arXiv subjects

Meena Jagadeesan

Publications and source records attributed to Meena Jagadeesan.

30 records · Page 2Linked to original sources

arXiv2022

Performative Power

We introduce the notion of performative power, which measures the ability of a firm operating an algorithmic system, such as a digital content recommendation platform, to cause change in a population of participants. We relate performative power to the economic study of competition in digital economies. Traditional economic concepts struggle with identifying anti-competitive patterns in digital platforms not least due to the complexity of market definition. In contrast, performative power is a causal notion that is identifiable with minimal knowledge of the market, its internals, participants, products, or prices. Low performative power implies that a firm can do no better than to optimize their objective on current data. In contrast, firms of high performative power stand to benefit from steering the population towards more profitable behavior. We confirm in a simple theoretical model that monopolies maximize performative power. A firm's ability to personalize increases performative power, while competition and outside options decrease performative power. On the empirical side, we propose an observational causal design to identify performative power from discontinuities in how digital platforms display content. This allows to repurpose causal effects from various studies about digital platforms as lower bounds on performative power. Finally, we speculate about the role that performative power might play in competition policy and antitrust enforcement in digital marketplaces.

cs.LG↗

arXiv2022

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 an algorithm that achieves regret bounds scaling only with the complexity of the distribution shifts and not that of the reward function. The algorithm only relies on smoothness of the shifts and does not assume convexity. Moreover, its final iterate is guaranteed to be near-optimal. 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. More broadly, our work establishes a conceptual approach for leveraging tools from the bandits literature for the purpose of regret minimization with performative feedback.

cs.LG↗

arXiv2022

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

We provide a function space characterization of the inductive bias resulting from minimizing the $\ell_2$ norm of the weights in multi-channel convolutional neural networks with linear activations and empirically test our resulting hypothesis on ReLU networks trained using gradient descent. We define an induced regularizer in the function space as the minimum $\ell_2$ norm of weights of a network required to realize a 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 are single channeled, the induced regularizer for any $K$ is independent of the number of output channels $C$. Furthermore, we derive the regularizer is a norm given by a semidefinite program (SDP). (b) In contrast, for 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 $C$. However, for sufficiently large $C$, the induced regularizer is again given by an SDP that is independent of $C$. In particular, the induced regularizer for $K=1$ and $K=D$ (input dimension) is given in closed form as the nuclear norm and the $\ell_{2,1}$ group-sparse norm, respectively, of the Fourier coefficients of the linear predictor. We investigate the broader applicability of our theoretical results to implicit regularization from gradient descent on linear and ReLU networks through experiments on MNIST and CIFAR-10 datasets.

cs.LG↗

arXiv2021

Individual Fairness in Advertising Auctions through Inverse Proportionality

Recent empirical work demonstrates that online advertisement can exhibit bias in the delivery of ads across users even when all advertisers bid in a non-discriminatory manner. We study the design of ad auctions that, given fair bids, are guaranteed to produce fair outcomes. Following the works of Dwork and Ilvento (2019) and Chawla et al. (2020), our goal is to design a truthful auction that satisfies ``individual fairness'' in its outcomes: informally speaking, users that are similar to each other should obtain similar allocations of ads. Within this framework we quantify the tradeoff between social welfare maximization and fairness. This work makes two conceptual contributions. First, we express the fairness constraint as a kind of stability condition: any two users that are assigned multiplicatively similar values by all the advertisers must receive additively similar allocations for each advertiser. This value stability constraint is expressed as a function that maps the multiplicative distance between value vectors to the maximum allowable $\ell_{\infty}$ distance between the corresponding allocations. Standard auctions do not satisfy this kind of value stability. Second, we introduce a new class of allocation algorithms called Inverse Proportional Allocation that achieve a near optimal tradeoff between fairness and social welfare for a broad and expressive class of value stability conditions. These allocation algorithms are truthful and prior-free, and achieve a constant factor approximation to the optimal (unconstrained) social welfare. In particular, the approximation ratio is independent of the number of advertisers in the system. In this respect, these allocation algorithms greatly surpass the guarantees achieved in previous work. We also extend our results to broader notions of fairness that we call subset fairness.

cs.GT↗

arXiv2021

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.

cs.LG↗

arXiv2020

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 naïve 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.

