Title: Strategic Classification with Non-Linear Classifiers
Venue: NeurIPS 2025
Accepted: yes
Keywords: strategic classification
Abstract: In strategic classification, the standard supervised learning setting is extended to support the notion of strategic user behavior in the form of costly feature manipulations made in response to a classifier. While standard learning supports a broad range of model classes, the study of strategic classification has, so far, been dedicated mostly to linear classifiers. This work aims to expand the horizon by exploring how strategic behavior manifests under non-linear classifiers and what this implies for learning. We take a bottom-up approach showing how non-linearity affects decision boundary points, classifier expressivity, and model class complexity. Our results show how, unlike the linear case, strategic behavior may either increase or decrease effective class complexity, and that the complexity decrease may be arbitrarily large. Another key finding is that universal approximators (e.g., neural nets) are no longer universal once the environment is strategic. We demonstrate empirically how this can create performance gaps even on an unrestricted model class.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies strategic classification in the offline geometric setting where the true underlying classifier is nonlinear. The authors take a \"bottom up\" approach that first considers changes to individual points on the boundary, then aggregate them to the effective classifier under manipulation and subsequently the effective model class. For individual points $x$, they discussed four cases of the mapping from $x$ to the set of points that manipulate to $x$ at maximum cost: one-to-one, wipeout, expansion and collision. For the effective classifiers, they identified two mechanisms that can limit their expressivity --- curvature and containment --- and derive some conditions about effective classifiers that cannot exist. The authors then move on to the model class level and compares the strategic VC dimension and the original one, and show that universal approximators may fail to capture the samples after manipulation.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\nThe authors systematically study strategic classification with nonlinear classifiers in the continuous geometric setting where agents can move in a bounded radius ball. In contrast, existing work either studies non-linear classifiers in a discrete setting via manipulation graphs, or exclusively focus on linear classifiers in the geometric spaces. The paper is clearly written, nicely builds up intuition via a bottom-up approach, and comprehensively study the structure and complexity of this problem.\n\nWeaknesses:\n\n1. Limited technical depth. While the paper presents many results, most of them feel like simple observations that have been written down formally, rather than deep technical insights. Since the paper takes a bottom-up approach, the results about individual points and classifiers are mostly just listing observations and examples --- this does help build intuition but doesn't really tell us anything new. The most interesting part is the model class-level results in Section 6 (which I consider to be the meat of this paper) where they quantitatively study the complexity and expressivity of the effective model class versus the original ones. But this analysis is incomplete since they only make a conjecture about the VC gap in the general case, but did not provide a satisfactory answer. \n\n2. The authors study both VC dimension/sample complexity and universality of model classes, but these results feel mostly pessimistic and do not provide clear guidance on how to select model classes or design learning algorithms in practice. The experimental results in section 7.2 are also somewhat unsurprising: showing improved accuracy with bigger separation between positive and negative examples and smaller manipulation radius, with H^{lin} being a lower bound. Overall, the practical implications and takeaways from this paper is a bit unclear.\n\nQUESTIONS:\nCould you comment on how the results in section 6 help the learner to choose model classes in practice?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-30"}

comments: SUMMARY:
This paper studies strategic classification, where agents can intentionally modify their input features (while incurring an L-2 cost) to receive favorable outcomes from a classifier (e.g. in credit approval). This paper explores the implications of non-linear classifiers in such a setting.

Thanks to the assumption that the L-2 cost is the same for all agents, Sec 4, 5 and 6.2 effectively reduces the problem to study the behavior of offset curves/surfaces in the context of classification.

Sec 6.1 studies the behavior of offset curves in the language of VC dimension.

Edit: Author's response to reviewer YJeu has made me feel more positive about the paper. I'm raising my score to a 3.

STRENGTHS AND WEAKNESSES:
Pros:

The paper is extremely well-written, with intuitions and/or illustrations provided for almost every proposition and proof. The analysis is well-structured and engaging.

