Jon Kleinberg

Emmanuel Abbe, Samy Bengio, Elisabetta Cornacchia et al. · apple-ml

This paper considers the Pointer Value Retrieval (PVR) benchmark introduced in [ZRKB21], where a 'reasoning' function acts on a string of digits to produce the label. More generally, the paper considers the learning of logical functions with gradient descent (GD) on neural networks. It is first shown that in order to learn logical functions with gradient descent on symmetric neural networks, the generalization error can be lower-bounded in terms of the noise-stability of the target function, supporting a conjecture made in [ZRKB21]. It is then shown that in the distribution shift setting, when the data withholding corresponds to freezing a single feature (referred to as canonical holdout), the generalization error of gradient descent admits a tight characterization in terms of the Boolean influence for several relevant architectures. This is shown on linear models and supported experimentally on other models such as MLPs and Transformers. In particular, this puts forward the hypothesis that for such architectures and for learning logical functions such as PVR functions, GD tends to have an implicit bias towards low-degree representations, which in turn gives the Boolean influence for the generalization error under quadratic loss.

Benjamin Laufer, Jon Kleinberg, Hoda Heidari

Recent advances in Machine Learning (ML) and Artificial Intelligence (AI) follow a familiar structure: A firm releases a large, pretrained model. It is designed to be adapted and tweaked by other entities to perform particular, domain-specific functions. The model is described as `general-purpose,' meaning it can be transferred to a wide range of downstream tasks, in a process known as adaptation or fine-tuning. Understanding this process - the strategies, incentives, and interactions involved in the development of AI tools - is crucial for making conclusions about societal implications and regulatory responses, and may provide insights beyond AI about general-purpose technologies. We propose a model of this adaptation process. A Generalist brings the technology to a certain level of performance, and one or more Domain specialist(s) adapt it for use in particular domain(s). Players incur costs when they invest in the technology, so they need to reach a bargaining agreement on how to share the resulting revenue before making their investment decisions. We find that for a broad class of cost and revenue functions, there exists a set of Pareto-optimal profit-sharing arrangements where the players jointly contribute to the technology. Our analysis, which utilizes methods based on bargaining solutions and sub-game perfect equilibria, provides insights into the strategic behaviors of firms in these types of interactions. For example, profit-sharing can arise even when one firm faces significantly higher costs than another. After demonstrating findings in the case of one domain-specialist, we provide closed-form and numerical bargaining solutions in the generalized setting with $n$ domain specialists. We find that any potential domain specialization will either contribute, free-ride, or abstain in their uptake of the technology, and provide conditions yielding these different responses.

Reid McIlroy-Young, Jon Kleinberg, Siddhartha Sen et al.

An emerging theme in artificial intelligence research is the creation of models to simulate the decisions and behavior of specific people, in domains including game-playing, text generation, and artistic expression. These models go beyond earlier approaches in the way they are tailored to individuals, and the way they are designed for interaction rather than simply the reproduction of fixed, pre-computed behaviors. We refer to these as mimetic models, and in this paper we develop a framework for characterizing the ethical and social issues raised by their growing availability. Our framework includes a number of distinct scenarios for the use of such models, and considers the impacts on a range of different participants, including the target being modeled, the operator who deploys the model, and the entities that interact with it.

Marios Papachristou, Jon Kleinberg

We introduce a random hypergraph model for core-periphery structure. By leveraging our model's sufficient statistics, we develop a novel statistical inference algorithm that is able to scale to large hypergraphs with runtime that is practically linear wrt. the number of nodes in the graph after a preprocessing step that is almost linear in the number of hyperedges, as well as a scalable sampling algorithm. Our inference algorithm is capable of learning embeddings that correspond to the reputation (rank) of a node within the hypergraph. We also give theoretical bounds on the size of the core of hypergraphs generated by our model. We experiment with hypergraph data that range to $\sim 10^5$ hyperedges mined from the Microsoft Academic Graph, Stack Exchange, and GitHub and show that our model outperforms baselines wrt. producing good fits.

Hoda Heidari, Solon Barocas, Jon Kleinberg et al.

Prior work has provided strong evidence that, within organizational settings, teams that bring a diversity of information and perspectives to a task are more effective than teams that do not. If this form of informational diversity confers performance advantages, why do we often see largely homogeneous teams in practice? One canonical argument is that the benefits of informational diversity are in tension with affinity bias. To better understand the impact of this tension on the makeup of teams, we analyze a sequential model of team formation in which individuals care about their team's performance (captured in terms of accurately predicting some future outcome based on a set of features) but experience a cost as a result of interacting with teammates who use different approaches to the prediction task. Our analysis of this simple model reveals a set of subtle behaviors that team-growth dynamics can exhibit: (i) from certain initial team compositions, they can make progress toward better performance but then get stuck partway to optimally diverse teams; while (ii) from other initial compositions, they can also move away from this optimal balance as the majority group tries to crowd out the opinions of the minority. The initial composition of the team can determine whether the dynamics will move toward or away from performance optimality, painting a path-dependent picture of inefficiencies in team compositions. Our results formalize a fundamental limitation of utility-based motivations to drive informational diversity in organizations and hint at interventions that may improve informational diversity and performance simultaneously.

Yanbang Wang, Jon Kleinberg

We study an issue commonly seen with graph data analysis: many real-world complex systems involving high-order interactions are best encoded by hypergraphs; however, their datasets often end up being published or studied only in the form of their projections (with dyadic edges). To understand this issue, we first establish a theoretical framework to characterize this issue's implications and worst-case scenarios. The analysis motivates our formulation of the new task, supervised hypergraph reconstruction: reconstructing a real-world hypergraph from its projected graph, with the help of some existing knowledge of the application domain. To reconstruct hypergraph data, we start by analyzing hyperedge distributions in the projection, based on which we create a framework containing two modules: (1) to handle the enormous search space of potential hyperedges, we design a sampling strategy with efficacy guarantees that significantly narrows the space to a smaller set of candidates; (2) to identify hyperedges from the candidates, we further design a hyperedge classifier in two well-working variants that capture structural features in the projection. Extensive experiments validate our claims, approach, and extensions. Remarkably, our approach outperforms all baselines by an order of magnitude in accuracy on hard datasets. Our code and data can be downloaded from bit.ly/SHyRe.

Lydia T. Liu, Solon Barocas, Jon Kleinberg et al.

