Regret Minimization with Performative Feedback

In performative prediction, the deployment of a predictive model triggers a shift in the data distribution. As these shifts are typically unknown ahead of time, the learner needs to deploy a model to get feedback about the distribution it induces. We study the problem of finding near-optimal models under performativity while maintaining low regret. On the surface, this problem might seem equivalent to a bandit problem. However, it exhibits a fundamentally richer feedback structure that we refer to as performative feedback: after every deployment, the learner receives samples from the shifted distribution rather than only bandit feedback about the reward. Our main contribution is regret bounds that scale only with the complexity of the distribution shifts and not that of the reward function. The key algorithmic idea is careful exploration of the distribution shifts that informs a novel construction of confidence bounds on the risk of unexplored models. The construction only relies on smoothness of the shifts and does not assume convexity. More broadly, our work establishes a conceptual approach for leveraging tools from the bandits literature for the purpose of regret minimization with performative feedback.

Learning Equilibria in Matching Markets from Bandit Feedback

Large-scale, two-sided matching platforms must find market outcomes that align with user preferences while simultaneously learning these preferences from data. However, since preferences are inherently uncertain during learning, the classical notion of stability (Gale and Shapley, 1962; Shapley and Shubik, 1971) is unattainable in these settings. To bridge this gap, we develop a framework and algorithms for learning stable market outcomes under uncertainty. Our primary setting is matching with transferable utilities, where the platform both matches agents and sets monetary transfers between them. We design an incentive-aware learning objective that captures the distance of a market outcome from equilibrium. Using this objective, we analyze the complexity of learning as a function of preference structure, casting learning as a stochastic multi-armed bandit problem. Algorithmically, we show that "optimism in the face of uncertainty," the principle underlying many bandit algorithms, applies to a primal-dual formulation of matching with transfers and leads to near-optimal regret bounds. Our work takes a first step toward elucidating when and how stable matchings arise in large, data-driven marketplaces.

Alternative Microfoundations for Strategic Classification

When reasoning about strategic behavior in a machine learning context it is tempting to combine standard microfoundations of rational agents with the statistical decision theory underlying classification. In this work, we argue that a direct combination of these standard ingredients leads to brittle solution concepts of limited descriptive and prescriptive value. First, we show that rational agents with perfect information produce discontinuities in the aggregate response to a decision rule that we often do not observe empirically. Second, when any positive fraction of agents is not perfectly strategic, desirable stable points -- where the classifier is optimal for the data it entails -- cease to exist. Third, optimal decision rules under standard microfoundations maximize a measure of negative externality known as social burden within a broad class of possible assumptions about agent behavior. Recognizing these limitations we explore alternatives to standard microfoundations for binary classification. We start by describing a set of desiderata that help navigate the space of possible assumptions about how agents respond to a decision rule. In particular, we analyze a natural constraint on feature manipulations, and discuss properties that are sufficient to guarantee the robust existence of stable points. Building on these insights, we then propose the noisy response model. Inspired by smoothed analysis and empirical observations, noisy response incorporates imperfection in the agent responses, which we show mitigates the limitations of standard microfoundations. Our model retains analytical tractability, leads to more robust insights about stable points, and imposes a lower social burden at optimality.

Inductive Bias of Multi-Channel Linear Convolutional Networks with Bounded Weight Norm

We study the function space characterization of the inductive bias resulting from controlling the $\ell_2$ norm of the weights in linear convolutional networks. We view this in terms of an induced regularizer in the function space given by the minimum norm of weights required to realize a linear function. For two layer linear convolutional networks with $C$ output channels and kernel size $K$, we show the following: (a) If the inputs to the network have a single channel, the induced regularizer for any $K$ is a norm given by a semidefinite program (SDP) that is independent of the number of output channels $C$. We further validate these results through a binary classification task on MNIST. (b) In contrast, for networks with multi-channel inputs, multiple output channels can be necessary to merely realize all matrix-valued linear functions and thus the inductive bias does depend on $C$. Further, for sufficiently large $C$, the induced regularizer for $K=1$ and $K=D$ are the nuclear norm and the $\ell_{2,1}$ group-sparse norm, respectively, of the Fourier coefficients -- both of which promote sparse structures.

Individual Fairness in Pipelines

It is well understood that a system built from individually fair components may not itself be individually fair. In this work, we investigate individual fairness under pipeline composition. Pipelines differ from ordinary sequential or repeated composition in that individuals may drop out at any stage, and classification in subsequent stages may depend on the remaining "cohort" of individuals. As an example, a company might hire a team for a new project and at a later point promote the highest performer on the team. Unlike other repeated classification settings, where the degree of unfairness degrades gracefully over multiple fair steps, the degree of unfairness in pipelines can be arbitrary, even in a pipeline with just two stages. Guided by a panoply of real-world examples, we provide a rigorous framework for evaluating different types of fairness guarantees for pipelines. We show that na\"{i}ve auditing is unable to uncover systematic unfairness and that, in order to ensure fairness, some form of dependence must exist between the design of algorithms at different stages in the pipeline. Finally, we provide constructions that permit flexibility at later stages, meaning that there is no need to lock in the entire pipeline at the time that the early stage is constructed.