cs.CY↗

arXiv2020

Understanding Sparse JL for Feature Hashing

Feature hashing and other random projection schemes are commonly used 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 much 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. A line of work introduced by Weinberger et. al (ICML '09) analyzes the accuracy of sparse JL with sparsity 1 on feature vectors with small $\ell_\infty$-to-$\ell_2$ norm ratio. Recently, Freksen, Kamma, and Larsen (NeurIPS '18) closed this line of work by proving a tight tradeoff between $\ell_\infty$-to-$\ell_2$ norm ratio and accuracy for sparse JL with sparsity $1$. In this paper, we demonstrate the benefits of using sparsity $s$ greater than $1$ in sparse JL on feature vectors. Our main result is a tight tradeoff between $\ell_\infty$-to-$\ell_2$ norm ratio and accuracy for a general sparsity $s$, that significantly generalizes the result of Freksen et. al. Our result theoretically demonstrates that sparse JL with $s > 1$ can have significantly better norm-preservation properties on feature vectors than sparse JL with $s = 1$; we also empirically demonstrate this finding.

stat.ML↗

arXiv2019

Simple Analysis of Sparse, Sign-Consistent JL

Allen-Zhu, Gelashvili, Micali, and Shavit construct a sparse, sign-consistent Johnson-Lindenstrauss distribution, and prove that this distribution yields an essentially optimal dimension for the correct choice of sparsity. However, their analysis of the upper bound on the dimension and sparsity requires a complicated combinatorial graph-based argument similar to Kane and Nelson's analysis of sparse JL. We present a simple, combinatorics-free analysis of sparse, sign-consistent JL that yields the same dimension and sparsity upper bounds as the original analysis. Our analysis also yields dimension/sparsity tradeoffs, which were not previously known. As with previous proofs in this area, our analysis is based on applying Markov's inequality to the pth moment of an error term that can be expressed as a quadratic form of Rademacher variables. Interestingly, we show that, unlike in previous work in the area, the traditionally used Hanson-Wright bound is not strong enough to yield our desired result. Indeed, although the Hanson-Wright bound is known to be optimal for gaussian degree-2 chaos, it was already shown to be suboptimal for Rademachers. Surprisingly, we are able to show a simple moment bound for quadratic forms of Rademachers that is sufficiently tight to achieve our desired result, which given the ubiquity of moment and tail bounds in theoretical computer science, is likely to be of broader interest.

cs.DS↗

arXiv2019

Multi-Category Fairness in Sponsored Search Auctions

Fairness in advertising is a topic of particular concern motivated by theoretical and empirical observations in both the computer science and economics literature. We examine the problem of fairness in advertising for general purpose platforms that service advertisers from many different categories. First, we propose inter-category and intra-category fairness desiderata that take inspiration from individual fairness and envy-freeness. Second, we investigate the "platform utility" (a proxy for the quality of the allocation) achievable by mechanisms satisfying these desiderata. More specifically, we compare the utility of fair mechanisms against the unfair optimal, and we show by construction that our fairness desiderata are compatible with utility. That is, we construct a family of fair mechanisms with high utility that perform close to optimally within a class of fair mechanisms. Our mechanisms also enjoy nice implementation properties including metric-obliviousness, which allows the platform to produce fair allocations without needing to know the specifics of the fairness requirements.

cs.GT↗

arXiv2016

The Exchange Graphs of Weakly Separated Collections

Weakly separated collections arise in the cluster algebra derived from the Plücker coordinates on the nonnegative Grassmannian. Oh, Postnikov, and Speyer studied weakly separated collections over a general Grassmann necklace $\mathcal{I}$ and proved the connectivity of every exchange graph. Oh and Speyer later introduced a generalization of exchange graphs that we call $\mathcal{C}$-constant graphs. They characterized these graphs in the smallest two cases. We prove an isomorphism between exchange graphs and a certain class of $\mathcal{C}$-constant graphs. We use this to extend Oh and Speyer's characterization of these graphs to the smallest four cases, and we present a conjecture on a bound on the maximal order of these graphs. In addition, we fully characterize certain classes of these graphs in the special cases of cycles and trees.

math.CO↗

arXiv2016

Dyson's Partition Ranks and their Multiplicative Extensions

We study the Dyson rank function $N(r,3;n)$, the number of partitions of $n$ with rank $\equiv r \pmod 3$. We investigate the convexity of these functions. We extend $N(r,3;n)$ multiplicatively to the set of partitions, and we determine the maximum value when taken over all partitions of size $n$.

math.NT↗

arXiv2014

Möbius Polynomials of Face Posets of Convex Polytopes

The Möbius polynomial is an invariant of ranked posets, closely related to the Möbius function. In this paper, we study the Möbius polynomial of face posets of convex polytopes. We present formulas for computing the Möbius polynomial of the face poset of a pyramid or a prism over an existing polytope, or of the gluing of two or more polytopes in terms of the Möbius polynomials of the original polytopes. We also present general formulas for calculating Möbius polynomials of face posets of simplicial polytopes and of Eulerian posets in terms of their $f$-vectors and some additional constraints.

math.CO↗