Predicting future outcomes is a prevalent application of machine learning in social impact domains. Examples range from predicting student success in education to predicting disease risk in healthcare. Practitioners recognize that the ultimate goal is not just to predict but to act effectively. Increasing evidence suggests that relying on outcome predictions for downstream interventions may not have desired results. In most domains there exists a multitude of possible interventions for each individual, making the challenge of taking effective action more acute. Even when causal mechanisms connecting the individual's latent states to outcomes is well understood, in any given instance (a specific student or patient), practitioners still need to infer -- from budgeted measurements of latent states -- which of many possible interventions will be most effective for this individual. With this in mind, we ask: when are accurate predictors of outcomes helpful for identifying the most suitable intervention? Through a simple model encompassing actions, latent states, and measurements, we demonstrate that pure outcome prediction rarely results in the most effective policy for taking actions, even when combined with other measurements. We find that except in cases where there is a single decisive action for improving the outcome, outcome prediction never maximizes "action value", the utility of taking actions. Making measurements of actionable latent states, where specific actions lead to desired outcomes, considerably enhances the action value compared to outcome prediction, and the degree of improvement depends on action costs and the outcome model. This analysis emphasizes the need to go beyond generic outcome prediction in interventional settings by incorporating knowledge of plausible actions and latent states.

Reid McIlroy-Young, Russell Wang, Siddhartha Sen et al.

The advent of machine learning models that surpass human decision-making ability in complex domains has initiated a movement towards building AI systems that interact with humans. Many building blocks are essential for this activity, with a central one being the algorithmic characterization of human behavior. While much of the existing work focuses on aggregate human behavior, an important long-range goal is to develop behavioral models that specialize to individual people and can differentiate among them. To formalize this process, we study the problem of behavioral stylometry, in which the task is to identify a decision-maker from their decisions alone. We present a transformer-based approach to behavioral stylometry in the context of chess, where one attempts to identify the player who played a set of games. Our method operates in a few-shot classification framework, and can correctly identify a player from among thousands of candidate players with 98% accuracy given only 100 labeled games. Even when trained on amateur play, our method generalises to out-of-distribution samples of Grandmaster players, despite the dramatic differences between amateur and world-class players. Finally, we consider more broadly what our resulting embeddings reveal about human style in chess, as well as the potential ethical implications of powerful methods for identifying individuals from behavioral data.

Zhenwei Tang, Difan Jiao, Reid McIlroy-Young et al.

There are an increasing number of domains in which artificial intelligence (AI) systems both surpass human ability and accurately model human behavior. This introduces the possibility of algorithmically-informed teaching in these domains through more relatable AI partners and deeper insights into human decision-making. Critical to achieving this goal, however, is coherently modeling human behavior at various skill levels. Chess is an ideal model system for conducting research into this kind of human-AI alignment, with its rich history as a pivotal testbed for AI research, mature superhuman AI systems like AlphaZero, and precise measurements of skill via chess rating systems. Previous work in modeling human decision-making in chess uses completely independent models to capture human style at different skill levels, meaning they lack coherence in their ability to adapt to the full spectrum of human improvement and are ultimately limited in their effectiveness as AI partners and teaching tools. In this work, we propose a unified modeling approach for human-AI alignment in chess that coherently captures human style across different skill levels and directly captures how people improve. Recognizing the complex, non-linear nature of human learning, we introduce a skill-aware attention mechanism to dynamically integrate players' strengths with encoded chess positions, enabling our model to be sensitive to evolving player skill. Our experimental results demonstrate that this unified framework significantly enhances the alignment between AI and human players across a diverse range of expertise levels, paving the way for deeper insights into human decision-making and AI-guided teaching tools.

Benjamin Laufer, Jon Kleinberg, Karen Levy et al.

A broad current application of algorithms is in formal and quantitative measures of murky concepts -- like merit -- to make decisions. When people strategically respond to these sorts of evaluations in order to gain favorable decision outcomes, their behavior can be subjected to moral judgments. They may be described as 'gaming the system' or 'cheating,' or (in other cases) investing 'honest effort' or 'improving.' Machine learning literature on strategic behavior has tried to describe these dynamics by emphasizing the efforts expended by decision subjects hoping to obtain a more favorable assessment -- some works offer ways to preempt or prevent such manipulations, some differentiate 'gaming' from 'improvement' behavior, while others aim to measure the effort burden or disparate effects of classification systems. We begin from a different starting point: that the design of an evaluation itself can be understood as furthering goals held by the evaluator which may be misaligned with broader societal goals. To develop the idea that evaluation represents a strategic interaction in which both the evaluator and the subject of their evaluation are operating out of self-interest, we put forward a model that represents the process of evaluation using three interacting agents: a decision subject, an evaluator, and society, representing a bundle of values and oversight mechanisms. We highlight our model's applicability to a number of social systems where one or two players strategically undermine the others' interests to advance their own. Treating evaluators as themselves strategic allows us to re-cast the scrutiny directed at decision subjects, towards the incentives that underpin institutional designs of evaluations. The moral standing of strategic behaviors often depend on the moral standing of the evaluations and incentives that provoke such behaviors.

Reid McIlroy-Young, Russell Wang, Siddhartha Sen et al.

AI systems that can capture human-like behavior are becoming increasingly useful in situations where humans may want to learn from these systems, collaborate with them, or engage with them as partners for an extended duration. In order to develop human-oriented AI systems, the problem of predicting human actions -- as opposed to predicting optimal actions -- has received considerable attention. Existing work has focused on capturing human behavior in an aggregate sense, which potentially limits the benefit any particular individual could gain from interaction with these systems. We extend this line of work by developing highly accurate predictive models of individual human behavior in chess. Chess is a rich domain for exploring human-AI interaction because it combines a unique set of properties: AI systems achieved superhuman performance many years ago, and yet humans still interact with them closely, both as opponents and as preparation tools, and there is an enormous corpus of recorded data on individual player games. Starting with Maia, an open-source version of AlphaZero trained on a population of human players, we demonstrate that we can significantly improve prediction accuracy of a particular player's moves by applying a series of fine-tuning methods. Furthermore, our personalized models can be used to perform stylometry -- predicting who made a given set of moves -- indicating that they capture human decision-making at an individual level. Our work demonstrates a way to bring AI systems into better alignment with the behavior of individual people, which could lead to large improvements in human-AI interaction.

Reid McIlroy-Young, Siddhartha Sen, Jon Kleinberg et al.