Cons:

The paper makes connection to the study of offset curves, yet largely dismiss it as it "does not convey direct implications for learning". However, I feel that the paper could have used more direct connection to and terminology from offset curves, e.g. self-intersections, tangent vectors, instead of reinventing terminology. It is a connection that should be highlighted and celebrated.

Every claim that is made up to (and include) Theorem 1 is somewhat immediate once the reader understand the basic mechanism of offset curves.

QUESTIONS:
See weaknesses

LIMITATIONS:
Yes
TITLE: Official Review
CLARITY: 3
IMPACT: 2
ORIGINALITY: 2
DATE: 2025-07-03

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper investigates strategic classification under non-linear models, extending prior work that largely focuses on linear classifiers. In strategic settings, users manipulate their input features to receive favorable outcomes, which fundamentally alters the learning problem. The authors make a bottom-up analytical and empirical approach to study how non-linear decision boundaries, classifier expressivity, and model class complexity behave in strategic environments. An interesting insight is that universal approximators like neural networks lose their universality when faced with strategic agents. The authors support their analysis with theoretical characterizations.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n\n1. Strategic classification is relevant in high-stakes applications, and extending its understanding beyond linear classifiers is both novel and necessary.\n\n2. The insight that universal approximators are no longer universal under strategic manipulation is intellectually significant and well-motivated.\n\n3. The paper rigorously analyzes the effects of strategic behavior at three levels: individual inputs, decision boundaries, and model classes.\n\n4. The exposition is clear and logical. Definitions, examples, and motivations are carefully laid out.\n\n**Weaknesses**\n\n\n1. The analysis focuses primarily on $l_2$ cost; broader conclusions might require considering other realistic cost structures (e.g., non-Euclidean, asymmetric, feature-dependent).\n\n2. While the claim that neural networks are no longer universal is well-motivated and empirically observed, a more formal characterization  would strengthen the main claim.\n\n3. The paper focuses on theoretical behavior but gives limited discussion on how this insight could influence real-world tasks in strategic settings.\n\nQUESTIONS:\n1. Could the non-universality of neural networks under strategic manipulation be quantified more rigorously?\n\n2. How sensitive are your findings to the assumption of optimal best-response from agents? Would bounded rationality change the picture?\n\nLIMITATIONS:\nSee weakness", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-05"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThey study Strategic Classification (SC) in a non-linear setting. Unlike SC in a linear setting, when the decision boundary is non-linear, datapoints move in different directions, and hence the effective decision boundary would get distorted. Their results are in 3 different categories: 1)points, 2)classifiers 3)model classes in a non-linear setting. \n\nSection 4 describes what would happen to points, how a point on h gets mapped to corresponding points in the effective decision boundary h_{\\delta}, cases 1-4 describe 4 possibilities in this case, there might be a 1-1 mapping, wipeout, expansion, or collission.\n\nSection 5 demonstrates what would happen to individual classifiers, in terms of their curvature and containment (number of positive regions in some sense).\n\nSection 6 explains what would happen to the VC dimension of the model classes; they might increase or decrease.\n\nSTRENGTHS AND WEAKNESSES:\nAlthough it is very important to study SC in non-linear setting, my biggest complain about this work is that it is mostly structural results, it does not give any algorithms for learning in non-linear settings.\n\nQUESTIONS:\nEq 1, shouldn't it be h(x')?\n\nLines 114 - 117, confusing since you are using x as a modified point in line 114 and then as an original point in line 117.\n\n\nFigure 2 needs more explanation. I do not understand what all the colors represent. It seems like you are addressing 4 categories, but there\nare 5 figures. It is very hard to understand what is going on.\n\nObs3 for what points x? The ones on the decision boundary?\n\nConfused about the wording of section 6.1. Do you mean that in some cases, the  VC-dimension of the effective classifier sometimes decreases, and sometimes it increases?\n\n\nWhat does line 292 mean?  For what point x?\n\nIn your experiments, how do you compute the effective decision boundary?\n\nLIMITATIONS:\nAlthough it is very important to study SC in non-linear setting, my biggest complain about this work is that it is mostly structural results, it does not give any algorithms for learning in non-linear settings.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-06"}