---
type: Research concept
title: Influence Functions
description: Concept page for estimating how training examples affect learned models.
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timestamp: '2026-05-05T19:52:29+08:00'
modified: '2026-07-02T20:03:20+08:00'
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review_due: '2026-12-29'
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---
**Influence Functions** are analytical tools for estimating how a training point affects a fitted model or a downstream prediction. In modern machine learning they are often used as approximations: instead of retraining after changing one point, the method estimates the effect through gradients and curvature information.[^influence]

## Role in this wiki

This page explains why influence-based reasoning appears across data-centric ML. If a researcher can estimate the effect of a point, group, or weighted subset, they can ask which data should be removed, downweighted, kept, or inspected. Influence functions therefore connect [data selection](./Data_Selection.md), [machine unlearning](./Machine_Unlearning.md), fairness correction, and robustness analysis. The same idea also motivates why Hessian-vector products and second-order approximations appear in unlearning papers.

## Connection to Qiao's work

Qiao's unlearning work uses influence-style reasoning in several forms. [Hessian-Free Online Certified Unlearning](./Hessian_Free_Online_Certified_Unlearning.md) relies on efficient updates without explicit Hessian inversion. [Beyond Binary Erasure](./Soft_Weighted_Machine_Unlearning.md) uses weighted influence to turn a deletion problem into a corrective intervention for fairness and robustness. In this wiki, influence functions are therefore not a standalone mathematical curiosity; they are the local sensitivity language behind Qiao's data-operation papers.

## See also

- [Machine Unlearning](./Machine_Unlearning.md)
- [Data Centric ML](./Data_Centric_Machine_Learning.md)
- [Fairness and Robustness](./Fairness_and_Robustness.md)
- [Hessian Free Online Certified Unlearning](./Hessian_Free_Online_Certified_Unlearning.md)

[^influence]: Koh and Liang, "Understanding Black-box Predictions via Influence Functions", ICML 2017, reintroduced classical influence-function ideas for explaining predictions in modern machine-learning models.