As artificial intelligence becomes increasingly intelligent---in some cases, achieving superhuman performance---there is growing potential for humans to learn from and collaborate with algorithms. However, the ways in which AI systems approach problems are often different from the ways people do, and thus may be uninterpretable and hard to learn from. A crucial step in bridging this gap between human and artificial intelligence is modeling the granular actions that constitute human behavior, rather than simply matching aggregate human performance. We pursue this goal in a model system with a long history in artificial intelligence: chess. The aggregate performance of a chess player unfolds as they make decisions over the course of a game. The hundreds of millions of games played online by players at every skill level form a rich source of data in which these decisions, and their exact context, are recorded in minute detail. Applying existing chess engines to this data, including an open-source implementation of AlphaZero, we find that they do not predict human moves well. We develop and introduce Maia, a customized version of Alpha-Zero trained on human chess games, that predicts human moves at a much higher accuracy than existing engines, and can achieve maximum accuracy when predicting decisions made by players at a specific skill level in a tuneable way. For a dual task of predicting whether a human will make a large mistake on the next move, we develop a deep neural network that significantly outperforms competitive baselines. Taken together, our results suggest that there is substantial promise in designing artificial intelligence systems with human collaboration in mind by first accurately modeling granular human decision-making.

Maithra Raghu, Alex Irpan, Jacob Andreas et al.

Deep reinforcement learning has achieved many recent successes, but our understanding of its strengths and limitations is hampered by the lack of rich environments in which we can fully characterize optimal behavior, and correspondingly diagnose individual actions against such a characterization. Here we consider a family of combinatorial games, arising from work of Erdos, Selfridge, and Spencer, and we propose their use as environments for evaluating and comparing different approaches to reinforcement learning. These games have a number of appealing features: they are challenging for current learning approaches, but they form (i) a low-dimensional, simply parametrized environment where (ii) there is a linear closed form solution for optimal behavior from any state, and (iii) the difficulty of the game can be tuned by changing environment parameters in an interpretable way. We use these Erdos-Selfridge-Spencer games not only to compare different algorithms, but test for generalization, make comparisons to supervised learning, analyse multiagent play, and even develop a self play algorithm. Code can be found at: https://github.com/rubai5/ESS_Game

Jon Kleinberg, Sendhil Mullainathan

Although current large language models are complex, the most basic specifications of the underlying language generation problem itself are simple to state: given a finite set of training samples from an unknown language, produce valid new strings from the language that don't already appear in the training data. Here we ask what we can conclude about language generation using only this specification, without further assumptions. In particular, suppose that an adversary enumerates the strings of an unknown target language L that is known only to come from one of a possibly infinite list of candidates. A computational agent is trying to learn to generate from this language; we say that the agent generates from L in the limit if after some finite point in the enumeration of L, the agent is able to produce new elements that come exclusively from L and that have not yet been presented by the adversary. Our main result is that there is an agent that is able to generate in the limit for every countable list of candidate languages. This contrasts dramatically with negative results due to Gold and Angluin in a well-studied model of language learning where the goal is to identify an unknown language from samples; the difference between these results suggests that identifying a language is a fundamentally different problem than generating from it.

Kenny Peng, Rajiv Movva, Jon Kleinberg et al.

While sparse autoencoders (SAEs) have generated significant excitement, a series of negative results have added to skepticism about their usefulness. Here, we establish a conceptual distinction that reconciles competing narratives surrounding SAEs. We argue that while SAEs may be less effective for acting on known concepts, SAEs are powerful tools for discovering unknown concepts. This distinction cleanly separates existing negative and positive results, and suggests several classes of SAE applications. Specifically, we outline use cases for SAEs in (i) ML interpretability, explainability, fairness, auditing, and safety, and (ii) social and health sciences.

Karim Hamade, Reid McIlroy-Young, Siddhartha Sen et al.

Powerful artificial intelligence systems are often used in settings where they must interact with agents that are computationally much weaker, for example when they work alongside humans or operate in complex environments where some tasks are handled by algorithms, heuristics, or other entities of varying computational power. For AI agents to successfully interact in these settings, however, achieving superhuman performance alone is not sufficient; they also need to account for suboptimal actions or idiosyncratic style from their less-skilled counterparts. We propose a formal evaluation framework for assessing the compatibility of near-optimal AI with interaction partners who may have much lower levels of skill; we use popular collaborative chess variants as model systems to study and develop AI agents that can successfully interact with lower-skill entities. Traditional chess engines designed to output near-optimal moves prove to be inadequate partners when paired with engines of various lower skill levels in this domain, as they are not designed to consider the presence of other agents. We contribute three methodologies to explicitly create skill-compatible AI agents in complex decision-making settings, and two chess game frameworks designed to foster collaboration between powerful AI agents and less-skilled partners. On these frameworks, our agents outperform state-of-the-art chess AI (based on AlphaZero) despite being weaker in conventional chess, demonstrating that skill-compatibility is a tangible trait that is qualitatively and measurably distinct from raw performance. Our evaluations further explore and clarify the mechanisms by which our agents achieve skill-compatibility.

Benjamin Laufer, Jon Kleinberg, Hoda Heidari

Recent policy proposals aim to improve the safety of general-purpose AI, but there is little understanding of the efficacy of different regulatory approaches to AI safety. We present a strategic model that explores the interactions between safety regulation, the general-purpose AI creators, and domain specialists--those who adapt the technology for specific applications. Our analysis examines how different regulatory measures, targeting different parts of the AI development chain, affect the outcome of this game. In particular, we assume AI technology is characterized by two key attributes: safety and performance. The regulator first sets a minimum safety standard that applies to one or both players, with strict penalties for non-compliance. The general-purpose creator then invests in the technology, establishing its initial safety and performance levels. Next, domain specialists refine the AI for their specific use cases, updating the safety and performance levels and taking the product to market. The resulting revenue is then distributed between the specialist and generalist through a revenue-sharing parameter. Our analysis reveals two key insights: First, weak safety regulation imposed predominantly on domain specialists can backfire. While it might seem logical to regulate AI use cases, our analysis shows that weak regulations targeting domain specialists alone can unintentionally reduce safety. This effect persists across a wide range of settings. Second, in sharp contrast to the previous finding, we observe that stronger, well-placed regulation can in fact mutually benefit all players subjected to it. When regulators impose appropriate safety standards on both general-purpose AI creators and domain specialists, the regulation functions as a commitment device, leading to safety and performance gains, surpassing what is achieved under no regulation or regulating one player alone.

Sophie Greenwood, Karen Levy, Solon Barocas et al.

As AI technologies improve, people are increasingly willing to delegate tasks to AI agents. In many cases, the human decision-maker chooses whether to delegate to an AI agent based on properties of the specific instance of the decision-making problem they are facing. Since humans typically lack full awareness of all the factors relevant to this choice for a given decision-making instance, they perform a kind of categorization by treating indistinguishable instances -- those that have the same observable features -- as the same. In this paper, we define the problem of designing the optimal algorithmic delegate in the presence of categories. This is an important dimension in the design of algorithms to work with humans, since we show that the optimal delegate can be an arbitrarily better teammate than the optimal standalone algorithmic agent. The solution to this optimal delegation problem is not obvious: we discover that this problem is fundamentally combinatorial, and illustrate the complex relationship between the optimal design and the properties of the decision-making task even in simple settings. Indeed, we show that finding the optimal delegate is computationally hard in general. However, we are able to find efficient algorithms for producing the optimal delegate in several broad cases of the problem, including when the optimal action may be decomposed into functions of features observed by the human and the algorithm. Finally, we run computational experiments to simulate a designer updating an algorithmic delegate over time to be optimized for when it is actually adopted by users, and show that while this process does not recover the optimal delegate in general, the resulting delegate often performs quite well.

