arXiv subjects

Jon Kleinberg

Publications and source records attributed to Jon Kleinberg.

At least 91 records · Page 5Linked to original sources

arXiv2022

Pointer Value Retrieval: A new benchmark for understanding the limits of neural network generalization

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.

cs.LG↗

arXiv2021

Hypergraph Ego-networks and Their Temporal Evolution

Interactions involving multiple objects simultaneously are ubiquitous across many domains. The systems these interactions inhabit can be modelled using hypergraphs, a generalization of traditional graphs in which each edge can connect any number of nodes. Analyzing the global and static properties of these hypergraphs has led to a plethora of novel findings regarding how these modelled systems are structured. However, less is known about the localized structure of these systems and how they evolve over time. In this paper, we propose the study of hypergraph ego-networks, a structure that can be used to model higher-order interactions involving a single node. We also propose the temporal reconstruction of hypergraph ego-networks as a benchmark problem for models that aim to predict the local temporal structure of hypergraphs. By combining a deep learning binary classifier with a hill-climbing algorithm, we will present a model for reconstructing hypergraph ego-networks by incorporating structural patterns found across multiple domains.

cs.SI↗

arXiv2021

Approximate Decomposable Submodular Function Minimization for Cardinality-Based Components

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.

cs.LG↗

arXiv2021

Using a Cross-Task Grid of Linear Probes to Interpret CNN Model Predictions On Retinal Images

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.

cs.LG↗

arXiv2021

Optimizing the order of actions in contact tracing

Contact tracing is a key tool for managing epidemic diseases like HIV, tuberculosis, and COVID-19. Manual investigations by human contact tracers remain a dominant way in which this is carried out. This process is limited by the number of contact tracers available, who are often overburdened during an outbreak or epidemic. As a result, a crucial decision in any contact tracing strategy is, given a set of contacts, which person should a tracer trace next? In this work, we develop a formal model that articulates these questions and provides a framework for comparing contact tracing strategies. Through analyzing our model, we give provably optimal prioritization policies via a clean connection to a tool from operations research called a "branching bandit". Examining these policies gives qualitative insight into trade-offs in contact tracing applications.

cs.DS↗

arXiv2021

The Paradox of Second-Order Homophily in Networks

Homophily -- the tendency of nodes to connect to others of the same type -- is a central issue in the study of networks. Here we take a local view of homophily, defining notions of first-order homophily of a node (its individual tendency to link to similar others) and second-order homophily of a node (the aggregate first-order homophily of its neighbors). Through this view, we find a surprising result for homophily values that applies with only minimal assumptions on the graph topology. It can be phrased most simply as "in a graph of red and blue nodes, red friends of red nodes are on average more homophilous than red friends of blue nodes." This gap in averages defies simple intuitive explanations, applies to globally heterophilous and homophilous networks and is reminiscent of but structurally distinct from the Friendship Paradox. The existence of this gap suggests intrinsic biases in homophily measurements between groups, and hence is relevant to empirical studies of homophily in networks.

cs.SI↗

arXiv2021

Augmented Sparsifiers for Generalized Hypergraph Cuts with Applications to Decomposable Submodular Function Minimization

In recent years, hypergraph generalizations of many graph cut problems have been introduced and analyzed as a way to better explore and understand complex systems and datasets characterized by multiway relationships. Recent work has made use of a generalized hypergraph cut function which for a hypergraph $\mathcal{H} = (V,E)$ can be defined by associating each hyperedge $e \in E$ with a splitting function ${\bf w}_e$, which assigns a penalty to each way of separating the nodes of $e$. When each ${\bf w}_e$ is a submodular cardinality-based splitting function, meaning that ${\bf w}_e(S) = g(|S|)$ for some concave function $g$, previous work has shown that a generalized hypergraph cut problem can be reduced to a directed graph cut problem on an augmented node set. However, existing reduction procedures often result in a dense graph, even when the hypergraph is sparse, which leads to slow runtimes for algorithms that run on the reduced graph. We introduce a new framework of sparsifying hypergraph-to-graph reductions, where a hypergraph cut defined by submodular cardinality-based splitting functions is $(1+\varepsilon)$-approximated by a cut on a directed graph. Our techniques are based on approximating concave functions using piecewise linear curves. For $\varepsilon > 0$ we need at most $O(\varepsilon^{-1}|e| \log |e|)$ edges to reduce any hyperedge $e$, which leads to faster runtimes for approximating generalized hypergraph $s$-$t$ cut problems. For the machine learning heuristic of a clique splitting function, our approach requires only $O(|e| \varepsilon^{-1/2} \log \log \frac{1}{\varepsilon})$ edges. This sparsification leads to faster approximate min $s$-$t$ graph cut algorithms for certain classes of co-occurrence graphs. Finally, we apply our sparsification techniques to develop approximation algorithms for minimizing sums of cardinality-based submodular functions.

