Almost all of the work in graphical models for game theory has mirrored previous work in probabilistic graphical models. Our work considers the opposite direction: Taking advantage of recent advances in equilibrium computation for probabilistic inference. We present formulations of inference problems in Markov random fields (MRFs) as computation of equilibria in a certain class of game-theoretic graphical models. We concretely establishes the precise connection between variational probabilistic inference in MRFs and correlated equilibria. No previous work exploits recent theoretical and empirical results from the literature on algorithmic and computational game theory on the tractable, polynomial-time computation of exact or approximate correlated equilibria in graphical games with arbitrary, loopy graph structure. We discuss how to design new algorithms with equally tractable guarantees for the computation of approximate variational inference in MRFs. Also, inspired by a previously stated game-theoretic view of state-of-the-art tree-reweighed (TRW) message-passing techniques for belief inference as zero-sum game, we propose a different, general-sum potential game to design approximate fictitious-play techniques. We perform synthetic experiments evaluating our proposed approximation algorithms with standard methods and TRW on several classes of classical Ising models (i.e., with binary random variables). We also evaluate the algorithms using Ising models learned from the MNIST dataset. Our experiments show that our global approach is competitive, particularly shinning in a class of Ising models with constant, "highly attractive" edge-weights, in which it is often better than all other alternatives we evaluated. With a notable exception, our more local approach was not as effective. Yet, in fairness, almost all of the alternatives are often no better than a simple baseline: estimate 0.5.

A current challenge for data management systems is to support the construction and maintenance of machine learning models over data that is large, multi-dimensional, and evolving. While systems that could support these tasks are emerging, the need to scale to distributed, streaming data requires new models and algorithms. In this setting, as well as computational scalability and model accuracy, we also need to minimize the amount of communication between distributed processors, which is the chief component of latency. We study Bayesian networks, the workhorse of graphical models, and present a communication-efficient method for continuously learning and maintaining a Bayesian network model over data that is arriving as a distributed stream partitioned across multiple processors. We show a strategy for maintaining model parameters that leads to an exponential reduction in communication when compared with baseline approaches to maintain the exact MLE (maximum likelihood estimation). Meanwhile, our strategy provides similar prediction errors for the target distribution and for classification tasks.

A well-calibrated estimator for the conditional probabilities should obey this equation. Once we have derived a statistical classifier, we need to validate it on some test data. This data should be different from that used to train the classifier, otherwise skill scores will be unduly optimistic. This is known as cross-validation. The confusion matrix expresses everything about the accuracy of a discrete classifier over a given database and you can use it to compose any possible skill score. Here, we are going to cover two that are rarely seen in the literature, but are nonetheless important for reasons that will become clear.

We introduce a novel kernel that models input-dependent couplings across multiple latent processes. The pairwise joint kernel measures covariance along inputs and across different latent signals in a mutually-dependent fashion. A latent correlation Gaussian process (LCGP) model combines these non-stationary latent components into multiple outputs by an input-dependent mixing matrix. Probit classification and support for multiple observation sets are derived by Variational Bayesian inference. Results on several datasets indicate that the LCGP model can recover the correlations between latent signals while simultaneously achieving state-of-the-art performance. We highlight the latent covariances with an EEG classification dataset where latent brain processes and their couplings simultaneously emerge from the model.

Recent advances in artificial intelligence are encouraging governments and corporations to deploy AI in high-stakes settings including driving cars autonomously, managing the power grid, trading on stock exchanges, and controlling autonomous weapons systems. Such applications require AI methods to be robust to both the known unknowns (those uncertain aspects of the world about which the computer can reason explicitly) and the unknown unknowns (those aspects of the world that are not captured by the system’s models). This article discusses recent progress in AI and then describes eight ideas related to robustness that are being pursued within the AI research community. While these ideas are a start, we need to devote more attention to the challenges of dealing with the known and unknown unknowns. These issues are fascinating, because they touch on the fundamental question of how finite systems can survive and thrive in a complex and dangerous world

