Approximate Regions of Attraction in Learning with Decision-Dependent Distributions
This work addresses the challenge of closed-loop behavior in strategic classification for applications like loan approvals, but it is incremental, building on existing performative prediction frameworks.
The paper tackles the problem of learning with decision-dependent distributions, where data sources react to the learner's decisions, by analyzing repeated risk minimization as perturbed trajectories of gradient flows in performative prediction. It provides sufficient conditions to characterize regions of attraction for multiple local minimizers and introduces performative alignment for convergence, but does not include concrete numerical results.
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.