STRATEGIC CLASSIFICATION WITH GRAPH NEURAL NETWORKS

Abstract

Strategic classification studies learning in settings where users can modify their features to obtain favorable predictions. Most current works focus on simple classifiers that trigger independent user responses. Here we examine the implications of learning with more elaborate models that break the independence assumption. Motivated by the idea that applications of strategic classification are often social in nature, we focus on graph neural networks, which make use of social relations between users to improve predictions. Using a graph for learning introduces inter-user dependencies in prediction; our key point is that strategic users can exploit these to promote their own goals. As we show through analysis and simulation, this can work either against the system-or for it. Based on this, we propose a differentiable framework for strategically-robust learning of graph-based classifiers. Experiments on several real networked datasets demonstrate the utility of our approach.

1. INTRODUCTION

Machine learning is increasingly being used to inform decisions about humans. But when users of a system stand to gain from certain predictive outcomes, they may be prone to "game" the system by strategically modifying their features (at some cost). The literature on strategic classification (Brückner & Scheffer, 2011; Hardt et al., 2016) studies learning in this setting, with emphasis on how to learn classifiers that are robust to strategic user behavior. The idea that users may respond to a decision rule applies broadly and across many domains, from hiring, admissions, and scholarships to loan approval, insurance, welfare benefits, and medical eligibility (McCrary, 2008; Almond et al., 2010; Camacho & Conover, 2011; Lee & Lemieux, 2010) . This, along with its clean formulation as a learning problem, have made strategic classification the target of much recent interest (Sundaram et al., 2021; Zhang & Conitzer, 2021; Levanon & Rosenfeld, 2021; Ghalme et al., 2021; Jagadeesan et al., 2021; Zrnic et al., 2021; Estornell et al., 2021; Lechner & Urner, 2021; Harris et al., 2021; Levanon & Rosenfeld, 2022; Liu et al., 2022; Ahmadi et al., 2022; Barsotti et al., 2022a) . But despite these advances, most works in strategic classification remain to follow the original problem formulation in assuming independence across users responses. From a technical perspective, this assumption greatly simplifies the learning task, as it allows the classifier to consider each user's response in isolation: user behavior is modeled via a response mapping ∆ h (x) determining how users modify their features x in response to the classifier h, and learning aims to find an h for which y ≈ h(∆ h (x)). Intuitively, a user will modify her features if this 'moves' her across the decision boundary, as long as this is worthwhile (i.e., gains from prediction exceed modification costs). Knowing ∆ h allows the system to anticipate user responses and learn an h that is robust. For a wide range of settings, learning under independent user responses has been shown to be theoretically possible (Hardt et al., 2016; Zhang & Conitzer, 2021; Sundaram et al., 2021) and practically feasible (Levanon & Rosenfeld, 2021; 2022) . Unfortunately, once this assumption of independence is removed-results no longer hold. One reason is that current approaches can safely assume independence because the decision rules they consider induce independence: when predictions inform decisions for each user independently, users have no incentive to account for the behavior of others. This limits the scope of predictive models to include only simple functions of single inputs. In this paper, we aim to extend the literature on strategic classification to support richer learning paradigms that enable inter-dependent user responses, with particular focus on the domain of Graph Neural Networks (GNNs) (Monti et al., 2017; Wang et al., 2019; Bronstein et al., 2017; Hamilton et al., 2017) . Generally, user responses can become dependent through the classifier if predictions for one user rely also on information regarding other users, i.e., if h(x i ) is also a function of other x j . In this way, the affects of a user modifying her features via x j → ∆ h (x j ) can propagate to other users and affect their decisions (since h(x i ) now relies on ∆ h (x j ) rather than x j ). For GNNS, this expresses through their relience on the graph. GNNs take as input a weighted graph whose nodes correspond to featurized examples, and whose edges indicate relations that are believed to be useful for prediction (e.g., if j→i indicates that y i = y j is likely). In our case, nodes represent users, and edges represent social links. The conventional approach is to first embed nodes in a way that depends on their neighbors' features, ϕ i = ϕ(x i ; x nei(i) ), and then perform classification (typically linear) in embedded space, ŷi = sign(w ⊤ ϕ i ). Notice ŷi depends on x i , but also on all other x j ∈ x nei(i) ; hence, in deciding how to respond, user i must also account for the strategic responses of her neighbors j ∈ nei(i). We aim to establish the affects of such dependencies on learning. As a concrete example, consider Lenddofoot_1 , a company that provides credit scoring services to lending institutions. Lenddo specializes in consumer-focused microlending for emerging economies, where many applicants lack credible financial records. To circumvent the need to rely on historical records, Lenddo uses applicants' social connections, which are easier to obtain, as a factor in their scoring system.foot_2 As an algorithmic approach for this task, GNNs are an adequate choice (Gao et al., 2021) . Once loan decisions become dependent on social relations, the incentives for acting strategically change (Wei et al., 2016) . To see how, consider that a user who lies far to the negative side of the decision boundary (and so independently cannot cross) may benefit from the graph if her neighbors "pull" her embedding towards the decision boundary and close enough for her to cross. Conversely, the graph can also suppress strategic behavior, since neighbors can "hold back" nodes and prevent them from crossing. Whether this is helpful to the system or not depends on the true label of the node. This presents a tradeoff: In general, graphs are useful if they are informative of labels in a way that complements features; the many success stories of GNNs suggest that this is often the case (Zhou et al., 2020) . But even if this holds sans strategic behavior-once introduced, graphs inadvertently create dependencies through user representations, which strategic users can exploit. Graphs therefore hold the potential to benefit the system, but also its users. Here we study the natural question: who does the graph help more? Through analysis and experimentation, we show that learning in a way that neglects to account for strategic behavior not only jeopardizes performance, but becomes worse as reliance on the graph increases. In this sense, the graph becomes a vulnerability which users can utilize for their needs, turning it from an asset to the system-to a potential threat. As a solution, we propose a practical approach to learning GNNs in strategic environments. We show that for a key neural architecture (SGC; Wu et al. (2019) ) and certain cost functions, graph-dependent user responses can be expressed as a 'projection-like' operator. This operator admits a simple and differentiable closed form; with additional smoothing, this allows us to implement responses as a neural layer, and learn robust predictors h using gradient methods. Experiments on synthetic and real data (with simulated responses) demonstrate that our approach not only effectively accounts for strategic behavior, but in some cases, can harness the efforts of self-interested users to promote the system's goals. Our code is publicly available at: http://github.com/StrategicGNNs/Code. 2013)), the literature on strategic classification has been growing at a rapid pace. Various aspects of learning have been studied, including: generalization behavior (Zhang & Conitzer, 2021; Sundaram et al., 2021; Ghalme et al., 2021 ), algorithmic hardness (Hardt et al., 2016) , practical optimization methods (Levanon & Rosenfeld, 2021; 2022), and societal implications (Milli et al., 2019; Hu et al., 2019; Chen et al., 2020; Levanon & Rosenfeld, 2021) . Some efforts have been made to extend beyond the conventional user models, e.g., by adding noise (Jagadeesan et al., 2021) , relying on partial information (Ghalme



* Equal contribution, alphabetical order http://lenddoefl.com; see also http://www.wired.com/2014/05/lenddo-facebook/. For a discussion on ethics, see final section. For similar initiatives, see https://en.wikipedia.org/wiki/Lenddo.



RELATED WORK Strategic classification. Since its introduction in Hardt et al. (2016) (and based on earlier formulations in Brückner & Scheffer (2009); Brückner et al. (2012); Großhans et al. (