Suppose there exist three machine learning experts that would like to understand the fundamental limits of classification (Bayes error) [1] for a specific dataset. Since the true distribution that generates the data is unknown, they take three different approaches: 1) Expert A: given empirical training samples, produce an estimate of the Bayes error that is (near) optimal statistically; 2) Expert B: construct a (near) optimal classifier based on the training sample, and then use its performance on the test set (may have infinite size) to estimate the Bayes error; 3) Expert C: use the training error of a (near) optimal classification algorithm to estimate the Bayes error. We ask the question: are there any fundamental differences between experts A, B, and C? Evidently, expert A is not constrained by any specific approaches as experts B and C are, but if B and C are using (near) optimal classification algorithms, would B or C achieve the same performance of A if A chooses to act optimally? Similar situations arise in the understanding of fundamental limits of data compression and sequential prediction under logarithmic loss, which is given by the Shannon entropy rate [2]. In this situation, there could exist four different experts: 1) A: would like to estimate the limits of compression (near) optimally; 2) B: would like to construct a predictor based on training samples and use its prediction accuracy under logarithmic loss on the test set (may have infinite size) to estimate the limits; 3) C: would like to use the training error of a (near) optimal sequential predictor to estimate the limits; 4) D: would like to construct a (near) optimal data compressor and use its normalized code length to estimate the limits. In this situation, are there any fundamental differences between the tasks of these four experts?

Recently, nonnegative matrix factorization (NMF) [1, 2] has been applied to text mining [3], signal processing [4, 5, 6], bioinformatics [7], and consumer analysis [8]. Experiments has shown that a new knowledge discovery method is derived by NMF, however, its mathematical property as a learning machine is not yet clarified, since it is not a regular statistical model. A statistical model is called regular if a function from a parameter to a probability density function is one-to-one and if the likelihood function can be approximated by a Gaussian function. It is proved that, if a statistical model is regular and if a true distribution is realizable by a statistical model, then the generalization error is asymptotically equal to d/(2n), where d, n, and the generalization error are the dimension of the parameter, the sample size, and the expected Kullback-Leibler divergence of the true distribution and the estimated learning machine, respectively. However, the statistical model used in NMF is not regular because the map from a parameter to a probability density function is not injective.

We propose an efficient meta-algorithm for Bayesian estimation problems that is based on low-degree polynomials, semidefinite programming, and tensor decomposition. The algorithm is inspired by recent lower bound constructions for sum-of-squares and related to the method of moments. Our focus is on sample complexity bounds that are as tight as possible (up to additive lower-order terms) and often achieve statistical thresholds or conjectured computational thresholds. Our algorithm recovers the best known bounds for community detection in the sparse stochastic block model, a widely-studied class of estimation problems for community detection in graphs. We obtain the first recovery guarantees for the mixed-membership stochastic block model (Airoldi et el.) in constant average degree graphs---up to what we conjecture to be the computational threshold for this model. We show that our algorithm exhibits a sharp computational threshold for the stochastic block model with multiple communities beyond the Kesten--Stigum bound---giving evidence that this task may require exponential time. The basic strategy of our algorithm is strikingly simple: we compute the best-possible low-degree approximation for the moments of the posterior distribution of the parameters and use a robust tensor decomposition algorithm to recover the parameters from these approximate posterior moments.

A well-calibrated estimator for the conditional probabilities should obey this equation. Once we have derived a statistical classifier, we need to validate it on some test data. This data should be different from that used to train the classifier, otherwise skill scores will be unduly optimistic. This is known as cross-validation. The confusion matrix expresses everything about the accuracy of a discrete classifier over a given database and you can use it to compose any possible skill score. Here, we are going to cover two that are rarely seen in the literature, but are nonetheless important for reasons that will become clear.

Finally, the main aim of this blog post is to give a well-intentioned advice about the importance of Mathematics in Machine Learning and the necessary topics and useful resources for a mastery of these topics. However, some Machine Learning enthusiasts are novice in Maths and will probably find this post disheartening (seriously, this is not my aim). For beginners, you don't need a lot of Mathematics to start doing Machine Learning. The fundamental prerequisite is data analysis as described in this blog post and you can learn the maths on the go as you master more techniques and algorithms. This entry was originally published on my LinkedIn page.