cs.DS↗

arXiv2021

Polarization in Geometric Opinion Dynamics

In light of increasing recent attention to political polarization, understanding how polarization can arise poses an important theoretical question. While more classical models of opinion dynamics seem poorly equipped to study this phenomenon, a recent novel approach by Hązła, Jin, Mossel, and Ramnarayan (HJMR) proposes a simple geometric model of opinion evolution that provably exhibits strong polarization in specialized cases. Moreover, polarization arises quite organically in their model: in each time step, each agent updates opinions according to their correlation/response with an issue drawn at random. However, their techniques do not seem to extend beyond a set of special cases they identify, which benefit from fragile symmetry or contractiveness assumptions, leaving open how general this phenomenon really is. In this paper, we further the study of polarization in related geometric models. We show that the exact form of polarization in such models is quite nuanced: even when strong polarization does not hold, it is possible for weaker notions of polarization to nonetheless attain. We provide a concrete example where weak polarization holds, but strong polarization provably fails. However, we show that strong polarization provably holds in many variants of the HJMR model, which are also robust to a wider array of distributions of random issues -- this indicates that the form of polarization introduced by HJMR is more universal than suggested by their special cases. We also show that the weaker notions connect more readily to the theory of Markov chains on general state spaces.

cs.SI↗

arXiv2021

Stochastic Model for Sunk Cost Bias

We present a novel model for capturing the behavior of an agent exhibiting sunk-cost bias in a stochastic environment. Agents exhibiting sunk-cost bias take into account the effort they have already spent on an endeavor when they evaluate whether to continue or abandon it. We model planning tasks in which an agent with this type of bias tries to reach a designated goal. Our model structures this problem as a type of Markov decision process: loosely speaking, the agent traverses a directed acyclic graph with probabilistic transitions, paying costs for its actions as it tries to reach a target node containing a specified reward. The agent's sunk cost bias is modeled by a cost that it incurs for abandoning the traversal: if the agent decides to stop traversing the graph, it incurs a cost of $λ\cdot C_{sunk}$, where ${λ\geq 0}$ is a parameter that captures the extent of the bias and $C_{sunk}$ is the sum of costs already invested. We analyze the behavior of two types of agents: naive agents that are unaware of their bias, and sophisticated agents that are aware of it. Since optimal (bias-free) behavior in this problem can involve abandoning the traversal before reaching the goal, the bias exhibited by these types of agents can result in sub-optimal behavior by shifting their decisions about abandonment. We show that in contrast to optimal agents, it is computationally hard to compute the optimal policy for a sophisticated agent. Our main results quantify the loss exhibited by these two types of agents with respect to an optimal agent. We present both general and topology-specific bounds.

cs.GT↗

arXiv2021

Optimality and Stability in Federated Learning: A Game-theoretic Approach

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).

cs.GT↗

arXiv2021

The Generalized Mean Densest Subgraph Problem

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.

cs.DS↗

arXiv2021

Optimal Stopping with Behaviorally Biased Agents: The Role of Loss Aversion and Changing Reference Points

People are often reluctant to sell a house, or shares of stock, below the price at which they originally bought it. While this is generally not consistent with rational utility maximization, it does reflect two strong empirical regularities that are central to the behavioral science of human decision-making: a tendency to evaluate outcomes relative to a reference point determined by context (in this case the original purchase price), and the phenomenon of loss aversion in which people are particularly prone to avoid outcomes below the reference point. Here we explore the implications of reference points and loss aversion in optimal stopping problems, where people evaluate a sequence of options in one pass, either accepting the option and stopping the search or giving up on the option forever. The best option seen so far sets a reference point that shifts as the search progresses, and a biased decision-maker's utility incurs an additional penalty when they accept a later option that is below this reference point. We formulate and study a behaviorally well-motivated version of the optimal stopping problem that incorporates these notions of reference dependence and loss aversion. We obtain tight bounds on the performance of a biased agent in this model relative to the best option obtainable in retrospect (a type of prophet inequality for biased agents), as well as tight bounds on the ratio between the performance of a biased agent and the performance of a rational one. We further establish basic monotonicity results, and show an exponential gap between the performance of a biased agent in a stopping problem with respect to a worst-case versus a random order. As part of this, we establish fundamental differences between optimal stopping problems for rational versus biased agents, and these differences inform our analysis.

cs.GT↗

arXiv2021

Algorithmic Monoculture and Social Welfare

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.

