Potential games, originally introduced in the early 1990's by Lloyd Shapley, the 2012 Nobel Laureate in Economics, and his colleague Dov Monderer, are a very important class of models in game theory. They have special properties such as the existence of Nash equilibria in pure strategies. This note introduces graphical versions of potential games. Special cases of graphical potential games have already found applicability in many areas of science and engineering beyond economics, including artificial intelligence, computer vision, and machine learning. They have been effectively applied to the study and solution of important real-world problems such as routing and congestion in networks, distributed resource allocation (e.g., public goods), and relaxation-labeling for image segmentation. Implicit use of graphical potential games goes back at least 40 years. Several classes of games considered standard in the literature, including coordination games, local interaction games, lattice games, congestion games, and party-affiliation games, are instances of graphical potential games. This note provides several characterizations of graphical potential games by leveraging well-known results from the literature on probabilistic graphical models. A major contribution of the work presented here that particularly distinguishes it from previous work is establishing that the convergence of certain type of game-playing rules implies that the agents/players must be embedded in some graphical potential game.

Understanding how housing values evolve over time is important to policy makers, consumers and real estate professionals. Existing methods for constructing housing indices are computed at a coarse spatial granularity, such as metropolitan regions, which can mask or distort price dynamics apparent in local markets, such as neighborhoods and census tracts. A challenge in moving to estimates at, for example, the census tract level is the sparsity of spatiotemporally localized house sales observations. Our work aims at addressing this challenge by leveraging observations from multiple census tracts discovered to have correlated valuation dynamics. Our proposed Bayesian nonparametric approach builds on the framework of latent factor models to enable a flexible, data-driven method for inferring the clustering of correlated census tracts. We explore methods for scalability and parallelizability of computations, yielding a housing valuation index at the level of census tract rather than zip code, and on a monthly basis rather than quarterly. Our analysis is provided on a large Seattle metropolitan housing dataset.

We introduce an adaptive output-sensitive Metropolis-Hastings algorithm for probabilistic models expressed as programs, Adaptive Lightweight Metropolis-Hastings (AdLMH). The algorithm extends Lightweight Metropolis-Hastings (LMH) by adjusting the probabilities of proposing random variables for modification to improve convergence of the program output. We show that AdLMH converges to the correct equilibrium distribution and compare convergence of AdLMH to that of LMH on several test problems to highlight different aspects of the adaptation scheme. We observe consistent improvement in convergence on the test problems.

Kernel matrices are popular in machine learning and scientific computing, but they are limited by their quadratic complexity in both construction and storage. It is well-known that as one varies the kernel parameter, e.g., the width parameter in radial basis function kernels, the kernel matrix changes from a smooth low-rank kernel to a diagonally-dominant and then fully-diagonal kernel. Low-rank approximation methods have been widely-studied, mostly in the first case, to reduce the memory storage and the cost of computing matrix-vector products. Here, we use ideas from scientific computing to propose an extension of these methods to situations where the matrix is not well-approximated by a low-rank matrix. In particular, we construct an efficient block low-rank approximation method---which we call the Block Basis Factorization---and we show that it has $\mathcal{O}(n)$ complexity in both time and memory. Our method works for a wide range of kernel parameters, extending the domain of applicability of low-rank approximation methods, and our empirical results demonstrate the stability (small standard deviation in error) and superiority over current state-of-art kernel approximation algorithms.

We consider learning, from strictly behavioral data, the structure and parameters of linear influence games (LIGs), a class of parametric graphical games introduced by Irfan and Ortiz (2014). LIGs facilitate causal strategic inference (CSI): Making inferences from causal interventions on stable behavior in strategic settings. Applications include the identification of the most influential individuals in large (social) networks. Such tasks can also support policy-making analysis. Motivated by the computational work on LIGs, we cast the learning problem as maximum-likelihood estimation (MLE) of a generative model defined by pure-strategy Nash equilibria (PSNE). Our simple formulation uncovers the fundamental interplay between goodness-of-fit and model complexity: good models capture equilibrium behavior within the data while controlling the true number of equilibria, including those unobserved. We provide a generalization bound establishing the sample complexity for MLE in our framework. We propose several algorithms including convex loss minimization (CLM) and sigmoidal approximations. We prove that the number of exact PSNE in LIGs is small, with high probability; thus, CLM is sound. We illustrate our approach on synthetic data and real-world U.S. congressional voting records. We briefly discuss our learning framework's generality and potential applicability to general graphical games.

