We reduce a broad class of fundamental machine learning problems, usually addressed by EM or sampling, to the problem of finding the k extreme rays spanning the conical hull of a1 data point set. These k "anchors" lead to a global solution and a more interpretable model that can even outperform EM and sampling on generalization error. To find the k anchors, we propose a novel divide-andconquer learning scheme "DCA" that distributes the problem to O(k log k) sametype sub-problems on different low-D random hyperplanes, each can be solved independently by any existing solver. For the 2D sub-problem, we instead present a non-iterative solver that only needs to compute an array of cosine values and its max/min entries. DCA also provides a faster subroutine inside other algorithms to check whether a point is covered in a conical hull, and thus improves these algorithms by providing significant speedups. We apply our method to GMM, HMM, LDA, NMF and subspace clustering, then show its competitive performance and scalability over other methods on large datasets.

We propose a technique for decomposing the parameter learning problem in Bayesian networks into independent learning problems. Our technique applies to incomplete datasets and exploits variables that are either hidden or observed in the given dataset. We show empirically that the proposed technique can lead to orders-of-magnitude savings in learning time. We explain, analytically and empirically, the reasons behind our reported savings, and compare the proposed technique to related ones that are sometimes used by inference algorithms.

We propose a distributed Markov chain Monte Carlo (MCMC) inference algorithm for large scale Bayesian posterior simulation. We assume that the dataset is partitioned and stored across nodes of a cluster. Our procedure involves an independent MCMC posterior sampler at each node based on its local partition of the data. Moment statistics of the local posteriors are collected from each sampler and propagated across the cluster using expectation propagation message passing with low communication costs. The moment sharing scheme improves posterior estimation quality by enforcing agreement among the samplers. We demonstrate the speed and inference quality of our method with empirical studies on Bayesian logistic regression and sparse linear regression with a spike-and-slab prior.

Sampling from hierarchical Bayesian models is often difficult for MCMC methods, because of the strong correlations between the model parameters and the hyperparameters. Recent Riemannian manifold Hamiltonian Monte Carlo (RMHMC) methods have significant potential advantages in this setting, but are computationally expensive. We introduce a new RMHMC method, which we call semi-separable Hamiltonian Monte Carlo, which uses a specially designed mass matrix that allows the joint Hamiltonian over model parameters and hyperparameters to decompose into two simpler Hamiltonians. This structure is exploited by a new integrator which we call the alternating blockwise leapfrog algorithm. The resulting method can mix faster than simpler Gibbs sampling while being simpler and more efficient than previous instances of RMHMC.

Multiple lines of evidence support the notion that the brain performs probabilistic inference in multiple cognitive domains, including perception and decision making. There is also evidence that probabilistic inference may be implemented in the brain through the (quasi-)stochastic activity of neural circuits, producing samples from the appropriate posterior distributions, effectively implementing a Markov chain Monte Carlo algorithm. However, time becomes a fundamental bottleneck in such sampling-based probabilistic representations: the quality of inferences depends on how fast the neural circuit generates new, uncorrelated samples from its stationary distribution (the posterior). We explore this bottleneck in a simple, linear-Gaussian latent variable model, in which posterior sampling can be achieved by stochastic neural networks with linear dynamics. The well-known Langevin sampling (LS) recipe, so far the only sampling algorithm for continuous variables of which a neural implementation has been suggested, naturally fits into this dynamical framework. However, we first show analytically and through simulations that the symmetry of the synaptic weight matrix implied by LS yields critically slow mixing when the posterior is high-dimensional. Next, using methods from control theory, we construct and inspect networks that are optimally fast, and hence orders of magnitude faster than LS, while being far more biologically plausible. In these networks, strong - but transient - selective amplification of external noise generates the spatially correlated activity fluctuations prescribed by the posterior. Intriguingly, although a detailed balance of excitation and inhibition is dynamically maintained, detailed balance of Markov chain steps in the resulting sampler is violated, consistent with recent findings on how statistical irreversibility can overcome the speed limitation of random walks in other domains.

The prediction of time-changing variances is an important task in the modeling of financial data. Standard econometric models are often limited as they assume rigid functional relationships for the evolution of the variance. Moreover, functional parameters are usually learned by maximum likelihood, which can lead to overfitting. To address these problems we introduce GP-Vol, a novel non-parametric model for time-changing variances based on Gaussian Processes. This new model can capture highly flexible functional relationships for the variances. Furthermore, we introduce a new online algorithm for fast inference in GP-Vol. This method is much faster than current offline inference procedures and it avoids overfitting problems by following a fully Bayesian approach. Experiments with financial data show that GP-Vol performs significantly better than current standard alternatives.

The idea that animals might use information-driven planning to explore an unknown environment and build an internal model of it has been proposed for quite some time. Recent work has demonstrated that agents using this principle can efficiently learn models of probabilistic environments with discrete, bounded state spaces. However, animals and robots are commonly confronted with unbounded environments. To address this more challenging situation, we study informationbased learning strategies of agents in unbounded state spaces using non-parametric Bayesian models. Specifically, we demonstrate that the Chinese Restaurant Process (CRP) model is able to solve this problem and that an Empirical Bayes version is able to efficiently explore bounded and unbounded worlds by relying on little prior information.

The integration of partially redundant information from multiple sensors is a standard computational problem for agents interacting with the world. In man and other primates, integration has been shown psychophysically to be nearly optimal in the sense of error minimization. An influential generalization of this notion of optimality is that populations of multisensory neurons should retain all the information from their unisensory afferents about the underlying, common stimulus [1]. More recently, it was shown empirically that a neural network trained to perform latent-variable density estimation, with the activities of the unisensory neurons as observed data, satisfies the information-preservation criterion, even though the model architecture was not designed to match the true generative process for the data [2]. We prove here an analytical connection between these seemingly different tasks, density estimation and sensory integration; that the former implies the latter for the model used in [2]; but that this does not appear to be true for all models.

Humans are able to segregate auditory objects in a complex acoustic scene, through an interplay of bottom-up feature extraction and top-down selective attention in the brain. The detailed mechanism underlying this process is largely unknown and the ability to mimic this procedure is an important problem in artificial intelligence and computational neuroscience. We consider the problem of decoding the attentional state of a listener in a competing-speaker environment from magnetoencephalographic (MEG) recordings from the human brain. We develop a behaviorally inspired state-space model to account for the modulation of the MEG with respect to attentional state of the listener. We construct a decoder based on the maximum a posteriori (MAP) estimate of the state parameters via the Expectation-Maximization (EM) algorithm. The resulting decoder is able to track the attentional modulation of the listener with multi-second resolution using only the envelopes of the two speech streams as covariates.

Gaussian processes (GPs) are a powerful tool for probabilistic inference over functions. They have been applied to both regression and non-linear dimensionality reduction, and offer desirable properties such as uncertainty estimates, robustness to over-fitting, and principled ways for tuning hyper-parameters. However the scalability of these models to big datasets remains an active topic of research. We introduce a novel re-parametrisation of variational inference for sparse GP regression and latent variable models that allows for an efficient distributed algorithm. This is done by exploiting the decoupling of the data given the inducing points to re-formulate the evidence lower bound in a Map-Reduce setting. We show that the inference scales well with data and computational resources, while preserving a balanced distribution of the load among the nodes. We further demonstrate the utility in scaling Gaussian processes to big data. We show that GP performance improves with increasing amounts of data in regression (on flight data with 2 million records) and latent variable modelling (on MNIST). The results show that GPs perform better than many common models often used for big data.