention mechanisms converge to the kernel regime is foundational to the validity of influence functions for transformer accountability. Exact NTK characterization of softmax attention is precluded by its exponential nonlinearity; linearized attention is the canonical tractable proxy and the object of study here. This paper establishes that even this proxy does not converge to its NTK limit at any practical width, revealing a fundamental trade-off in the learning dynamics of attention. An exact correspondence is established between parameter-free linearized attention and a data-dependent Gram-induced kernel; spectral amplification analysis shows that the attention transformation cubes the Gram matrix's condition number, requiring width $m = Ω(κ_d(\mathbf{G})^6 n\log n)$ for NTK convergence, where $κ_d(\mathbf{G})$ is the effective condition number of the rank-$\min(n,d)$ truncation of the input Gram matrix; for natural image datasets this threshold is physically infeasible ($m \gg 10^{24}$ for MNIST and $m \gg 10^{29}$ for CIFAR-10, 12--17 orders of magnitude beyond the largest known architectures). Influence malleability is introduced to characterize this non-convergence: linearized attention exhibits 2--9$\times$ higher malleability than ReLU networks under adversarial data perturbation, with the gap depending on dataset condition number and task setting. A dual implication is established: the same data-dependent kernel is shown theoretically to reduce approximation error when targets align with the data geometry, while, empirically, creating vulnerability to adversarial manipulation of the training data. The structural argument extends to trainable QKV attention under standard initialization, with direct consequences for influence methods applied to deployed transformer architectures.

Learning from demonstrations has emerged as a promising paradigm for end-to-end robot control, particularly when scaled to diverse and large datasets. However, the quality of demonstration data, often collected through human teleoperation, remains a critical bottleneck for effective data-driven robot learning. Human errors, operational constraints, and teleoperator variability introduce noise and suboptimal behaviors, making data curation essential yet largely manual and heuristic-driven. In this work, we propose Quality over Quantity (QoQ), a grounded and systematic approach to identifying high-quality data by defining data quality as the contribution of each training sample to reducing loss on validation demonstrations. To efficiently estimate this contribution, we leverage influence functions, which quantify the impact of individual training samples on model performance. We further introduce two key techniques to adapt influence functions for robot demonstrations: (i) using maximum influence across validation samples to capture the most relevant state-action pairs, and (ii) aggregating influence scores of state-action pairs within the same trajectory to reduce noise and improve data coverage. Experiments in both simulated and real-world settings show that QoQ consistently improves policy performances over prior data selection methods.

Labeling bias arises during data collection due to resource limitations or unconscious bias, leading to unequal label error rates across subgroups or misrepresentation of subgroup prevalence. Most fairness constraints assume training labels reflect the true distribution, rendering them ineffective when labeling bias is present; leaving a challenging question, that how can we detect such labeling bias? In this work, we investigate whether influence functions can be used to detect labeling bias. Influence functions estimate how much each training sample affects a model's predictions by leveraging the gradient and Hessian of the loss function -- when labeling errors occur, influence functions can identify wrongly labeled samples in the training set, revealing the underlying failure mode. We develop a sample valuation pipeline and test it first on the MNIST dataset, then scaled to the more complex CheXpert medical imaging dataset. To examine label noise, we introduced controlled errors by flipping 20% of the labels for one class in the dataset. Using a diagonal Hessian approximation, we demonstrated promising results, successfully detecting nearly 90% of mislabeled samples in MNIST. On CheXpert, mislabeled samples consistently exhibit higher influence scores. These results highlight the potential of influence functions for identifying label errors.

Supervised fine-tuning (SFT) relies critically on selecting training data that most benefits a model's downstream performance. Gradient-based data selection methods such as TracIn and Influence Functions leverage influence to identify useful samples, but their computational cost scales poorly, making them impractical for multi-billion-parameter large language models (LLMs). A common alternative is to use off-the-shelf smaller models as proxies, but they remain suboptimal since their learning dynamics are unclear, their sizes cannot be flexibly adjusted, and they cannot be further aligned with the target model in terms of gradient-based influence estimation. To address these challenges, we introduce Iprox, a two-stage framework that derives influence-preserving proxies directly from the target model. It first applies a low-rank compression stage to preserve influence information of the target model, and then an aligning stage to align both model gradients and logits, thereby constructing proxies that flexibly control computational cost while retaining the target model's influence. Experimental results across diverse LLM families and evaluation tasks show that Iprox consistently outperforms off-the-shelf proxies and baseline methods. On Qwen3-4B, a 1.5B proxy constructed with Iprox achieves stronger performance than the larger 1.7B off-the-shelf proxy. Notably, on Llama3.2, Iprox achieves better performance than baselines while reducing computational cost by more than half relative to the full 3B model. These results show that Iprox provides effective influence-preserving proxies, making gradient-based data selection more scalable for LLMs.

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.