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Clusters, Games and Axioms
The Axiomatic Method in Social Choice Theory: Preference Aggregation, Judgment Aggregation, Graph Aggregation
Social choice theory is the study of the aggregation of the views of several individuals into a single collective view, with the prime example being the aggregation of preferences in an election. Axioms in this context are mathematically rigorous encodings of philosophically motivated normative requirements on aggregation methods, such as the requirement to give equal weight to every individual. Social choice theorists have explored the consequences of imposing such axioms to fully characterise certain aggregation methods or to show that certain combinations of axioms are impossible to satisfy. Famous examples, in preference aggregation, are May's characterisation of the simple majority rule, Arrow's Theorem on the impossibility of "reasonable" preference aggregation, and the Gibbard-Satterthwaite Theorem showing that all "reasonable" voting rule are subject to strategic manipulation. This talk will be an introduction to the axiomatic method in social choice theory, with examples drawn mostly from the well established areas of preference aggregation and judgment aggregation. I will also briefly report on recent work using the axiomatic method in graph aggregation, where we try to compute a single graph that is a good compromise between the graphs reported by several individuals.
Computational Game Theory and Clustering
This tutorial will provide an introduction to some of the fundamental questions and ideas studied in computational game theory. The field can be roughly divided into several main branches -- computation of equilibria, bounding the inefficiency of equilibria, and mechanism design -- and the tutorial will highlight a selection of central concepts and analysis tools from these areas. We'll also show how to apply some of them to analyze games related to clustering.
Clustering - classic methods and modern views.
In the first part of this tutorial, I will given an overview of the
many dimensions of clustering: hard vs soft, vectors vs. similarities, cost-based vs probability based, parametric vs non-parametric.
In the second part, I will consider clustering of weighted graphs,
also known as similarity based clustering, and I will describe several paradigms for this problem. Among them, spectral clustering maps the graph data back to real-valued vectors, and from a graph clustering problem to the well-know K-means/Least squares problem. I will show that in fact this is a two-way relationship, and highlight some properties of the venerable but widely used K-means and EM algorithms.
Axiomatic vs statistical approaches to theoretical foundations of clustering.
Clustering is an area of huge practical relevance but very meager theoretical foundations.
I will start by discussing some fundamental challenges that the development of a theory of clustering faces. I will then describe two different approaches to addressing those challenges;
an axiomatic approach and a statistical/machine-learning perspective.
I will outline recent progress made along these directions, as well as allude to some common misconceptions and potential pitfalls. The talk is geared towards stimulating discussions and highlighting open questions more than to providing answers or boasting definitive results.
Joachim M. Buhmann
Information theory of algorithms
Algorithms as selection procedures for mathematical structures are often exposed to uncertainty in the input or randomness during computation. The achievable precision of an algorithm, i.e., the attainable resolution in its output space, is determined by the predictive information in the input of an algorithm relative to its output space. Therefore, algorithms can be compared and optimized w.r.t. its runtime, its memory consumption and its informativeness, meaning its robustness to stochastic influences in the input and during computation. I will present an information theoretic framework for algorithm analysis where an algorithm is characterized as computational evolution of a (possibly contracting) posterior distribution over the output space. The tradeoff between precision and stability is controlled by an input sensitive generalization capacity (GC). GC measures how much the posteriors on two different prpoblem instances agree despite the noise in the input. Thereby, GC objectively ranks different algorithms for the same data processing task based on the bit rate of their respective capacities. Information theoretic algorithm selection is rigorously demonstrated for minimum spanning tree algorithms and for greedy MaxCut algorithms. The method also allows us to rank centroid based and spectral clustering methods, e.g. k-means, pairwise clustering, normalized cut, adaptive ratio cut and dominant set clustering.
Crowdsourcing Societal Tradeoffs
It would be desirable if, as a society, we could reduce the amount of landfill trash we create, the amount of carbon dioxide we emit, the amount of forest we clear, etc. Since we cannot (or are in any case not willing to) simultaneously pursue all these objectives to their maximum extent, we must prioritize among them. Currently, this is done mostly in an ad-hoc manner, with people, companies, local governments, and other entities deciding on an individual basis which of these objectives to pursue, and to what extent.
A more systematic approach would be to set, at a global level, exact numerical tradeoffs: using one gallon of gasoline is as bad as creating x bags of landfill trash. Having such tradeoffs available would greatly facilitate decision making, and reduce inefficiencies resulting from inconsistent decisions across agents. But how could we arrive at a reasonable value for x? I will discuss this from the viewpoint of computational social choice.
