Strategyproof linear regression in high dimensions: an overview
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paper_title: Strategyproof Linear Regression in High Dimensions

paper_content:

This paper is part of an emerging line of work at the intersection of machine learning and mechanism design, which aims to avoid noise in training data by correctly aligning the incentives of data sources. Specifically, we focus on the ubiquitous problem of linear regression, where strategyproof mechanisms have previously been identified in two dimensions. In our setting, agents have single-peaked preferences and can manipulate only their response variables. Our main contribution is the discovery of a family of group strategyproof linear regression mechanisms in any number of dimensions, which we call generalized resistant hyperplane mechanisms. The game-theoretic properties of these mechanisms -- and, in fact, their very existence -- are established through a connection to a discrete version of the Ham Sandwich Theorem.
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paper_title: Strategic Classification from Revealed Preferences

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We study an online linear classification problem, in which the data is generated by strategic agents who manipulate their features in an effort to change the classification outcome. In rounds, the learner deploys a classifier, and an adversarially chosen agent arrives, possibly manipulating her features to optimally respond to the learner. The learner has no knowledge of the agents' utility functions or"real"features, which may vary widely across agents. Instead, the learner is only able to observe their"revealed preferences"--- i.e. the actual manipulated feature vectors they provide. For a broad family of agent cost functions, we give a computationally efficient learning algorithm that is able to obtain diminishing"Stackelberg regret"--- a form of policy regret that guarantees that the learner is obtaining loss nearly as small as that of the best classifier in hindsight, even allowing for the fact that agents will best-respond differently to the optimal classifier.
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paper_title: Optimum Statistical Estimation with Strategic Data Sources

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We propose an optimum mechanism for providing monetary incentives to the data sources of a statistical estimator such as linear regression, so that high quality data is provided at low cost, in the sense that the sum of payments and estimation error is minimized. The mechanism applies to a broad range of estimators, including linear and polynomial regression, kernel regression, and, under some additional assumptions, ridge regression. It also generalizes to several objectives, including minimizing estimation error subject to budget constraints. Besides our concrete results for regression problems, we contribute a mechanism design framework through which to design and analyze statistical estimators whose examples are supplied by workers with cost for labeling said examples.
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paper_title: Truthful Linear Regression

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We consider the problem of fitting a linear model to data held by individuals who are concerned about their privacy. Incentivizing most players to truthfully report their data to the analyst constrains our design to mechanisms that provide a privacy guarantee to the participants; we use differential privacy to model individuals' privacy losses. This immediately poses a problem, as differentially private computation of a linear model necessarily produces a biased estimation, and existing approaches to design mechanisms to elicit data from privacy-sensitive individuals do not generalize well to biased estimators. We overcome this challenge through an appropriate design of the computation and payment scheme.
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paper_title: Incentive compatible regression learning

paper_content:

We initiate the study of incentives in a general machine learning framework. We focus on a game-theoretic regression learning setting where private information is elicited from multiple agents with different, possibly conflicting, views on how to label the points of an input space. This conflict potentially gives rise to untruthfulness on the part of the agents. In the restricted but important case when every agent cares about a single point, and under mild assumptions, we show that agents are motivated to tell the truth. In a more general setting, we study the power and limitations of mechanisms without payments. We finally establish that, in the general setting, the VCG mechanism goes a long way in guaranteeing truthfulness and economic efficiency.
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paper_title: Strategy-Proof Estimators for Simple Regression

paper_content:

Abstract This paper introduces a whole class of estimators (clockwise repeated median estimators or CRM) for the simple regression model that are immune to strategic manipulation by the agents generating the data. We find that some well-known robust estimators proposed in the literature like the resistant line method are included in our family. Finally, we also undertake a Monte Carlo study to compare the distribution of some estimators that are robust to data manipulation with the OLS estimators under different scenarios.
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paper_title: Algorithms for strategyproof classification

paper_content:

The strategyproof classification problem deals with a setting where a decision maker must classify a set of input points with binary labels, while minimizing the expected error. The labels of the input points are reported by self-interested agents, who might lie in order to obtain a classifier that more closely matches their own labels, thereby creating a bias in the data; this motivates the design of truthful mechanisms that discourage false reports. In this paper we give strategyproof mechanisms for the classification problem in two restricted settings: (i) there are only two classifiers, and (ii) all agents are interested in a shared set of input points. We show that these plausible assumptions lead to strong positive results. In particular, we demonstrate that variations of a random dictator mechanism, that are truthful, can guarantee approximately optimal outcomes with respect to any family of classifiers. Moreover, these results are tight in the sense that they match the best possible approximation ratio that can be guaranteed by any truthful mechanism. We further show how our mechanisms can be used for learning classifiers from sampled data, and provide PAC-style generalization bounds on their expected error. Interestingly, our results can be applied to problems in the context of various fields beyond classification, including facility location and judgment aggregation.
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paper_title: Strategic Classification