Gali Noti, Kate Donahue, Jon Kleinberg et al.

AI systems increasingly support human decision-making. In many cases, despite the algorithm's superior performance, the final decision remains in human hands. For example, an AI may assist doctors in determining which diagnostic tests to run, but the doctor ultimately makes the diagnosis. This paper studies such AI-assisted decision-making settings, where the human learns through repeated interactions with the algorithm. In our framework, the algorithm -- designed to maximize decision accuracy according to its own model -- determines which features the human can consider. The human then makes a prediction based on their own less accurate model. We observe that the discrepancy between the algorithm's model and the human's model creates a fundamental tradeoff. Should the algorithm prioritize recommending more informative features, encouraging the human to recognize their importance, even if it results in less accurate predictions in the short term until learning occurs? Or is it preferable to forgo educating the human and instead select features that align more closely with their existing understanding, minimizing the immediate cost of learning? This tradeoff is shaped by the algorithm's time-discounted objective and the human's learning ability. Our results show that optimal feature selection has a surprisingly clean combinatorial characterization, reducible to a stationary sequence of feature subsets that is tractable to compute. As the algorithm becomes more "patient" or the human's learning improves, the algorithm increasingly selects more informative features, enhancing both prediction accuracy and the human's understanding. Notably, early investment in learning leads to the selection of more informative features than a later investment. We complement our analysis by showing that the impact of errors in the algorithm's knowledge is limited as it does not make the prediction directly.

Yanbang Wang, Hejie Cui, Jon Kleinberg

Graphs data is crucial for many applications, and much of it exists in the relations described in textual format. As a result, being able to accurately recall and encode a graph described in earlier text is a basic yet pivotal ability that LLMs need to demonstrate if they are to perform reasoning tasks that involve graph-structured information. Human performance at graph recall has been studied by cognitive scientists for decades, and has been found to often exhibit certain structural patterns of bias that align with human handling of social relationships. To date, however, we know little about how LLMs behave in analogous graph recall tasks: do their recalled graphs also exhibit certain biased patterns, and if so, how do they compare with humans and affect other graph reasoning tasks? In this work, we perform the first systematical study of graph recall by LLMs, investigating the accuracy and biased microstructures (local structural patterns) in their recall. We find that LLMs not only underperform often in graph recall, but also tend to favor more triangles and alternating 2-paths. Moreover, we find that more advanced LLMs have a striking dependence on the domain that a real-world graph comes from -- by yielding the best recall accuracy when the graph is narrated in a language style consistent with its original domain.

Keyon Vafa, Justin Y. Chen, Ashesh Rambachan et al.

Recent work suggests that large language models may implicitly learn world models. How should we assess this possibility? We formalize this question for the case where the underlying reality is governed by a deterministic finite automaton. This includes problems as diverse as simple logical reasoning, geographic navigation, game-playing, and chemistry. We propose new evaluation metrics for world model recovery inspired by the classic Myhill-Nerode theorem from language theory. We illustrate their utility in three domains: game playing, logic puzzles, and navigation. In all domains, the generative models we consider do well on existing diagnostics for assessing world models, but our evaluation metrics reveal their world models to be far less coherent than they appear. Such incoherence creates fragility: using a generative model to solve related but subtly different tasks can lead to failures. Building generative models that meaningfully capture the underlying logic of the domains they model would be immensely valuable; our results suggest new ways to assess how close a given model is to that goal.

Yanbang Wang, Jon Kleinberg

We study the implications of the modeling choice to use a graph, instead of a hypergraph, to represent real-world interconnected systems whose constituent relationships are of higher order by nature. Such a modeling choice typically involves an underlying projection process that maps the original hypergraph onto a graph, and is common in graph-based analysis. While hypergraph projection can potentially lead to loss of higher-order relations, there exists very limited studies on the consequences of doing so, as well as its remediation. This work fills this gap by doing two things: (1) we develop analysis based on graph and set theory, showing two ubiquitous patterns of hyperedges that are root to structural information loss in all hypergraph projections; we also quantify the combinatorial impossibility of recovering the lost higher-order structures if no extra help is provided; (2) we still seek to recover the lost higher-order structures in hypergraph projection, and in light of (1)'s findings we propose to relax the problem into a learning-based setting. Under this setting, we develop a learning-based hypergraph reconstruction method based on an important statistic of hyperedge distributions that we find. Our reconstruction method is evaluated on 8 real-world datasets under different settings, and exhibits consistently good performance. We also demonstrate benefits of the reconstructed hypergraphs via use cases of protein rankings and link predictions.

Kate Donahue, Jon Kleinberg

In many real-world situations, data is distributed across multiple self-interested agents. These agents can collaborate to build a machine learning model based on data from multiple agents, potentially reducing the error each experiences. However, sharing models in this way raises questions of fairness: to what extent can the error experienced by one agent be significantly lower than the error experienced by another agent in the same coalition? In this work, we consider two notions of fairness that each may be appropriate in different circumstances: "egalitarian fairness" (which aims to bound how dissimilar error rates can be) and "proportional fairness" (which aims to reward players for contributing more data). We similarly consider two common methods of model aggregation, one where a single model is created for all agents (uniform), and one where an individualized model is created for each agent. For egalitarian fairness, we obtain a tight multiplicative bound on how widely error rates can diverge between agents collaborating (which holds for both aggregation methods). For proportional fairness, we show that the individualized aggregation method always gives a small player error that is upper bounded by proportionality. For uniform aggregation, we show that this upper bound is guaranteed for any individually rational coalition (where no player wishes to leave to do local learning).

Nate Veldt, Austin R. Benson, Jon Kleinberg

Minimizing a sum of simple submodular functions of limited support is a special case of general submodular function minimization that has seen numerous applications in machine learning. We develop fast techniques for instances where components in the sum are cardinality-based, meaning they depend only on the size of the input set. This variant is one of the most widely applied in practice, encompassing, e.g., common energy functions arising in image segmentation and recent generalized hypergraph cut functions. We develop the first approximation algorithms for this problem, where the approximations can be quickly computed via reduction to a sparse graph cut problem, with graph sparsity controlled by the desired approximation factor. Our method relies on a new connection between sparse graph reduction techniques and piecewise linear approximations to concave functions. Our sparse reduction technique leads to significant improvements in theoretical runtimes, as well as substantial practical gains in problems ranging from benchmark image segmentation tasks to hypergraph clustering problems.

