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Ben Green

Publications and source records attributed to Ben Green.

At least 37 records · Page 2Linked to original sources
arXiv2022

The Flaws of Policies Requiring Human Oversight of Government Algorithms

As algorithms become an influential component of government decision-making around the world, policymakers have debated how governments can attain the benefits of algorithms while preventing the harms of algorithms. One mechanism that has become a centerpiece of global efforts to regulate government algorithms is to require human oversight of algorithmic decisions. Despite the widespread turn to human oversight, these policies rest on an uninterrogated assumption: that people are able to effectively oversee algorithmic decision-making. In this article, I survey 41 policies that prescribe human oversight of government algorithms and find that they suffer from two significant flaws. First, evidence suggests that people are unable to perform the desired oversight functions. Second, as a result of the first flaw, human oversight policies legitimize government uses of faulty and controversial algorithms without addressing the fundamental issues with these tools. Thus, rather than protect against the potential harms of algorithmic decision-making in government, human oversight policies provide a false sense of security in adopting algorithms and enable vendors and agencies to shirk accountability for algorithmic harms. In light of these flaws, I propose a shift from human oversight to institutional oversight as the central mechanism for regulating government algorithms. This institutional approach operates in two stages. First, agencies must justify that it is appropriate to incorporate an algorithm into decision-making and that any proposed forms of human oversight are supported by empirical evidence. Second, these justifications must receive democratic public review and approval before the agency can adopt the algorithm.

cs.HC↗
arXiv2022

"If it didn't happen, why would I change my decision?": How Judges Respond to Counterfactual Explanations for the Public Safety Assessment

Many researchers and policymakers have expressed excitement about algorithmic explanations enabling more fair and responsible decision-making. However, recent experimental studies have found that explanations do not always improve human use of algorithmic advice. In this study, we shed light on how people interpret and respond to counterfactual explanations (CFEs) -- explanations that show how a model's output would change with marginal changes to its input(s) -- in the context of pretrial risk assessment instruments (PRAIs). We ran think-aloud trials with eight sitting U.S. state court judges, providing them with recommendations from a PRAI that includes CFEs. We found that the CFEs did not alter the judges' decisions. At first, judges misinterpreted the counterfactuals as real -- rather than hypothetical -- changes to defendants. Once judges understood what the counterfactuals meant, they ignored them, stating their role is only to make decisions regarding the actual defendant in question. The judges also expressed a mix of reasons for ignoring or following the advice of the PRAI without CFEs. These results add to the literature detailing the unexpected ways in which people respond to algorithms and explanations. They also highlight new challenges associated with improving human-algorithm collaborations through explanations.

cs.HC↗
arXiv2022

New lower bounds for van der Waerden numbers

We show that there is a red-blue colouring of $[N]$ with no blue 3-term arithmetic progression and no red arithmetic progression of length $e^{C(\log N)^{3/4}(\log \log N)^{1/4}}$. Consequently, the two-colour van der Waerden number $w(3,k)$ is bounded below by $k^{b(k)}$, where $b(k) = c \big( \frac{\log k}{\log\log k} \big)^{1/3}$. Previously it had been speculated, supported by data, that $w(3,k) = O(k^2)$.

math.CO↗
arXiv2022

Technology Ethics in Action: Critical and Interdisciplinary Perspectives

This special issue interrogates the meaning and impacts of "tech ethics": the embedding of ethics into digital technology research, development, use, and governance. In response to concerns about the social harms associated with digital technologies, many individuals and institutions have articulated the need for a greater emphasis on ethics in digital technology. Yet as more groups embrace the concept of ethics, critical discourses have emerged questioning whose ethics are being centered, whether "ethics" is the appropriate frame for improving technology, and what it means to develop "ethical" technology in practice. This interdisciplinary issue takes up these questions, interrogating the relationships among ethics, technology, and society in action. This special issue engages with the normative and contested notions of ethics itself, how ethics has been integrated with technology across domains, and potential paths forward to support more just and egalitarian technology. Rather than starting from philosophical theories, the authors in this issue orient their articles around the real-world discourses and impacts of tech ethics--i.e., tech ethics in action.

cs.CY↗
arXiv2022

The Contestation of Tech Ethics: A Sociotechnical Approach to Technology Ethics in Practice

This article introduces the special issue "Technology Ethics in Action: Critical and Interdisciplinary Perspectives". In response to recent controversies about the harms of digital technology, discourses and practices of "tech ethics" have proliferated across the tech industry, academia, civil society, and government. Yet despite the seeming promise of ethics, tech ethics in practice suffers from several significant limitations: tech ethics is vague and toothless, has a myopic focus on individual engineers and technology design, and is subsumed into corporate logics and incentives. These limitations suggest that tech ethics enables corporate "ethics-washing": embracing the language of ethics to defuse criticism and resist government regulation, without committing to ethical behavior. Given these dynamics, I describe tech ethics as a terrain of contestation where the central debate is not whether ethics is desirable, but what "ethics" entails and who gets to define it. Current approaches to tech ethics are poised to enable technologists and technology companies to label themselves as "ethical" without substantively altering their practices. Thus, those striving for structural improvements in digital technologies must be mindful of the gap between ethics as a mode of normative inquiry and ethics as a practical endeavor. In order to better evaluate the opportunities and limits of tech ethics, I propose a sociotechnical approach that analyzes tech ethics in light of who defines it and what impacts it generates in practice.