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Machine learning relies on the assumption that unseen test instances of a classification problem follow the same distribution as observed training data. However, this principle can break down when machine learning is used to make important decisions about the welfare (employment, education, health) of strategic individuals. Knowing information about the classifier, such individuals may manipulate their attributes in order to obtain a better classification outcome. As a result of this behavior---often referred to as gaming---the performance of the classifier may deteriorate sharply. Indeed, gaming is a well-known obstacle for using machine learning methods in practice; in financial policy-making, the problem is widely known as Goodhart's law. In this paper, we formalize the problem, and pursue algorithms for learning classifiers that are robust to gaming. We model classification as a sequential game between a player named"Jury"and a player named"Contestant."Jury designs a classifier, and Contestant receives an input to the classifier, which he may change at some cost. Jury's goal is to achieve high classification accuracy with respect to Contestant's original input and some underlying target classification function. Contestant's goal is to achieve a favorable classification outcome while taking into account the cost of achieving it. For a natural class of cost functions, we obtain computationally efficient learning algorithms which are near-optimal. Surprisingly, our algorithms are efficient even on concept classes that are computationally hard to learn. For general cost functions, designing an approximately optimal strategy-proof classifier, for inverse-polynomial approximation, is NP-hard.
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paper_title: The Resistant Line and Related Regression Methods

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Abstract In a bivariate (x, y) scatterplot the three-group resistant line is that line for which the median residual in each outer third of the data (ordered on x) is zero. It was proposed by Tukey as an exploratory method resistant to outliers in x and y and suited to hand calculation. A dual plot representation of the procedure yields a fast, convergent algorithm, nonparametric confidence intervals for the slope, consistency, influence function, and asymptotic normality results. Monte Carlo results show that small sample efficiencies exceed their asymptotic values in important cases. Both breakdown and efficiency are adequate for exploratory work. Replacing medians by other M-estimators of location increases efficiency substantially without affecting breakdown or computational complexity. The method thus combines bounded influence and acceptable efficiency with conceptual and computational simplicity.
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paper_title: Strategy-Proof Estimators for Simple Regression

paper_content:

Abstract This paper introduces a whole class of estimators (clockwise repeated median estimators or CRM) for the simple regression model that are immune to strategic manipulation by the agents generating the data. We find that some well-known robust estimators proposed in the literature like the resistant line method are included in our family. Finally, we also undertake a Monte Carlo study to compare the distribution of some estimators that are robust to data manipulation with the OLS estimators under different scenarios.
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paper_title: Generalized Ham-Sandwich Cuts

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Barany, Hubard, and Jeronimo recently showed that for given well-separated convex bodies S 1,…,S d in R d and constants β i ∈[0,1], there exists a unique hyperplane h with the property that Vol (h +∩S i )=β i ⋅Vol (S i ); h + is the closed positive transversal halfspace of h, and h is a “generalized ham-sandwich cut.” We give a discrete analogue for a set S of n points in R d which are partitioned into a family S=P 1∪⋅⋅⋅∪P d of well-separated sets and are in weak general position. The combinatorial proof inspires an O(n(log n)d−3) algorithm which, given positive integers a i ≤|P i |, finds the unique hyperplane h incident with a point in each P i and having |h +∩P i |=a i . Finally we show two other consequences of the direct combinatorial proof: the first is a stronger result, namely that in the discrete case, the conditions assuring existence and uniqueness of generalized cuts are also necessary; the second is an alternative and simpler proof of the theorem in Barany et al., and in addition, we strengthen the result via a partial converse.
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paper_title: Uneven Splitting of Ham Sandwiches