cs.GT↗

arXiv2021

On Modeling Human Perceptions of Allocation Policies with Uncertain Outcomes

Many policies allocate harms or benefits that are uncertain in nature: they produce distributions over the population in which individuals have different probabilities of incurring harm or benefit. Comparing different policies thus involves a comparison of their corresponding probability distributions, and we observe that in many instances the policies selected in practice are hard to explain by preferences based only on the expected value of the total harm or benefit they produce. In cases where the expected value analysis is not a sufficient explanatory framework, what would be a reasonable model for societal preferences over these distributions? Here we investigate explanations based on the framework of probability weighting from the behavioral sciences, which over several decades has identified systematic biases in how people perceive probabilities. We show that probability weighting can be used to make predictions about preferences over probabilistic distributions of harm and benefit that function quite differently from expected-value analysis, and in a number of cases provide potential explanations for policy preferences that appear hard to motivate by other means. In particular, we identify optimal policies for minimizing perceived total harm and maximizing perceived total benefit that take the distorting effects of probability weighting into account, and we discuss a number of real-world policies that resemble such allocational strategies. Our analysis does not provide specific recommendations for policy choices, but is instead fundamentally interpretive in nature, seeking to describe observed phenomena in policy choices.

cs.CY↗

arXiv2021

Random Graphs with Prescribed $K$-Core Sequences: A New Null Model for Network Analysis

In the analysis of large-scale network data, a fundamental operation is the comparison of observed phenomena to the predictions provided by null models: when we find an interesting structure in a family of real networks, it is important to ask whether this structure is also likely to arise in random networks with similar characteristics to the real ones. A long-standing challenge in network analysis has been the relative scarcity of reasonable null models for networks; arguably the most common such model has been the configuration model, which starts with a graph $G$ and produces a random graph with the same node degrees as $G$. This leads to a very weak form of null model, since fixing the node degrees does not preserve many of the crucial properties of the network, including the structure of its subgraphs. Guided by this challenge, we propose a new family of network null models that operate on the $k$-core decomposition. For a graph $G$, the $k$-core is its maximal subgraph of minimum degree $k$; and the core number of a node $v$ in $G$ is the largest $k$ such that $v$ belongs to the $k$-core of $G$. We provide the first efficient sampling algorithm to solve the following basic combinatorial problem: given a graph $G$, produce a random graph sampled nearly uniformly from among all graphs with the same sequence of core numbers as $G$. This opens the opportunity to compare observed networks $G$ with random graphs that exhibit the same core numbers, a comparison that preserves aspects of the structure of $G$ that are not captured by more local measures like the degree sequence. We illustrate the power of this core-based null model on some fundamental tasks in network analysis, including the enumeration of networks motifs.

cs.SI↗

arXiv2021

Allocating Opportunities in a Dynamic Model of Intergenerational Mobility

Opportunities such as higher education can promote intergenerational mobility, leading individuals to achieve levels of socioeconomic status above that of their parents. We develop a dynamic model for allocating such opportunities in a society that exhibits bottlenecks in mobility; the problem of optimal allocation reflects a trade-off between the benefits conferred by the opportunities in the current generation and the potential to elevate the socioeconomic status of recipients, shaping the composition of future generations in ways that can benefit further from the opportunities. We show how optimal allocations in our model arise as solutions to continuous optimization problems over multiple generations, and we find in general that these optimal solutions can favor recipients of low socioeconomic status over slightly higher-performing individuals of high socioeconomic status -- a form of socioeconomic affirmative action that the society in our model discovers in the pursuit of purely payoff-maximizing goals. We characterize how the structure of the model can lead to either temporary or persistent affirmative action, and we consider extensions of the model with more complex processes modulating the movement between different levels of socioeconomic status.

cs.CY↗

arXiv2020

Model-sharing Games: Analyzing Federated Learning Under Voluntary Participation

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.

cs.GT↗

arXiv2020

Opinion Dynamics with Varying Susceptibility to Persuasion via Non-Convex Local Search

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, motivated by this fact we propose a new framework for social influence. Specifically, we adopt a popular model for social opinion dynamics, where each agent has some fixed innate opinion, and a resistance that measures the importance it places on its innate opinion; agents influence one another's opinions through an iterative process. Under non-trivial conditions, this iterative process converges to some equilibrium opinion vector. For the unbudgeted variant of the problem, the goal is to select the resistance of each agent (from some given range) such that the sum of the equilibrium opinions is minimized. We prove that the objective function is in general non-convex. Hence, formulating the problem as a convex program as in an early version of this work (Abebe et al., KDD'18) might have potential correctness issues. We instead analyze the structure of the objective function, and show that any local optimum is also a global optimum, which is somehow surprising as the objective function might not be convex. Furthermore, we combine the iterative process and the local search paradigm to design very efficient algorithms that can solve the unbudgeted variant of the problem optimally on large-scale graphs containing millions of nodes. Finally, we propose and evaluate experimentally a family of heuristics for the budgeted variation of the problem.

cs.SI↗