cs.CY↗
arXiv2022

Data Science as Political Action: Grounding Data Science in a Politics of Justice

In response to public scrutiny of data-driven algorithms, the field of data science has adopted ethics training and principles. Although ethics can help data scientists reflect on certain normative aspects of their work, such efforts are ill-equipped to generate a data science that avoids social harms and promotes social justice. In this article, I argue that data science must embrace a political orientation. Data scientists must recognize themselves as political actors engaged in normative constructions of society and evaluate their work according to its downstream impacts on people's lives. I first articulate why data scientists must recognize themselves as political actors. In this section, I respond to three arguments that data scientists commonly invoke when challenged to take political positions regarding their work. In confronting these arguments, I describe why attempting to remain apolitical is itself a political stance--a fundamentally conservative one--and why data science's attempts to promote "social good" dangerously rely on unarticulated and incrementalist political assumptions. I then propose a framework for how data science can evolve toward a deliberative and rigorous politics of social justice. I conceptualize the process of developing a politically engaged data science as a sequence of four stages. Pursuing these new approaches will empower data scientists with new methods for thoughtfully and rigorously contributing to social justice.

cs.CY↗
arXiv2021

Quadratic forms in 8 prime variables

We give an asymptotic for the number of prime solutions to $Q(x_1,\dots, x_8) = N$, subject to a mild non-degeneracy condition on the homogeneous quadratic form $Q$. The argument initially proceeds via the circle method, but this does not suffice by itself. To obtain a nontrivial bound on certain averages of exponential sums, we interpret these sums as matrix coefficients for the Weil representation of the symplectic group $\operatorname{Sp}_8(\mathbf{Z}/q\mathbf{Z})$. Averages of such matrix coefficients are then bounded using an amplification argument and a convergence result for convolutions of measures, which reduces matters to understanding the action of certain 12-dimensional subgroups in the Weil representation. Sufficient understanding can be gained by using the basic represention theory of $\operatorname{SL}_2(k)$, $k$ a finite field.

math.NT↗
arXiv2021

Algorithmic Risk Assessments Can Alter Human Decision-Making Processes in High-Stakes Government Contexts

Governments are increasingly turning to algorithmic risk assessments when making important decisions, such as whether to release criminal defendants before trial. Policymakers assert that providing public servants with algorithmic advice will improve human risk predictions and thereby lead to better (e.g., fairer) decisions. Yet because many policy decisions require balancing risk-reduction with competing goals, improving the accuracy of predictions may not necessarily improve the quality of decisions. If risk assessments make people more attentive to reducing risk at the expense of other values, these algorithms would diminish the implementation of public policy even as they lead to more accurate predictions. Through an experiment with 2,140 lay participants simulating two high-stakes government contexts, we provide the first direct evidence that risk assessments can systematically alter how people factor risk into their decisions. These shifts counteracted the potential benefits of improved prediction accuracy. In the pretrial setting of our experiment, the risk assessment made participants more sensitive to increases in perceived risk; this shift increased the racial disparity in pretrial detention by 1.9%. In the government loans setting of our experiment, the risk assessment made participants more risk-averse; this shift reduced government aid by 8.3%. These results demonstrate the potential limits and harms of attempts to improve public policy by incorporating predictive algorithms into multifaceted policy decisions. If these observed behaviors occur in practice, presenting risk assessments to public servants would generate unexpected and unjust shifts in public policy without being subject to democratic deliberation or oversight.

cs.HC↗
arXiv2021

Lower bounds for corner-free sets

A corner is a set of three points in $\mathbf{Z}^2$ of the form $(x, y), (x + d, y), (x, y + d)$ with $d \neq 0$. We show that for infinitely many $N$ there is a set $A \subset [N]^2$ of size $2^{-(c + o(1)) \sqrt{\log_2 N}} N^2$ not containing any corner, where $c = 2 \sqrt{2 \log_2 \frac{4}{3}} \approx 1.822\dots$.

math.CO↗
arXiv2021

On the width of transitive sets: bounds on matrix coefficients of finite groups

We say that a finite subset of the unit sphere in $\mathbf{R}^d$ is transitive if there is a group of isometries which acts transitively on it. We show that the width of any transitive set is bounded above by a constant times $(\log d)^{-1/2}$. This is a consequence of the following result: If $G$ is a finite group and $ρ: G \rightarrow \mbox{U}_d(\mathbf{C})$ a unitary representation, and if $v \in \mathbf{C}^d$ is a unit vector, there is another unit vector $w \in \mathbf{C}^d$ such that \[ \sup_{g \in G} |\langle ρ(g) v, w \rangle| \leq (1 + c \log d)^{-1/2}.\] These results answer a question of Yufei Zhao. An immediate consequence of our result is that the diameter of any quotient $S(\mathbf{R}^d)/G$ of the unit sphere by a finite group $G$ of isometries is at least $π/2 - o_{d \rightarrow \infty}(1)$.