(joint work with Markus Brill and Rupert Freeman)
We consider approval-based committee voting, i.e., the setting where each voter approves a subset of candidates, and these votes are then used to select a fixed-size set of winners (committee). We propose a natural axiom for this setting, which we call justified representation (JR). This axiom requires that if a large enough group of voters exhibits agreement by supporting the same candidate, then at least one voter in this group has an approved candidate in the winning committee. We show that for every list of ballots it is possible to select a committee that provides JR. We then check if this axiom is fulfilled by well-known approval-based voting rules. We show that the answer is negative for most of these rules, with notable exceptions of PAV (Proportional Approval Voting), an extreme version of RAV (Reweighted Approval Voting), and, for a restricted preference domain, MAV (Minimax Approval Voting). We then introduce a stronger version of the JR axiom, which we call extended justified representation (EJR), and show that PAV satisfies EJR, while other rules do not. We also consider several other questions related to JR and EJR, including the relationship between JR/EJR and unanimity, and the complexity of the associated algorithmic problems.
Based on joint work with Haris Aziz, Markus Brill, Vincent Conitzer, Rupert Freeman, and Toby Walsh.
Twan van Laarhoven
Local quality functions for graph clustering
Many graph clustering quality functions suffer from a resolution limit, the inability to find small clusters in large graphs. This limit stems from a dependence of the quality function on global properties of the graph. In this talk I will discuss locality, which roughly states that changing one part of the graph does not affect the clustering of other parts of the graph. Local quality functions are a way to avoid the resolution limit. I will look at locality in different graph clustering settings. In particular, I will look at hard clustering, where a clustering is a partition of the set of nodes. Then I will extend the notion of locality to soft clustering, where nodes can belong to more than one cluster, with varying degrees of membership. Several common methods do not satisfy locality, but local variants exist.
Coordination Games on Graphs
We propose and study a natural class of games, which we call coordination games on graphs: We are given a finite graph whose nodes correspond to the players of the game. Each player chooses a colour from a set of colours available to her. The payoff of a player is the number of neighbours who choose the same colour. These games capture the idea of decentralized coordination in a local setting. We study their equilibria when groups of players may jointly deviate. We analyze the question of when those equilibria exist; how to recognize and compute them; and, finally, their inefficiency compared to an optimal colouring.
Samuel Rota Bulò
A Game-Theoretic Framework for Similarity-Based Data Clustering
Similarity-based clustering refers to the process of extracting maximally coherent groups from a set of data points using their observed, similarity relations. Traditional approaches to this problem are based on the idea of partitioning the input data into a predetermined number of classes, thereby obtaining the clusters as a by-product of the partitioning process. In this talk, I present a radically different view of the clustering problem that attempts to derive a rigorous formulation of the very notion of a cluster using game-theoretic principles. Clusters are derived as a result of the competition of individuals playing the so-called “clustering game", which is a non-cooperative game involving two or more players according to whether similarities are pairwise relations or higher-order ones. The competition fostered by this game induces the players engaged in the dispute to agree on a common notion of cluster, which turns out to be equivalent to the concept of evolutionary stable strategy (ESS), a classical equilibrium concept from evolutionary game theory. Computationally, ESS-clusters can be found using, e.g., replicator dynamics, a well-known formalization of natural selection processes. The proposed game-theoretic framework has already found applications in a variety of application fields, including bioinformatics, computer vision, image processing, security and video surveillance, etc., and it is general by applying also to asymmetric, negative, or high-order similarity relations.
The human in the loop and the co-evolution of processes: tackling under-specified goals
Many desirable goals are under-specified.
A first such goal is to deal with a non-digital native in a friendly and effective way. In order to do so, the computational agent should build a model of the human in the loop
(HiL), estimate his preferences and account for his unavoidable inconsistencies, and behave in a way that i) complies with the estimated HiL preferences and ii) enables to refine the HiL preference model.
A second such goal is to control the dynamics of two interacting processes (e.g. identifying and repairing faults in search-based softare engineering), in a sustainable way: each process should make progress and support the other process' progress.
The talk will propose some principles for tackling such under specified goals: make it a game; define rewards.
Based on joint work with Riad Akrour and Francois-Michel de Rainville: Programming by Feedback, Int. Conf. on Machine Learning 2014, R. Akrour et al. Sustainable cooperative coevolution with a multi-armed bandit, GECCO 2013, FM De Rainville et al.