Chiyuan Zhang, Maithra Raghu, Jon Kleinberg et al.

Central to the success of artificial neural networks is their ability to generalize. But does neural network generalization primarily rely on seeing highly similar training examples (memorization)? Or are neural networks capable of human-intelligence styled reasoning, and if so, to what extent? These remain fundamental open questions on artificial neural networks. In this paper, as steps towards answering these questions, we introduce a new benchmark, Pointer Value Retrieval (PVR) to study the limits of neural network reasoning. The PVR suite of tasks is based on reasoning about indirection, a hallmark of human intelligence, where a first stage (task) contains instructions for solving a second stage (task). In PVR, this is done by having one part of the task input act as a pointer, giving instructions on a different input location, which forms the output. We show this simple rule can be applied to create a diverse set of tasks across different input modalities and configurations. Importantly, this use of indirection enables systematically varying task difficulty through distribution shifts and increasing functional complexity. We conduct a detailed empirical study of different PVR tasks, discovering large variations in performance across dataset sizes, neural network architectures and task complexity. Further, by incorporating distribution shift and increased functional complexity, we develop nuanced tests for reasoning, revealing subtle failures and surprising successes, suggesting many promising directions of exploration on this benchmark.

Katy Blumer, Subhashini Venugopalan, Michael P. Brenner et al.

We analyze a dataset of retinal images using linear probes: linear regression models trained on some "target" task, using embeddings from a deep convolutional (CNN) model trained on some "source" task as input. We use this method across all possible pairings of 93 tasks in the UK Biobank dataset of retinal images, leading to ~164k different models. We analyze the performance of these linear probes by source and target task and by layer depth. We observe that representations from the middle layers of the network are more generalizable. We find that some target tasks are easily predicted irrespective of the source task, and that some other target tasks are more accurately predicted from correlated source tasks than from embeddings trained on the same task.

Kate Donahue, Jon Kleinberg

Federated learning is a distributed learning paradigm where multiple agents, each only with access to local data, jointly learn a global model. There has recently been an explosion of research aiming not only to improve the accuracy rates of federated learning, but also provide certain guarantees around social good properties such as total error. One branch of this research has taken a game-theoretic approach, and in particular, prior work has viewed federated learning as a hedonic game, where error-minimizing players arrange themselves into federating coalitions. This past work proves the existence of stable coalition partitions, but leaves open a wide range of questions, including how far from optimal these stable solutions are. In this work, we motivate and define a notion of optimality given by the average error rates among federating agents (players). First, we provide and prove the correctness of an efficient algorithm to calculate an optimal (error minimizing) arrangement of players. Next, we analyze the relationship between the stability and optimality of an arrangement. First, we show that for some regions of parameter space, all stable arrangements are optimal (Price of Anarchy equal to 1). However, we show this is not true for all settings: there exist examples of stable arrangements with higher cost than optimal (Price of Anarchy greater than 1). Finally, we give the first constant-factor bound on the performance gap between stability and optimality, proving that the total error of the worst stable solution can be no higher than 9 times the total error of an optimal solution (Price of Anarchy bound of 9).

Nate Veldt, Austin R. Benson, Jon Kleinberg

Finding dense subgraphs of a large graph is a standard problem in graph mining that has been studied extensively both for its theoretical richness and its many practical applications. In this paper we introduce a new family of dense subgraph objectives, parameterized by a single parameter $p$, based on computing generalized means of degree sequences of a subgraph. Our objective captures both the standard densest subgraph problem and the maximum $k$-core as special cases, and provides a way to interpolate between and extrapolate beyond these two objectives when searching for other notions of dense subgraphs. In terms of algorithmic contributions, we first show that our objective can be minimized in polynomial time for all $p \geq 1$ using repeated submodular minimization. A major contribution of our work is analyzing the performance of different types of peeling algorithms for dense subgraphs both in theory and practice. We prove that the standard peeling algorithm can perform arbitrarily poorly on our generalized objective, but we then design a more sophisticated peeling method which for $p \geq 1$ has an approximation guarantee that is always at least $1/2$ and converges to 1 as $p \rightarrow \infty$. In practice, we show that this algorithm obtains extremely good approximations to the optimal solution, scales to large graphs, and highlights a range of different meaningful notions of density on graphs coming from numerous domains. Furthermore, it is typically able to approximate the densest subgraph problem better than the standard peeling algorithm, by better accounting for how the removal of one node affects other nodes in its neighborhood.

Jon Kleinberg, Manish Raghavan

As algorithms are increasingly applied to screen applicants for high-stakes decisions in employment, lending, and other domains, concerns have been raised about the effects of algorithmic monoculture, in which many decision-makers all rely on the same algorithm. This concern invokes analogies to agriculture, where a monocultural system runs the risk of severe harm from unexpected shocks. Here we show that the dangers of algorithmic monoculture run much deeper, in that monocultural convergence on a single algorithm by a group of decision-making agents, even when the algorithm is more accurate for any one agent in isolation, can reduce the overall quality of the decisions being made by the full collection of agents. Unexpected shocks are therefore not needed to expose the risks of monoculture; it can hurt accuracy even under "normal" operations, and even for algorithms that are more accurate when used by only a single decision-maker. Our results rely on minimal assumptions, and involve the development of a probabilistic framework for analyzing systems that use multiple noisy estimates of a set of alternatives.

Kate Donahue, Jon Kleinberg

Federated learning is a setting where agents, each with access to their own data source, combine models from local data to create a global model. If agents are drawing their data from different distributions, though, federated learning might produce a biased global model that is not optimal for each agent. This means that agents face a fundamental question: should they choose the global model or their local model? We show how this situation can be naturally analyzed through the framework of coalitional game theory. We propose the following game: there are heterogeneous players with different model parameters governing their data distribution and different amounts of data they have noisily drawn from their own distribution. Each player's goal is to obtain a model with minimal expected mean squared error (MSE) on their own distribution. They have a choice of fitting a model based solely on their own data, or combining their learned parameters with those of some subset of the other players. Combining models reduces the variance component of their error through access to more data, but increases the bias because of the heterogeneity of distributions. Here, we derive exact expected MSE values for problems in linear regression and mean estimation. We then analyze the resulting game in the framework of hedonic game theory; we study how players might divide into coalitions, where each set of players within a coalition jointly construct model(s). We analyze three methods of federation, modeling differing degrees of customization. In uniform federation, the agents collectively produce a single model. In coarse-grained federation, each agent can weight the global model together with their local model. In fine-grained federation, each agent can flexibly combine models from all other agents in the federation. For each method, we analyze the stable partitions of players into coalitions.

