Both learning and inference tasks on Bayesian networks are NP-hard in general. Bounded tree-width Bayesian networks have recently received a lot of attention as a way to circumvent this complexity issue; however, while inference on bounded tree-width networks is tractable, the learning problem remains NP-hard even for tree-width~2. In this paper, we propose bounded vertex cover number Bayesian networks as an alternative to bounded tree-width networks. In particular, we show that both inference and learning can be done in polynomial time for any fixed vertex cover number bound $k$, in contrast to the general and bounded tree-width cases; on the other hand, we also show that learning problem is W[1]-hard in parameter $k$. Furthermore, we give an alternative way to learn bounded vertex cover number Bayesian networks using integer linear programming (ILP), and show this is feasible in practice.

We consider the problem of finding M best diverse solutions of energy minimization problems for graphical models. Contrary to the sequential method of Batra et al., which greedily finds one solution after another, we infer all $M$ solutions jointly. It was shown recently that such jointly inferred labelings not only have smaller total energy but also qualitatively outperform the sequentially obtained ones. The only obstacle for using this new technique is the complexity of the corresponding inference problem, since it is considerably slower algorithm than the method of Batra et al. In this work we show that the joint inference of $M$ best diverse solutions can be formulated as a submodular energy minimization if the original MAP-inference problem is submodular, hence fast inference techniques can be used. In addition to the theoretical results we provide practical algorithms that outperform the current state-of-the art and can be used in both submodular and non-submodular case.

The modern scale of data has brought new challenges to Bayesian inference. In particular, conventional MCMC algorithms are computationally very expensive for large data sets. A promising approach to solve this problem is embarrassingly parallel MCMC (EP-MCMC), which first partitions the data into multiple subsets and runs independent sampling algorithms on each subset. The subset posterior draws are then aggregated via some combining rules to obtain the final approximation. Existing EP-MCMC algorithms are limited by approximation accuracy and difficulty in resampling. In this article, we propose a new EP-MCMC algorithm PART that solves these problems. The new algorithm applies random partition trees to combine the subset posterior draws, which is distribution-free, easy to resample from and can adapt to multiple scales. We provide theoretical justification and extensive experiments illustrating empirical performance.

We reexamine in this article the conceptual and mathematical framework for understanding the organization of plasticity in spiking neural networks. We propose that inherent stochasticity enables synaptic plasticity to carry out probabilistic inference by sampling from a posterior distribution of synaptic parameters. This view provides a viable alternative to existing models that propose convergence of synaptic weights to maximum likelihood parameters. It explains how priors on weight distributions and connection probabilities can be merged optimally with learned experience. In simulations we show that our model for synaptic plasticity allows spiking neural networks to compensate continuously for unforeseen disturbances. Furthermore it provides a normative mathematical framework to better understand the permanent variability and rewiring observed in brain networks.

This paper presents a methodology for creating streaming, distributed inference algorithms for Bayesian nonparametric (BNP) models. In the proposed framework, processing nodes receive a sequence of data minibatches, compute a variational posterior for each, and make asynchronous streaming updates to a central model. In contrast to previous algorithms, the proposed framework is truly streaming, distributed, asynchronous, learning-rate-free, and truncation-free. The key challenge in developing the framework, arising from fact that BNP models do not impose an inherent ordering on their components, is finding the correspondence between minibatch and central BNP posterior components before performing each update. To address this, the paper develops a combinatorial optimization problem over component correspondences, and provides an efficient solution technique. The paper concludes with an application of the methodology to the DP mixture model, with experimental results demonstrating its practical scalability and performance.

