The Active and Noise-Tolerant Strategic Perceptron

Thank you for your thoughtful review. We address your suggestions and questions below.

1. Clarifying the iterative setting.

Thank you for this suggestion. We will add a short summary in Section 3.2 explaining, for each iteration, which example the learner selects, what is observed, what the true example is, what the agent does, and how the oracle returns the label. This will make the process easier to follow.

2. Assumptions and generalization.

You are correct that our guarantees depend on the assumptions of the utility function and data distribution. These assumptions (linear cost utility and uniform/smooth data distributions) are standard in the active learning and strategic classification literature (e.g., Hardt et al. 2016; Yan & Zhang 2017; Ahmadi et al. 2021) and allow us to obtain tight, exponential label savings. We will add text clarifying why these assumptions are necessary for our analysis and how they can be generalized:

• The uniform distribution assumption can be relaxed to smooth generalizations where density ratios along rays are bounded (as outlined in Section 1.3 and Appendix B).

• Our approach can also handle unknown costs. We will add this to the revised version.

3. Novelty relative to Yan & Zhang (2017).

While our algorithm is inspired by Yan & Zhang (2017), adapting it to the strategic setting is non-trivial. We had to address manipulation in feature space, identify unmanipulated points for reliable updates, and extend the analysis from the unit sphere to the unit ball using new coupling arguments. Furthermore, we do not view the similarity of our algorithm to Yan & Zhang (2017) as a downside; on the contrary, it is both surprising and appealing that nonsignificant changes to the algorithm can yield strong guarantees in the strategic setting. We will make these distinctions clearer in the introduction and Section 3.

Thank you for your constructive review. We address your points below.

Perceived small step over Yan & Zhang (2017).

Our contributions are orthogonal and non-trivial:

1. We adapt active learning guarantees to a setting with strategic manipulation, a key departure from non-strategic analysis.

2. We identify a structural property unique to strategic classification: only negative-classified points are guaranteed unmanipulated and thus safe for updates.

3. We extend the analysis from the unit-sphere to the unit-ball distribution.

4. We provide noise-tolerant guarantees in the strategic setting, addressing an open problem from Ahmadi et al. (2021).

We would like to emphasize that our contribution is primarily conceptual rather than introducing a highly complex technique. We do not view it as a drawback that our results are obtained through relatively simple modifications of existing algorithms and hence following a similar proof structure; on the contrary, it is both surprising and appealing that small, clean changes can transform strong guarantees to the strategic setting.

Writing and structure.

We will: (i) present Algorithms 1 and 2 earlier, before Theorem 2; (ii) include proof sketches in the main text; (iii) bring the Appendix C discussion on relaxing the initial-halfspace assumption into the main text; and (iv) better separate prior work from our contributions.

Assumptions.

We acknowledge that the uniform distribution, bounded noise, and initial halfspace assumptions are strong. These are standard in active learning theory. We now clarify that:

• our arguments extend to smooth non-isotropic distributions (Section 1.3, Appendix B);

• the initial halfspace assumption can be removed using a bootstrapping strategy (Appendix C, to be referenced in the main text);

• the noise tolerance we achieve is essentially optimal for this class of algorithms.

Margin-based connection.

We believe our techniques work for margin-based active variants of the perceptron algorithm. In fact, we first verified this for the variant from Dasgupta et al. (2005), but then we realized that if we use Yan and Zhang (2017)’s algorithm, our final algorithm will be cleaner and the proof will be much simpler. We will explicitly highlight how the margin-based analysis is adapted to the strategic setting.

Noise parameter condition.

Thank you for the suggestion. We will rewrite the noise condition more intuitively: if $\nu < \nu^\star(\epsilon, d)$ the algorithm succeeds; if $\nu > \nu^\star$, no active learner can guarantee error $\epsilon$ (Hanneke & Yang 2015).

Uniform on the surface.