Katherine Van Koevering, Austin R. Benson, Jon Kleinberg

There is inherent information captured in the order in which we write words in a list. The orderings of binomials --- lists of two words separated by `and' or `or' --- has been studied for more than a century. These binomials are common across many areas of speech, in both formal and informal text. In the last century, numerous explanations have been given to describe what order people use for these binomials, from differences in semantics to differences in phonology. These rules describe primarily `frozen' binomials that exist in exactly one ordering and have lacked large-scale trials to determine efficacy. Online text provides a unique opportunity to study these lists in the context of informal text at a very large scale. In this work, we expand the view of binomials to include a large-scale analysis of both frozen and non-frozen binomials in a quantitative way. Using this data, we then demonstrate that most previously proposed rules are ineffective at predicting binomial ordering. By tracking the order of these binomials across time and communities we are able to establish additional, unexplored dimensions central to these predictions. Expanding beyond the question of individual binomials, we also explore the global structure of binomials in various communities, establishing a new model for these lists and analyzing this structure for non-frozen and frozen binomials. Additionally, novel analysis of trinomials --- lists of length three --- suggests that none of the binomials analysis applies in these cases. Finally, we demonstrate how large data sets gleaned from the web can be used in conjunction with older theories to expand and improve on old questions.

Nate Veldt, Austin R. Benson, Jon Kleinberg

Hypergraphs are a useful abstraction for modeling multiway relationships in data, and hypergraph clustering is the task of detecting groups of closely related nodes in such data. Graph clustering has been studied extensively, and there are numerous methods for detecting small, localized clusters without having to explore an entire input graph. However, there are only a few specialized approaches for localized clustering in hypergraphs. Here we present a framework for local hypergraph clustering based on minimizing localized ratio cut objectives. Our framework takes an input set of reference nodes in a hypergraph and solves a sequence of hypergraph minimum $s$-$t$ cut problems in order to identify a nearby well-connected cluster of nodes that overlaps substantially with the input set. Our methods extend graph-based techniques but are significantly more general and have new output quality guarantees. First, our methods can minimize new generalized notions of hypergraph cuts, which depend on specific configurations of nodes within each hyperedge, rather than just on the number of cut hyperedges. Second, our framework has several attractive theoretical properties in terms of output cluster quality. Most importantly, our algorithm is strongly-local, meaning that its runtime depends only on the size of the input set, and does not need to explore the entire hypergraph to find good local clusters. We use our methodology to effectively identify clusters in hypergraphs of real-world data with millions of nodes, millions of hyperedges, and large average hyperedge size with runtimes ranging between a few seconds and a few minutes.

Drew Fudenberg, Jon Kleinberg, Annie Liang et al.

We use machine learning to provide a tractable measure of the amount of predictable variation in the data that a theory captures, which we call its "completeness." We apply this measure to three problems: assigning certain equivalents to lotteries, initial play in games, and human generation of random sequences. We discover considerable variation in the completeness of existing models, which sheds light on whether to focus on developing better models with the same features or instead to look for new features that will improve predictions. We also illustrate how and why completeness varies with the experiments considered, which highlights the role played in choosing which experiments to run.

Manish Raghavan, Solon Barocas, Jon Kleinberg et al.

There has been rapidly growing interest in the use of algorithms in hiring, especially as a means to address or mitigate bias. Yet, to date, little is known about how these methods are used in practice. How are algorithmic assessments built, validated, and examined for bias? In this work, we document and analyze the claims and practices of companies offering algorithms for employment assessment. In particular, we identify vendors of algorithmic pre-employment assessments (i.e., algorithms to screen candidates), document what they have disclosed about their development and validation procedures, and evaluate their practices, focusing particularly on efforts to detect and mitigate bias. Our analysis considers both technical and legal perspectives. Technically, we consider the various choices vendors make regarding data collection and prediction targets, and explore the risks and trade-offs that these choices pose. We also discuss how algorithmic de-biasing techniques interface with, and create challenges for, antidiscrimination law.

Ilya Amburg, Jon Kleinberg, Austin R. Benson

In various application areas, networked data is collected by measuring interactions involving some specific set of core nodes. This results in a network dataset containing the core nodes along with a potentially much larger set of fringe nodes that all have at least one interaction with a core node. In many settings, this type of data arises for structures that are richer than graphs, because they involve the interactions of larger sets; for example, the core nodes might be a set of individuals under surveillance, where we observe the attendees of meetings involving at least one of the core individuals. We model such scenarios using hypergraphs, and we study the problem of core recovery: if we observe the hypergraph but not the labels of core and fringe nodes, can we recover the "planted" set of core nodes in the hypergraph? We provide a theoretical framework for analyzing the recovery of such a set of core nodes and use our theory to develop a practical and scalable algorithm for core recovery. The crux of our analysis and algorithm is that the core nodes are a hitting set of the hypergraph, meaning that every hyperedge has at least one node in the set of core nodes. We demonstrate the efficacy of our algorithm on a number of real-world datasets, outperforming competitive baselines derived from network centrality and core-periphery measures.

Maithra Raghu, Katy Blumer, Greg Corrado et al.

In a wide array of areas, algorithms are matching and surpassing the performance of human experts, leading to consideration of the roles of human judgment and algorithmic prediction in these domains. The discussion around these developments, however, has implicitly equated the specific task of prediction with the general task of automation. We argue here that automation is broader than just a comparison of human versus algorithmic performance on a task; it also involves the decision of which instances of the task to give to the algorithm in the first place. We develop a general framework that poses this latter decision as an optimization problem, and we show how basic heuristics for this optimization problem can lead to performance gains even on heavily-studied applications of AI in medicine. Our framework also serves to highlight how effective automation depends crucially on estimating both algorithmic and human error on an instance-by-instance basis, and our results show how improvements in these error estimation problems can yield significant gains for automation as well.

Maithra Raghu, Chiyuan Zhang, Jon Kleinberg et al.

Transfer learning from natural image datasets, particularly ImageNet, using standard large models and corresponding pretrained weights has become a de-facto method for deep learning applications to medical imaging. However, there are fundamental differences in data sizes, features and task specifications between natural image classification and the target medical tasks, and there is little understanding of the effects of transfer. In this paper, we explore properties of transfer learning for medical imaging. A performance evaluation on two large scale medical imaging tasks shows that surprisingly, transfer offers little benefit to performance, and simple, lightweight models can perform comparably to ImageNet architectures. Investigating the learned representations and features, we find that some of the differences from transfer learning are due to the over-parametrization of standard models rather than sophisticated feature reuse. We isolate where useful feature reuse occurs, and outline the implications for more efficient model exploration. We also explore feature independent benefits of transfer arising from weight scalings.