Expectation Propagation is a very popular algorithm for variational inference, but comes with few theoretical guarantees. In this article, we prove that the approximation errors made by EP can be bounded. Our bounds have an asymptotic interpretation in the number n of datapoints, which allows us to study EP's convergence with respect to the true posterior. In particular, we show that EP converges at a rate of $O(n^{-2})$ for the mean, up to an order of magnitude faster than the traditional Gaussian approximation at the mode. We also give similar asymptotic expansions for moments of order 2 to 4, as well as excess Kullback-Leibler cost (defined as the additional KL cost incurred by using EP rather than the ideal Gaussian approximation). All these expansions highlight the superior convergence properties of EP. Our approach for deriving those results is likely applicable to many similar approximate inference methods. In addition, we introduce bounds on the moments of log-concave distributions that may be of independent interest.

To improve the efficiency of Monte Carlo estimation, practitioners are turning to biased Markov chain Monte Carlo procedures that trade off asymptotic exactness for computational speed. The reasoning is sound: a reduction in variance due to more rapid sampling can outweigh the bias introduced. However, the inexactness creates new challenges for sampler and parameter selection, since standard measures of sample quality like effective sample size do not account for asymptotic bias. To address these challenges, we introduce a new computable quality measure based on Stein's method that bounds the discrepancy between sample and target expectations over a large class of test functions. We use our tool to compare exact, biased, and deterministic sample sequences and illustrate applications to hyperparameter selection, convergence rate assessment, and quantifying bias-variance tradeoffs in posterior inference.

Neural population activity often exhibits rich variability. This variability can arise from single-neuron stochasticity, neural dynamics on short timescales, as well as from modulations of neural firing properties on long timescales, often referred to as neural non-stationarity. To better understand the nature of co-variability in neural circuits and their impact on cortical information processing, we introduce a hierarchical dynamics model that is able to capture both slow inter-trial modulations infiring rates as well as neural population dynamics. We derive a Bayesian Laplace propagation algorithm for joint inference of parameters and population states. On neural population recordings from primary visual cortex, we demonstrate thatour model provides a better account of the structure of neural firing than stationary dynamics models.

We develop a sequential low-complexity inference procedure for Dirichlet process mixtures of Gaussians for online clustering and parameter estimation when the number of clusters are unknown a-priori. We present an easily computable, closed form parametric expression for the conditional likelihood, in which hyperparameters are recursively updated as a function of the streaming data assuming conjugate priors. Motivated by large-sample asymptotics, we propose a noveladaptive low-complexity design for the Dirichlet process concentration parameter and show that the number of classes grow at most at a logarithmic rate. We further prove that in the large-sample limit, the conditional likelihood and datapredictive distribution become asymptotically Gaussian. We demonstrate through experiments on synthetic and real data sets that our approach is superior to otheronline state-of-the-art methods.

The rise of Big Data has led to new demands for Machine Learning (ML) systems to learn complex models with millions to billions of parameters, that promise adequate capacity to digest massive datasets and offer powerful predictive analytics thereupon. In order to run ML algorithms at such scales, on a distributed cluster with 10s to 1000s of machines, it is often the case that significant engineering efforts are required --- and one might fairly ask if such engineering truly falls within the domain of ML research or not. Taking the view that Big ML systems can benefit greatly from ML-rooted statistical and algorithmic insights --- and that ML researchers should therefore not shy away from such systems design --- we discuss a series of principles and strategies distilled from our recent efforts on industrial-scale ML solutions. These principles and strategies span a continuum from application, to engineering, and to theoretical research and development of Big ML systems and architectures, with the goal of understanding how to make them efficient, generally-applicable, and supported with convergence and scaling guarantees. They concern four key questions which traditionally receive little attention in ML research: How to distribute an ML program over a cluster? How to bridge ML computation with inter-machine communication? How to perform such communication? What should be communicated between machines? By exposing underlying statistical and algorithmic characteristics unique to ML programs but not typically seen in traditional computer programs, and by dissecting successful cases to reveal how we have harnessed these principles to design and develop both high-performance distributed ML software as well as general-purpose ML frameworks, we present opportunities for ML researchers and practitioners to further shape and grow the area that lies between ML and systems.