For large matrix factorisation problems, we develop a distributed Markov Chain Monte Carlo (MCMC) method based on stochastic gradient Langevin dynamics (SGLD) that we call Parallel SGLD (PSGLD). PSGLD has very favourable scaling properties with increasing data size and is comparable in terms of computational requirements to optimisation methods based on stochastic gradient descent. PSGLD achieves high performance by exploiting the conditional independence structure of the MF models to sub-sample data in a systematic manner as to allow paralleli-sation and distributed computation. We provide a convergence proof of the algorithm and verify its superior performance on various architectures such as Graphics Processing Units, shared memory multi-core systems and multi-computer clusters.

Consider the problem of a controller sampling sequentially from a finite number of $N \geq 2$ populations, specified by random variables $X^i_k$, $ i = 1,\ldots , N,$ and $k = 1, 2, \ldots$; where $X^i_k$ denotes the outcome from population $i$ the $k^{th}$ time it is sampled. It is assumed that for each fixed $i$, $\{ X^i_k \}_{k \geq 1}$ is a sequence of i.i.d. uniform random variables over some interval $[a_i, b_i]$, with the support (i.e., $a_i, b_i$) unknown to the controller. The objective is to have a policy $\pi$ for deciding, based on available data, from which of the $N$ populations to sample from at any time $n=1,2,\ldots$ so as to maximize the expected sum of outcomes of $n$ samples or equivalently to minimize the regret due to lack on information of the parameters $\{ a_i \}$ and $\{ b_i \}$. In this paper, we present a simple inflated sample mean (ISM) type policy that is asymptotically optimal in the sense of its regret achieving the asymptotic lower bound of Burnetas and Katehakis (1996). Additionally, finite horizon regret bounds are given.

IllinoisSL is a Java library for learning structured prediction models. It supports structured Support Vector Machines and structured Perceptron. The library consists of a core learning module and several applications, which can be executed from command-lines. Documentation is provided to guide users. In Comparison to other structured learning libraries, IllinoisSL is efficient, general, and easy to use.

Deep dynamic generative models are developed to learn sequential dependencies in time-series data. The multi-layered model is designed by constructing a hierarchy of temporal sigmoid belief networks (TSBNs), defined as a sequential stack of sigmoid belief networks (SBNs). Each SBN has a contextual hidden state, inherited from the previous SBNs in the sequence, and is used to regulate its hidden bias. Scalable learning and inference algorithms are derived by introducing a recognition model that yields fast sampling from the variational posterior. This recognition model is trained jointly with the generative model, by maximizing its variational lower bound on the log-likelihood. Experimental results on bouncing balls, polyphonic music, motion capture, and text streams show that the proposed approach achieves state-of-the-art predictive performance, and has the capacity to synthesize various sequences.

As a model for an on-line classification setting we consider a stochastic process $(X_{-n},Y_{-n})_{n}$, the present time-point being denoted by 0, with observables $ \ldots,X_{-n},X_{-n+1},\ldots, X_{-1}, X_0$ from which the pattern $Y_0$ is to be inferred. So in this classification setting, in addition to the present observation $X_0$ a number $l$ of preceding observations may be used for classification, thus taking a possible dependence structure into account as it occurs e.g. in an ongoing classification of handwritten characters. We treat the question how the performance of classifiers is improved by using such additional information. For our analysis, a hidden Markov model is used. Letting $R_l$ denote the minimal risk of misclassification using $l$ preceding observations we show that the difference $\sup_k |R_l - R_{l+k}|$ decreases exponentially fast as $l$ increases. This suggests that a small $l$ might already lead to a noticeable improvement. To follow this point we look at the use of past observations for kernel classification rules. Our practical findings in simulated hidden Markov models and in the classification of handwritten characters indicate that using $l=1$, i.e. just the last preceding observation in addition to $X_0$, can lead to a substantial reduction of the risk of misclassification. So, in the presence of stochastic dependencies, we advocate to use $ X_{-1},X_0$ for finding the pattern $Y_0$ instead of only $X_0$ as one would in the independent situation.