This paper studies the form and complexity of inference in graphical models using the abstraction offered by algebraic structures. In particular, we broadly formalize inference problems in graphical models by viewing them as a sequence of operations based on commutative semigroups. We then study the computational complexity of inference by organizing various problems into an "inference hierarchy". When the underlying structure of an inference problem is a commutative semiring -- i.e. a combination of two commutative semigroups with the distributive law -- a message passing procedure called belief propagation can leverage this distributive law to perform polynomial-time inference for certain problems. After establishing the NP-hardness of inference in any commutative semiring, we investigate the relation between algebraic properties in this setting and further show that polynomial-time inference using distributive law does not (trivially) extend to inference problems that are expressed using more than two commutative semigroups. We then extend the algebraic treatment of message passing procedures to survey propagation, providing a novel perspective using a combination of two commutative semirings. This formulation generalizes the application of survey propagation to new settings.

We present an approach to simultaneously reasoning about a video clip and an entire natural-language sentence. The compositional nature of language is exploited to construct models which represent the meanings of entire sentences composed out of the meanings of the words in those sentences mediated by a grammar that encodes the predicate-argument relations. We demonstrate that these models faithfully represent the meanings of sentences and are sensitive to how the roles played by participants (nouns), their characteristics (adjectives), the actions performed (verbs), the manner of such actions (adverbs), and changing spatial relations between participants (prepositions) affect the meaning of a sentence and how it is grounded in video. We exploit this methodology in three ways. In the first, a video clip along with a sentence are taken as input and the participants in the event described by the sentence are highlighted, even when the clip depicts multiple similar simultaneous events. In the second, a video clip is taken as input without a sentence and a sentence is generated that describes an event in that clip. In the third, a corpus of video clips is paired with sentences which describe some of the events in those clips and the meanings of the words in those sentences are learned. We learn these meanings without needing to specify which attribute of the video clips each word in a given sentence refers to. The learned meaning representations are shown to be intelligible to humans.

In this paper, we establish some theoretical connections between Sum-Product Networks (SPNs) and Bayesian Networks (BNs). We prove that every SPN can be converted into a BN in linear time and space in terms of the network size. The key insight is to use Algebraic Decision Diagrams (ADDs) to compactly represent the local conditional probability distributions at each node in the resulting BN by exploiting context-specific independence (CSI). The generated BN has a simple directed bipartite graphical structure. We show that by applying the Variable Elimination algorithm (VE) to the generated BN with ADD representations, we can recover the original SPN where the SPN can be viewed as a history record or caching of the VE inference process. To help state the proof clearly, we introduce the notion of {\em normal} SPN and present a theoretical analysis of the consistency and decomposability properties. We conclude the paper with some discussion of the implications of the proof and establish a connection between the depth of an SPN and a lower bound of the tree-width of its corresponding BN.

We introduce an approximate search algorithm for fast maximum a posteriori probability estimation in probabilistic programs, which we call Bayesian ascent Monte Carlo (BaMC). Probabilistic programs represent probabilistic models with varying number of mutually dependent finite, countable, and continuous random variables. BaMC is an anytime MAP search algorithm applicable to any combination of random variables and dependencies. We compare BaMC to other MAP estimation algorithms and show that BaMC is faster and more robust on a range of probabilistic models.

In this paper we introduce a novel method for linear system identification with quantized output data. We model the impulse response as a zero-mean Gaussian process whose covariance (kernel) is given by the recently proposed stable spline kernel, which encodes information on regularity and exponential stability. This serves as a starting point to cast our system identification problem into a Bayesian framework. We employ Markov Chain Monte Carlo (MCMC) methods to provide an estimate of the system. In particular, we show how to design a Gibbs sampler which quickly converges to the target distribution. Numerical simulations show a substantial improvement in the accuracy of the estimates over state-of-the-art kernel-based methods when employed in identification of systems with quantized data.