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Let m_1,...,m_n be continuous probability measures on R^n and a_1,...,a_n in [0,1]. When does there exist an oriented hyperplane H such that the positive half-space H^+ has m_i(H^+)=a_i for all i in [n]? It is well known that such a hyperplane does not exist in general. The famous ham sandwich theorem states that if a_i=1/2 for all i, then such a hyperplane always exists. In this paper we give sufficient criteria for the existence of H for general a_i in [0,1]. Let f_1,...,f_n:S^{n-1}->R^n denote auxiliary functions with the property that for all i the unique hyperplane H_i with normal v that contains the point f_i(v) has m_i(H_i^+)=a_i. Our main result is that if Im(f_1),...,Im(f_n) are bounded and can be separated by hyperplanes, then there exists a hyperplane H with m_i(H^+)=a_i for all i. This gives rise to several corollaries, for instance if the supports of m_1,...,m_n are bounded and can be separated by hyperplanes, then H exists for any choice of a_1,...,a_n in [0,1]. We also obtain results that can be applied if the supports of m_1,...,m_n overlap.
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paper_title: Incentive compatible regression learning

paper_content:

We initiate the study of incentives in a general machine learning framework. We focus on a game-theoretic regression learning setting where private information is elicited from multiple agents with different, possibly conflicting, views on how to label the points of an input space. This conflict potentially gives rise to untruthfulness on the part of the agents. In the restricted but important case when every agent cares about a single point, and under mild assumptions, we show that agents are motivated to tell the truth. In a more general setting, we study the power and limitations of mechanisms without payments. We finally establish that, in the general setting, the VCG mechanism goes a long way in guaranteeing truthfulness and economic efficiency.
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paper_title: Slicing convex sets and measures by a hyperplane

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We generalize the ham sandwich theorem for the case of well separated measures. Given convex bodies $K_1,...,K_d$ in $\mathbb{R_d}$ and numbers $\alpha_1,...,\alpha_d \in [0, 1]$, we give a sufficient condition for existence and uniqueness of an (oriented)halfspace H with Vol($H \cap K_i$)= $\alpha_i \dot$ Vol$K_i$ for every i. The result is extended from convex bodies to measures.
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paper_title: On strategy-proofness and single peakedness

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This paper investigates one of the possible weakening of the (too demanding) assumptions of the Gibbard-Satterthwaite theorem. Namely we deal with a class of voting schemes where at the same time the domain of possible preference preordering of any agent is limited to single-peaked preferences, and the message that this agent sends to the central authority is simply its ‘peak’ — his best preferred alternative. In this context we have shown that strategic considerations justify the central role given to the Condorcet procedure which amounts to elect the ‘median’ peak: namely all strategy-proof anonymous and efficient voting schemes can be derived from the Condorcet procedure by simply adding some fixed ballots to the agent's ballots (with the only restriction that the number of fixed ballots is strictly less than the number of agents).
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paper_title: Truthful univariate estimators

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We revisit the classic problem of estimating the population mean of an unknown single-dimensional distribution from samples, taking a game-theoretic viewpoint. In our setting, samples are supplied by strategic agents, who wish to pull the estimate as close as possible to their own value. In this setting, the sample mean gives rise to manipulation opportunities, whereas the sample median does not. Our key question is whether the sample median is the best (in terms of mean squared error) truthful estimator of the population mean. We show that when the underlying distribution is symmetric, there are truthful estimators that dominate the median. Our main result is a characterization of worst-case optimal truthful estimators, which provably outperform the median, for possibly asymmetric distributions with bounded support.
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paper_title: On strategy-proofness and single peakedness

paper_content:

This paper investigates one of the possible weakening of the (too demanding) assumptions of the Gibbard-Satterthwaite theorem. Namely we deal with a class of voting schemes where at the same time the domain of possible preference preordering of any agent is limited to single-peaked preferences, and the message that this agent sends to the central authority is simply its ‘peak’ — his best preferred alternative. In this context we have shown that strategic considerations justify the central role given to the Condorcet procedure which amounts to elect the ‘median’ peak: namely all strategy-proof anonymous and efficient voting schemes can be derived from the Condorcet procedure by simply adding some fixed ballots to the agent's ballots (with the only restriction that the number of fixed ballots is strictly less than the number of agents).
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paper_title: Learning from untrusted data

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The vast majority of theoretical results in machine learning and statistics assume that the training data is a reliable reflection of the phenomena to be learned. Similarly, most learning techniques used in practice are brittle to the presence of large amounts of biased or malicious data. Motivated by this, we consider two frameworks for studying estimation, learning, and optimization in the presence of significant fractions of arbitrary data. The first framework, list-decodable learning, asks whether it is possible to return a list of answers such that at least one is accurate. For example, given a dataset of n points for which an unknown subset of αn points are drawn from a distribution of interest, and no assumptions are made about the remaining (1 - α)n points, is it possible to return a list of poly(1/α) answers? The second framework, which we term the semi-verified model, asks whether a small dataset of trusted data (drawn from the distribution in question) can be used to extract accurate information from a much larger but untrusted dataset (of which only an α-fraction is drawn from the distribution). We show strong positive results in both settings, and provide an algorithm for robust learning in a very general stochastic optimization setting. This result has immediate implications for robustly estimating the mean of distributions with bounded second moments, robustly learning mixtures of such distributions, and robustly finding planted partitions in random graphs in which significant portions of the graph have been perturbed by an adversary.
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paper_title: Incentive compatible regression learning

paper_content:

