Approximate Regions of Attraction in Learning with Decision-Dependent Distributions

TL;DR

This paper analyzes the stability and convergence of learning algorithms in environments where data distributions react to decisions, providing conditions for the regions of attraction and introducing the concept of performative alignment.

Contribution

It offers a novel analysis of repeated risk minimization as perturbed gradient flows and introduces performative alignment to understand convergence in decision-dependent data settings.

Findings

Provided sufficient conditions for regions of attraction of equilibria.

Characterized the impact of initial conditions on long-term behavior.

Introduced the concept of performative alignment for convergence analysis.

Abstract

As data-driven methods are deployed in real-world settings, the processes that generate the observed data will often react to the decisions of the learner. For example, a data source may have some incentive for the algorithm to provide a particular label (e.g. approve a bank loan), and manipulate their features accordingly. Work in strategic classification and decision-dependent distributions seeks to characterize the closed-loop behavior of deploying learning algorithms by explicitly considering the effect of the classifier on the underlying data distribution. More recently, works in performative prediction seek to classify the closed-loop behavior by considering general properties of the mapping from classifier to data distribution, rather than an explicit form. Building on this notion, we analyze repeated risk minimization as the perturbed trajectories of the gradient flows of performative risk minimization. We consider the case where there may be multiple local minimizers of performative risk, motivated by situations where the initial conditions may have significant impact on the long-term behavior of the system. We provide sufficient conditions to characterize the region of attraction for the various equilibria in this settings. Additionally, we introduce the notion of performative alignment, which provides a geometric condition on the convergence of repeated risk minimization to performative risk minimizers.