Fairness in ad auctions through inverse proportionality

We study the tradeoff between social welfare maximization and fairness in the context of ad auctions. We study an ad auction setting where users arrive in an online fashion, $k$ advertisers submit bids for each user, and the auction assigns a distribution over ads to the user. Following the works of Dwork and Ilvento (2019) and Chawla et al. (2020), our goal is to design a truthful auction that satisfies multiple-task fairness in its outcomes: informally speaking, users that are similar to each other should obtain similar allocations of ads. We develop a new class of allocation algorithms that we call inverse-proportional allocation. These allocation algorithms are truthful, online, and do not explicitly need to know the fairness constraint over the users. In terms of fairness, they guarantee fair outcomes as long as every advertiser's bids are non-discriminatory across users. In terms of social welfare, inverse-proportional allocation achieves a constant factor approximation in social welfare against the optimal (unfair) allocation, independent of the number of advertisers in the system. In this respect, these allocation algorithms greatly surpass the guarantees achieved in previous work; in fact, they achieve the optimal tradeoffs between fairness and social welfare in some contexts. We also extend our results to broader notions of fairness that we call subset fairness.

Individual Fairness in Sponsored Search Auctions

Fairness in advertising is a topic of particular interest in both the computer science and economics literatures, supported by theoretical and empirical observations. We initiate the study of tradeoffs between individual fairness and performance in online advertising, where advertisers place bids on ad slots for each user and the platform must determine which ads to display. Our main focus is to investigate the "cost of fairness": more specifically, whether a fair allocation mechanism can achieve utility close to that of a utility-optimal unfair mechanism. Motivated by practice, we consider both the case of many advertisers in a single category, e.g. sponsored results on a job search website, and ads spanning multiple categories, e.g. personalized display advertising on a social networking site, and show the tradeoffs are inherently different in these settings. We prove lower and upper bounds on the cost of fairness for each of these settings. For the single category setting, we show constraints on the "fairness" of advertisers' bids are necessary to achieve good utility. Moreover, with bid fairness constraints, we construct a mechanism that simultaneously achieves a high utility and a strengthening of typical fairness constraints that we call total variation fairness. For the multiple category setting, we show that fairness relaxations are necessary to achieve good utility. We consider a relaxed definition based on user-specified category preferences that we call user-directed fairness, and we show that with this fairness notion a high utility is achievable. Finally, we show that our mechanisms in the single and multiple category settings compose well, yielding a high utility combined mechanism that satisfies user-directed fairness across categories and conditional total variation fairness within categories.

Understanding Sparse JL for Feature Hashing

Feature hashing and more general projection schemes are commonly used in machine learning to reduce the dimensionality of feature vectors. The goal is to efficiently project a high-dimensional feature vector living in $\mathbb{R}^n$ into a lower-dimensional space $\mathbb{R}^m$, while approximately preserving Euclidean norm. These schemes can be constructed using sparse random projections, for example using a sparse Johnson-Lindenstrauss (JL) transform. In practice, feature vectors often have a low $\ell_\infty$-to-$\ell_2$ norm ratio, and for this restricted set of vectors, many sparse JL-based schemes can achieve the norm-preserving objective with smaller dimension $m$ than is necessary for the scheme on the full space $\mathbb{R}^n$. A line of work introduced by Weinberger et. al (ICML '09) analyzes the sparse JL transform with one nonzero entry per column, which is a standard feature hashing scheme. Recently, Freksen, Kamma, and Larsen (NIPS '18) closed this line of work by proving an essentially tight tradeoff between $\ell_\infty$-to-$\ell_2$ norm ratio, distortion, failure probability, and dimension $m$ for this feature hashing scheme. We study more general projection schemes that are constructed using sparse JL transforms permitted to have more than one (but still a small fraction of) nonzero entries per column. Our main result is an essentially tight tradeoff between $\ell_\infty$-to-$\ell_2$ norm ratio, distortion, failure probability, and dimension $m$ for a general sparsity $s$, that generalizes the result of Freksen et. al. We also connect our result to the sparse JL literature by showing that it implies lower bounds on dimension-sparsity tradeoffs that essentially match upper bounds by Cohen (SODA '16). Moreover, our proof introduces a new perspective on bounding moments of certain random variables, that could be useful in other settings in theoretical computer science.