We initiate the study of incentives in a general machine learning framework. We focus on a game-theoretic regression learning setting where private information is elicited from multiple agents with different, possibly conflicting, views on how to label the points of an input space. This conflict potentially gives rise to untruthfulness on the part of the agents. In the restricted but important case when every agent cares about a single point, and under mild assumptions, we show that agents are motivated to tell the truth. In a more general setting, we study the power and limitations of mechanisms without payments. We finally establish that, in the general setting, the VCG mechanism goes a long way in guaranteeing truthfulness and economic efficiency.
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paper_title: Towards a theory of incentives in machine learning

paper_content:

The connection between machine learning and economics is, I feel, quite natural. There is a growing body of work that lies at the intersection of the two fields, but most of this work focuses on applying machine learning paradigms to economic problems. Examples include prediction of consumer behavior [Kalai 2003; Beigman and Vohra 2006], automated design of voting rules [Procaccia et al. 2007; Procaccia et al. 2008], and reduction of mechanism design problems to standard algorithmic questions [Balcan et al. 2005]. Nevertheless, there are preciously few papers investigating the incentives that, in some settings, govern the learning process itself (see, e.g., Perote and PerotePena [2004], Dalvi et al. [2004]); none of them do so in a general machine learning framework. Where, indeed, do strategic considerations come into play in the learning world? In general, a machine learning algorithm receives a (small but hopefully representative) training set consisting of points sampled from an input space and labeled according to some target function; the algorithm outputs a hypothesis that is presumably close to the target function. For simplicity, consider a basic setup where n selfish agents control n disjoint subsets of the input space. The label of each point in the training set is reported by the agent that controls it (whereas the identities of the points controlled by an agent are common knowledge). Crucially, each agent is interested only in the accuracy of the generated hypothesis on its own part of the input space. An agent can influence the outcome of the learning process by misreporting the labels of the points under its control. The above strategic setup seems relevant, for instance, in the context of decisions taken by a central bank, such as the European Central Bank (ECB). The governing council of the central bank collects information from national bankers (the agents), who in turn gather data on different economic parameters by means of their own institutions. The central bank decides on an economic policy (hypothesis) by using, say, regression learning on the examples provided by the national bankers. The national bankers may thus be motivated to manipulate their portion of the data
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Title: Strategyproof linear regression in high dimensions: an overview

Section 1: INTRODUCTION
Description 1: Provide an introduction to the dependency of machine learning models on training data quality, the study of robust algorithms to noise, and the concept of strategyproof learning algorithms in the presence of strategic noise.

Section 2: MODEL
Description 2: Present the formal model used in the study, including definitions, notations, and the problem setup involving agents controlling datapoints with private and public components.

Section 3: STRATEGYPROOF MECHANISMS
Description 3: Describe the family of strategyproof mechanisms, starting from the CRM mechanisms for simple linear regression and extending to Generalized Resistant Line (GRL) mechanisms and Generalized Resistant Hyperplane (GRH) mechanisms in higher dimensions.

Section 4: Generalizing in two dimensions
Description 4: Detail the process and mathematical constructs used to generalize the CRM family to GRL mechanisms in two dimensions, explaining how these mechanisms ensure strategyproofness and statistical efficiency.

Section 5: Generalizing to higher dimensions
Description 5: Explain the generalization from GRL mechanisms to GRH mechanisms in higher dimensions using concepts such as the Ham Sandwich theorem and separability conditions.

Section 6: Strategyproofness versus group-strategyproofness
Description 6: Discuss the distinction between strategyproof and group-strategyproof mechanisms, including examples and conditions under which strategyproof mechanisms may not be group-strategyproof.

Section 7: Efficiency versus strategyproofness
Description 7: Analyze the trade-off between statistical efficiency and strategyproofness, illustrating why it is challenging to design mechanisms that are both highly efficient and strategyproof.

Section 8: CONCLUSION AND OPEN QUESTIONS
Description 8: Summarize the findings and suggest future research directions to further explore and characterize strategyproof mechanisms for linear regression, considering various possible generalizations and practical applications.