Jon Kleinberg, Jens Ludwig, Sendhil Mullainathan et al.

The law forbids discrimination. But the ambiguity of human decision-making often makes it extraordinarily hard for the legal system to know whether anyone has actually discriminated. To understand how algorithms affect discrimination, we must therefore also understand how they affect the problem of detecting discrimination. By one measure, algorithms are fundamentally opaque, not just cognitively but even mathematically. Yet for the task of proving discrimination, processes involving algorithms can provide crucial forms of transparency that are otherwise unavailable. These benefits do not happen automatically. But with appropriate requirements in place, the use of algorithms will make it possible to more easily examine and interrogate the entire decision process, thereby making it far easier to know whether discrimination has occurred. By forcing a new level of specificity, the use of algorithms also highlights, and makes transparent, central tradeoffs among competing values. Algorithms are not only a threat to be regulated; with the right safeguards in place, they have the potential to be a positive force for equity.

Jon Kleinberg, Sendhil Mullainathan

Algorithms are increasingly used to aid, or in some cases supplant, human decision-making, particularly for decisions that hinge on predictions. As a result, two additional features in addition to prediction quality have generated interest: (i) to facilitate human interaction and understanding with these algorithms, we desire prediction functions that are in some fashion simple or interpretable; and (ii) because they influence consequential decisions, we also want them to produce equitable allocations. We develop a formal model to explore the relationship between the demands of simplicity and equity. Although the two concepts appear to be motivated by qualitatively distinct goals, we show a fundamental inconsistency between them. Specifically, we formalize a general framework for producing simple prediction functions, and in this framework we establish two basic results. First, every simple prediction function is strictly improvable: there exists a more complex prediction function that is both strictly more efficient and also strictly more equitable. Put another way, using a simple prediction function both reduces utility for disadvantaged groups and reduces overall welfare relative to other options. Second, we show that simple prediction functions necessarily create incentives to use information about individuals' membership in a disadvantaged group --- incentives that weren't present before simplification, and that work against these individuals. Thus, simplicity transforms disadvantage into bias against the disadvantaged group. Our results are not only about algorithms but about any process that produces simple models, and as such they connect to the psychology of stereotypes and to an earlier economics literature on statistical discrimination.

Jon Kleinberg, Manish Raghavan

Algorithms are often used to produce decision-making rules that classify or evaluate individuals. When these individuals have incentives to be classified a certain way, they may behave strategically to influence their outcomes. We develop a model for how strategic agents can invest effort in order to change the outcomes they receive, and we give a tight characterization of when such agents can be incentivized to invest specified forms of effort into improving their outcomes as opposed to "gaming" the classifier. We show that whenever any "reasonable" mechanism can do so, a simple linear mechanism suffices.

Maithra Raghu, Katy Blumer, Rory Sayres et al.

The issue of disagreements amongst human experts is a ubiquitous one in both machine learning and medicine. In medicine, this often corresponds to doctor disagreements on a patient diagnosis. In this work, we show that machine learning models can be trained to give uncertainty scores to data instances that might result in high expert disagreements. In particular, they can identify patient cases that would benefit most from a medical second opinion. Our central methodological finding is that Direct Uncertainty Prediction (DUP), training a model to predict an uncertainty score directly from the raw patient features, works better than Uncertainty Via Classification, the two-step process of training a classifier and postprocessing the output distribution to give an uncertainty score. We show this both with a theoretical result, and on extensive evaluations on a large scale medical imaging application.

Austin R. Benson, Jon Kleinberg

A typical way in which network data is recorded is to measure all the interactions among a specified set of core nodes; this produces a graph containing this core together with a potentially larger set of fringe nodes that have links to the core. Interactions between pairs of nodes in the fringe, however, are not recorded by this process, and hence not present in the resulting graph data. For example, a phone service provider may only have records of calls in which at least one of the participants is a customer; this can include calls between a customer and a non-customer, but not between pairs of non-customers. Knowledge of which nodes belong to the core is an important piece of metadata that is crucial for interpreting the network dataset. But in many cases, this metadata is not available, either because it has been lost due to difficulties in data provenance, or because the network consists of found data obtained in settings such as counter-surveillance. This leads to a natural algorithmic problem, namely the recovery of the core set. Since the core set forms a vertex cover of the graph, we essentially have a planted vertex cover problem, but with an arbitrary underlying graph. We develop a theoretical framework for analyzing this planted vertex cover problem, based on results in the theory of fixed-parameter tractability, together with algorithms for recovering the core. Our algorithms are fast, simple to implement, and out-perform several methods based on network core-periphery structure on various real-world datasets.

Manish Raghavan, Ashton Anderson, Jon Kleinberg

The surge in political information, discourse, and interaction has been one of the most important developments in social media over the past several years. There is rich structure in the interaction among different viewpoints on the ideological spectrum. However, we still have only a limited analytical vocabulary for expressing the ways in which these viewpoints interact. In this paper, we develop network-based methods that operate on the ways in which users share content; we construct \emph{invocation graphs} on Web domains showing the extent to which pages from one domain are invoked by users to reply to posts containing pages from other domains. When we locate the domains on a political spectrum induced from the data, we obtain an embedded graph showing how these interaction links span different distances on the spectrum. The structure of this embedded network, and its evolution over time, helps us derive macro-level insights about how political interaction unfolded through 2016, leading up to the US Presidential election. In particular, we find that the domains invoked in replies spanned increasing distances on the spectrum over the months approaching the election, and that there was clear asymmetry between the left-to-right and right-to-left patterns of linkage.

Rediet Abebe, Jon Kleinberg, David Parkes et al.

A long line of work in social psychology has studied variations in people's susceptibility to persuasion -- the extent to which they are willing to modify their opinions on a topic. This body of literature suggests an interesting perspective on theoretical models of opinion formation by interacting parties in a network: in addition to considering interventions that directly modify people's intrinsic opinions, it is also natural to consider interventions that modify people's susceptibility to persuasion. In this work, we adopt a popular model for social opinion dynamics, and we formalize the opinion maximization and minimization problems where interventions happen at the level of susceptibility. We show that modeling interventions at the level of susceptibility lead to an interesting family of new questions in network opinion dynamics. We find that the questions are quite different depending on whether there is an overall budget constraining the number of agents we can target or not. We give a polynomial-time algorithm for finding the optimal target-set to optimize the sum of opinions when there are no budget constraints on the size of the target-set. We show that this problem is NP-hard when there is a budget, and that the objective function is neither submodular nor supermodular. Finally, we propose a heuristic for the budgeted opinion optimization and show its efficacy at finding target-sets that optimize the sum of opinions compared on real world networks, including a Twitter network with real opinion estimates.