math.GR↗
arXiv2020

Extremal problems for GCDs

We prove that if $A \subseteq [X, 2X]$ and $B \subseteq [Y, 2Y]$ are sets of integers such that $\gcd(a,b) \geq D$ for at least $δ|A||B|$ pairs $(a,b) \in A \times B$ then $|A||B| \ll_{\varepsilon} δ^{-2 - \varepsilon} XY/D^2$. This is a new result even when $δ= 1$. The proof uses ideas of Koukoulopoulos and Maynard and some additional combinatorial arguments.

math.NT↗
arXiv2020

An arithmetic regularity lemma, associated counting lemma, and applications

Szemeredi's regularity lemma can be viewed as a rough structure theorem for arbitrary dense graphs, decomposing such graphs into a structured piece (a partition into cells with edge densities), a small error (corresponding to irregular cells), and a uniform piece (the pseudorandom deviations from the edge densities). We establish an arithmetic regularity lemma that similarly decomposes bounded functions f : [N] -> C, into a (well-equidistributed, virtual) -step nilsequence, an error which is small in L^2 and a further error which is miniscule in the Gowers U^{s+1}-norm, where s is a positive integer. We then establish a complementary arithmetic counting lemma that counts arithmetic patterns in the nilsequence component of f. We provide a number of applications of these lemmas: a proof of Szemeredi's theorem on arithmetic progressions, a proof of a conjecture of Bergelson, Host and Kra, and a generalisation of certain results of Gowers and Wolf. Our result is dependent on the inverse conjecture for the Gowers U^{s+1} norm, recently established for general s by the authors and T. Ziegler. REVISED November 2020: This paper has been revised so that only systems of linear forms satisfying a condition called the flag property are covered by the counting lemma. Translation-invariant systems, as well as systems of Cauchy-Schwarz complexity 1 and some other naturally-occurring systems have this property. We thank Daniel Altman for drawing our attention to what appeared to be a minor technical issue in one of our proofs but which ultimately led us to realise that the the counting lemma fails quite badly (with rather simple examples, which we shall describe) without some assumption of this type. Additionally, we supply a short appendix showing that the Gowers-Wolf complexity of a system is bounded by its Cauchy-Schwarz complexity.

math.CO↗
arXiv2020

Multiple correlation sequences not approximable by nilsequences

We show that there is a measure-preserving system $(X,\mathscr{B}, μ, T)$ together with functions $F_0, F_1, F_2 \in L^{\infty}(μ)$ such that the correlation sequence $C_{F_0, F_1, F_2}(n) = \int_X F_0 \cdot T^n F_1 \cdot T^{2n} F_2 dμ$ is not an approximate integral combination of $2$-step nilsequences.

math.DS↗
arXiv2020

A Weighted Prékopa-Leindler inequality and sumsets with quasicubes

We give a short, self-contained proof of two key results from a paper of four of the authors. The first is a kind of weighted discrete Prékopa-Leindler inequality. This is then applied to show that if $A, B \subseteq \mathbb{Z}^d$ are finite sets and $U$ is a subset of a "quasicube" then $|A + B + U| \geq |A|^{1/2} |B|^{1/2} |U|$. This result is a key ingredient in forthcoming work of the fifth author and Pälvölgyi on the sum-product phenomenon.

math.NT↗
arXiv2018

A note on multiplicative functions on progressions to large moduli

Let $f : \mathbf{N} \rightarrow \mathbf{C}$ be a bounded multiplicative function. Let $a$ be a fixed integer (say $a = 1$). Then $f$ is well-distributed on the progression $n \equiv a \pmod{q} \subset \{1,\dots, X\}$, for almost all primes $q \in [Q,2Q]$, for $Q$ as large as $X^{\frac{1}{2} + \frac{1}{78} - o(1)}$.

math.NT↗
arXiv2017

On the arithmetic Kakeya conjecture of Katz and Tao

The arithmetic Kakeya conjecture, formulated by Katz and Tao in 2002, is a statement about addition of finite sets. It is known to imply a form of the Kakeya conjecture, namely that the upper Minkowski dimension of a Besicovitch set in $\mathbb{R}^n$ is $n$. In this note we discuss this conjecture, giving a number of equivalent forms of it. We show that a natural finite field variant of it does hold. We also give some lower bounds.

math.NT↗
arXiv2017

An example concerning set addition in F_2^n

We construct sets $A, B$ in a vector space over $\mathbb{F}_2$ with the property that $A$ is "statistically" almost closed under addition by $B$ in the sense that $a + b$ almost always lies in $A$ when $a \in A, b \in B$, but which is extremely far from being "combinatorially" almost closed under addition by $B$: if $A' \subset A$, $B' \subset B$ and $A' + B'$ is comparable in size to $A'$ then $|B'| \lessapprox |B|^{1/2}$.

math.CO↗