A Unified Theory of Random Projection for Influence Functions

TL;DR

This paper develops a unified theoretical framework for understanding how random projection techniques preserve influence functions in high-dimensional models, accounting for regularization, factorization, and out-of-range effects.

Contribution

It provides the first comprehensive theory explaining when and how random projections preserve influence functions, including effects of regularization and structured curvature approximations.

Findings

Exact influence preservation requires injective sketches on the curvature range.

Regularization modifies the sketching barrier based on effective dimension.

Factorized curvature sketches maintain guarantees despite row correlations.

Abstract

Influence functions and related data attribution scores take the form of $g^{\top}F^{-1}g^{\prime}$, where $F\succeq 0$ is a curvature operator. In modern overparametrized models, forming or inverting $F\in\mathbb{R}^{d\times d}$ is prohibitive, motivating scalable influence computation via random projection with a sketch $P \in \mathbb{R}^{m\times d}$. This practice is commonly justified via the Johnson--Lindenstrauss (JL) lemma, which ensures approximate preservation of Euclidean geometry for a fixed dataset. However, JL does not address how sketching behaves under inversion. Furthermore, there is no existing theory that explains how sketching interacts with other widely-used techniques, such as ridge regularization and structured curvature approximations. We develop a unified theory characterizing when projection provably preserves influence functions. When $g,g^{\prime}\in\text{range}(F)$, we show that: 1) Unregularized projection: exact preservation holds iff $P$ is injective on $\text{range}(F)$, which necessitates $m\geq \text{rank}(F)$; 2) Regularized projection: ridge regularization fundamentally alters the sketching barrier, with approximation guarantees governed by the effective dimension of $F$ at the regularization scale; 3) Factorized influence: for Kronecker-factored curvatures $F=A\otimes E$, the guarantees continue to hold for decoupled sketches $P=P_A\otimes P_E$, even though such sketches exhibit row correlations that violate i.i.d. assumptions. Beyond this range-restricted setting, we analyze out-of-range test gradients and quantify a leakage term that arises when test gradients have components in $\ker(F)$. This yields guarantees for influence queries on general test points. Overall, this work develops a novel theory that characterizes when projection provably preserves influence and provides principled guidance for choosing the sketch size in practice.