Jon Kleinberg, Manish Raghavan

Over the past two decades, the notion of implicit bias has come to serve as an important component in our understanding of discrimination in activities such as hiring, promotion, and school admissions. Research on implicit bias posits that when people evaluate others -- for example, in a hiring context -- their unconscious biases about membership in particular groups can have an effect on their decision-making, even when they have no deliberate intention to discriminate against members of these groups. A growing body of experimental work has pointed to the effect that implicit bias can have in producing adverse outcomes. Here we propose a theoretical model for studying the effects of implicit bias on selection decisions, and a way of analyzing possible procedural remedies for implicit bias within this model. A canonical situation represented by our model is a hiring setting: a recruiting committee is trying to choose a set of finalists to interview among the applicants for a job, evaluating these applicants based on their future potential, but their estimates of potential are skewed by implicit bias against members of one group. In this model, we show that measures such as the Rooney Rule, a requirement that at least one of the finalists be chosen from the affected group, can not only improve the representation of this affected group, but also lead to higher payoffs in absolute terms for the organization performing the recruiting. However, identifying the conditions under which such measures can lead to improved payoffs involves subtle trade-offs between the extent of the bias and the underlying distribution of applicant characteristics, leading to novel theoretical questions about order statistics in the presence of probabilistic side information.

Geoff Pleiss, Manish Raghavan, Felix Wu et al.

The machine learning community has become increasingly concerned with the potential for bias and discrimination in predictive models. This has motivated a growing line of work on what it means for a classification procedure to be "fair." In this paper, we investigate the tension between minimizing error disparity across different population groups while maintaining calibrated probability estimates. We show that calibration is compatible only with a single error constraint (i.e. equal false-negatives rates across groups), and show that any algorithm that satisfies this relaxation is no better than randomizing a percentage of predictions for an existing classifier. These unsettling findings, which extend and generalize existing results, are empirically confirmed on several datasets.

Jon Kleinberg, Annie Liang, Sendhil Mullainathan

When we test a theory using data, it is common to focus on correctness: do the predictions of the theory match what we see in the data? But we also care about completeness: how much of the predictable variation in the data is captured by the theory? This question is difficult to answer, because in general we do not know how much "predictable variation" there is in the problem. In this paper, we consider approaches motivated by machine learning algorithms as a means of constructing a benchmark for the best attainable level of prediction. We illustrate our methods on the task of predicting human-generated random sequences. Relative to an atheoretical machine learning algorithm benchmark, we find that existing behavioral models explain roughly 15 percent of the predictable variation in this problem. This fraction is robust across several variations on the problem. We also consider a version of this approach for analyzing field data from domains in which human perception and generation of randomness has been used as a conceptual framework; these include sequential decision-making and repeated zero-sum games. In these domains, our framework for testing the completeness of theories provides a way of assessing their effectiveness over different contexts; we find that despite some differences, the existing theories are fairly stable across our field domains in their performance relative to the benchmark. Overall, our results indicate that (i) there is a significant amount of structure in this problem that existing models have yet to capture and (ii) there are rich domains in which machine learning may provide a viable approach to testing completeness.

Jon Kleinberg, Sendhil Mullainathan, Johan Ugander

A broad range of on-line behaviors are mediated by interfaces in which people make choices among sets of options. A rich and growing line of work in the behavioral sciences indicate that human choices follow not only from the utility of alternatives, but also from the choice set in which alternatives are presented. In this work we study comparison-based choice functions, a simple but surprisingly rich class of functions capable of exhibiting so-called choice-set effects. Motivated by the challenge of predicting complex choices, we study the query complexity of these functions in a variety of settings. We consider settings that allow for active queries or passive observation of a stream of queries, and give analyses both at the granularity of individuals or populations that might exhibit heterogeneous choice behavior. Our main result is that any comparison-based choice function in one dimension can be inferred as efficiently as a basic maximum or minimum choice function across many query contexts, suggesting that choice-set effects need not entail any fundamental algorithmic barriers to inference. We also introduce a class of choice functions we call distance-comparison-based functions, and briefly discuss the analysis of such functions. The framework we outline provides intriguing connections between human choice behavior and a range of questions in the theory of sorting.

Maithra Raghu, Ben Poole, Jon Kleinberg et al.

We survey results on neural network expressivity described in "On the Expressive Power of Deep Neural Networks". The paper motivates and develops three natural measures of expressiveness, which all display an exponential dependence on the depth of the network. In fact, all of these measures are related to a fourth quantity, trajectory length. This quantity grows exponentially in the depth of the network, and is responsible for the depth sensitivity observed. These results translate to consequences for networks during and after training. They suggest that parameters earlier in a network have greater influence on its expressive power -- in particular, given a layer, its influence on expressivity is determined by the remaining depth of the network after that layer. This is verified with experiments on MNIST and CIFAR-10. We also explore the effect of training on the input-output map, and find that it trades off between the stability and expressivity.

Rediet Abebe, Jon Kleinberg, David Parkes

In the classical cake cutting problem, a resource must be divided among agents with different utilities so that each agent believes they have received a fair share of the resource relative to the other agents. We introduce a variant of the problem in which we model an underlying social network on the agents with a graph, and agents only evaluate their shares relative to their neighbors' in the network. This formulation captures many situations in which it is unrealistic to assume a global view, and also exposes interesting phenomena in the original problem. Specifically, we say an allocation is locally envy-free if no agent envies a neighbor's allocation and locally proportional if each agent values her own allocation as much as the average value of her neighbor's allocations, with the former implying the latter. While global envy-freeness implies local envy-freeness, global proportionality does not imply local proportionality, or vice versa. A general result is that for any two distinct graphs on the same set of nodes and an allocation, there exists a set of valuation functions such that the allocation is locally proportional on one but not the other. We fully characterize the set of graphs for which an oblivious single-cutter protocol-- a protocol that uses a single agent to cut the cake into pieces --admits a bounded protocol with $O(n^2)$ query complexity for locally envy-free allocations in the Robertson-Webb model. We also consider the price of envy-freeness, which compares the total utility of an optimal allocation to the best utility of an allocation that is envy-free. We show that a lower bound of $Ω(\sqrt{n})$ on the price of envy-freeness for global allocations in fact holds for local envy-freeness in any connected undirected graph. Thus, sparse graphs surprisingly do not provide more flexibility with respect to the quality of envy-free allocations.