The Effect of Strategic Noise in Linear Regression

TL;DR

This paper investigates the strategic manipulation in linear regression algorithms, analyzing equilibrium outcomes and their quality, revealing connections between strategyproof and non-strategyproof methods.

Contribution

It characterizes the unique Nash equilibrium in non-strategyproof linear regression algorithms and links it to strategyproof algorithms, providing insights into equilibrium structure and efficiency.

Findings

Unique pure Nash equilibrium exists for bounded manipulations.

Connection established between strategyproof algorithms and Nash equilibria.

Analysis of the price of anarchy for equilibrium outcomes.

Abstract

We build on an emerging line of work which studies strategic manipulations in training data provided to machine learning algorithms. Specifically, we focus on the ubiquitous task of linear regression. Prior work focused on the design of strategyproof algorithms, which aim to prevent such manipulations altogether by aligning the incentives of data sources. However, algorithms used in practice are often not strategyproof, which induces a strategic game among the agents. We focus on a broad class of non-strategyproof algorithms for linear regression, namely $\ell_p$ norm minimization ($p > 1$) with convex regularization. We show that when manipulations are bounded, every algorithm in this class admits a unique pure Nash equilibrium outcome. We also shed light on the structure of this equilibrium by uncovering a surprising connection between strategyproof algorithms and pure Nash equilibria of non-strategyproof algorithms in a broader setting, which may be of independent interest. Finally, we analyze the quality of equilibria under these algorithms in terms of the price of anarchy.