In the case where points are uniformly distributed on the surface of the unit sphere, we can show that there is a one-to-one mapping between each observed point and its original point. To illustrate, consider the two-dimensional case where points lie on the circumference of a circle and the linear classifier is positioned at a distance $1/c$ from the origin. Half of the points (those on one semicircle) are too far to manipulate and will be classified as negative. A portion of the remaining points (on the opposite arc) already lie on the positive side of the classifier and are classified as positive without manipulation. The rest will manipulate to exactly reach the classification boundary; these points form two symmetric line segments, and their observed positions on the boundary correspond directly to their projections from the original points. This projection establishes a one-to-one mapping, allowing the original points to be recovered exactly.

In contrast, when points are uniformly distributed inside the unit ball, this projection is no longer one-to-one but instead many-to-one. We will include additional details and formal explanations of this distinction in the revised version.

Thank you for your review. Below we respond to your questions and comments.

Q1. Practical motivation and realism of assumptions.

Our framework applies to domains where strategic manipulation and costly labeling coexist, such as hiring and admissions (applicants modify resumes, labels require expert review), financial systems (credit scores can be manipulated, labels such as default events are expensive), and platform moderation (users adapt behavior, human labeling is costly). The cost parameter $c$ models known manipulation penalties (e.g., exam fees, regulation costs) and is standard in prior work (Hardt et al. 2016; Ahmadi et al. 2021). Learners can estimate $c$ from observed shifts or domain knowledge. While we assume isotropy for theoretical guarantees, our arguments extend to smooth non-isotropic distributions where density ratios along rays are bounded. We will clarify this generalization. Also, please see our response to Q4 that clarifies the algorithm can be made robust to unknown $c$ with a simple modification.

Q2. Noise condition compared to Ahmadi et al. (2021).

Our admissible noise level is

matching the best-known active Perceptron bounds (Yan & Zhang 2017). This is strictly stronger than Ahmadi et al. (2021), which fails for even a single noisy data-point in the data stream. Our algorithm thus achieves exponential label savings with noise tolerance, closing an open gap.

Q3. Notation in Theorem 2.

The $\Theta$ bounds merely drop the constant multipliers. We will clarify this explicitly.

Q4. Robustness to mismatched $c$.

Our algorithm uses $c$ as an input; however, it is relatively straightforward to determine whether $c$ is being overestimated or underestimated, and by how much, by examining unlabeled samples. In the case of a uniform distribution (or smoothed generalizations), if we are overestimating $c$, we will observe data points in a margin region around the origin on the positive side. This occurs because the classification threshold is set too conservatively, preventing many points that could otherwise move across the boundary from doing so. Conversely, if we are underestimating $c$, the margin region around the origin on the negative side will be essentially empty, as manipulated points can more easily cross the boundary. By conditioning on the observation of points in these margin regions, one can infer whether $c$ is being over- or underestimated and adjust it accordingly.

Q5. Uniform-ball assumption.

The uniform-ball setting is strictly harder than the unit-sphere case because true features have varying norms, and we cannot exactly recover unmanipulated points. If we assume points are on the surface, in our setting we are able to fully recover the original point. We will clarify why this extension matters.

Q6. Empirical validation.

We appreciate the suggestion. Our focus is a theoretical contribution, but empirical validation is a valuable future direction.

Novelty.

Although we build on Yan & Zhang (2017) and Ahmadi et al. (2021), our conceptual contributions are substantial:

1. first to bring active learning to strategic classification,

2. identifying how to learn solely from unmanipulated negative-classified points,

3. extending the analysis from the sphere to the ball,

4. closing the noise-tolerance gap for the strategic setting.

We will make these points clearer in the revised version.

We would like to emphasize that our contribution is primarily conceptual rather than based on introducing a highly complex technique. We do not view it as a limitation that our results are obtained through relatively simple modifications of existing algorithms; on the contrary, it is both surprising and appealing that small, clean adaptations can yield strong guarantees in the strategic setting. The simplicity of our algorithm is a natural consequence of selecting the most compatible active learning method from several possible alternatives in the literature. Our techniques can also be applied to other active variants of the Perceptron algorithm (e.g., Dasgupta et al., 2005). We chose Yan and Zhang (2017) as our baseline to highlight how strong guarantees in the strategic case can be achieved with small but carefully designed adaptations.