---
type: Research concept
title: Influence Functions
description: Concept page for estimating how training examples affect learned models.
tags:
  - en
  - research
  - concept
  - research-concept
timestamp: '2026-05-05T19:52:29+08:00'
modified: '2026-07-02T20:03:20+08:00'
content_hash: 'sha256:6b90d74a4b42b4f6c28de85fa0e85f92ca8e9988557827d5641e32265639d5f3'
reviewed_at: '2026-07-02T20:03:20+08:00'
review_due: '2026-12-29'
name: Influence Functions
summary: Concept page for estimating how training examples affect learned models.
occupation: Research concept
---
**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]], [[Machine_Unlearning|machine unlearning]], 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]] relies on efficient updates without explicit Hessian inversion. [[Soft_Weighted_Machine_Unlearning|Beyond Binary Erasure]] 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]]
- [[Data_Centric_Machine_Learning|Data Centric ML]]
- [[Fairness_and_Robustness]]
- [[Hessian_Free_Online_Certified_Unlearning]]

[^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.