Applying standard Markov chain Monte Carlo (MCMC) algorithms to large data sets is computationally infeasible. The recently proposed stochastic gradient Langevin dynamics (SGLD) method circumvents this problem in three ways: it generates proposed moves using only a subset of the data, it skips the Metropolis-Hastings accept-reject step, and it uses sequences of decreasing step sizes. In \cite{TehThierryVollmerSGLD2014}, we provided the mathematical foundations for the decreasing step size SGLD, including consistency and a central limit theorem. However, in practice the SGLD is run for a relatively small number of iterations, and its step size is not decreased to zero. The present article investigates the behaviour of the SGLD with fixed step size. In particular we characterise the asymptotic bias explicitly, along with its dependence on the step size and the variance of the stochastic gradient. On that basis a modified SGLD which removes the asymptotic bias due to the variance of the stochastic gradients up to first order in the step size is derived. Moreover, we are able to obtain bounds on the finite-time bias, variance and mean squared error (MSE). The theory is illustrated with a Gaussian toy model for which the bias and the MSE for the estimation of moments can be obtained explicitly. For this toy model we study the gain of the SGLD over the standard Euler method in the limit of large data sets.

Transferring knowledge across a sequence of related tasks is an important challenge in reinforcement learning (RL). Despite much encouraging empirical evidence, there has been little theoretical analysis. In this paper, we study a class of lifelong RL problems: the agent solves a sequence of tasks modeled as finite Markov decision processes (MDPs), each of which is from a finite set of MDPs with the same state/action sets and different transition/reward functions. Motivated by the need for cross-task exploration in lifelong learning, we formulate a novel online coupon-collector problem and give an optimal algorithm. This allows us to develop a new lifelong RL algorithm, whose overall sample complexity in a sequence of tasks is much smaller than single-task learning, even if the sequence of tasks is generated by an adversary. Benefits of the algorithm are demonstrated in simulated problems, including a recently introduced human-robot interaction problem.

Continuous vector representations of words and objects appear to carry surprisingly rich semantic content. In this paper, we advance both the conceptual and theoretical understanding of word embeddings in three ways. First, we ground embeddings in semantic spaces studied in cognitive-psychometric literature and introduce new evaluation tasks. Second, in contrast to prior work, we take metric recovery as the key object of study, unify existing algorithms as consistent metric recovery methods based on co-occurrence counts from simple Markov random walks, and propose a new recovery algorithm. Third, we generalize metric recovery to graphs and manifolds, relating co-occurence counts on random walks in graphs and random processes on manifolds to the underlying metric to be recovered, thereby reconciling manifold estimation and embedding algorithms. We compare embedding algorithms across a range of tasks, from nonlinear dimensionality reduction to three semantic language tasks, including analogies, sequence completion, and classification.

Gathering the most information by picking the least amount of data is a common task in experimental design or when exploring an unknown environment in reinforcement learning and robotics. A widely used measure for quantifying the information contained in some distribution of interest is its entropy. Greedily minimizing the expected entropy is therefore a standard method for choosing samples in order to gain strong beliefs about the underlying random variables. We show that this approach is prone to temporally getting stuck in local optima corresponding to wrongly biased beliefs. We suggest instead maximizing the expected cross entropy between old and new belief, which aims at challenging refutable beliefs and thereby avoids these local optima. We show that both criteria are closely related and that their difference can be traced back to the asymmetry of the Kullback-Leibler divergence. In illustrative examples as well as simulated and real-world experiments we demonstrate the advantage of cross entropy over simple entropy for practical applications.

Models for recommender systems use latent factors to explain the preferences and behaviors of users with respect to a set of items (e.g., movies, books, academic papers). Typically, the latent factors are assumed to be static and, given these factors, the observed preferences and behaviors of users are assumed to be generated without order. These assumptions limit the explorative and predictive capabilities of such models, since users' interests and item popularity may evolve over time. To address this, we propose dPF, a dynamic matrix factorization model based on the recent Poisson factorization model for recommendations. dPF models the time evolving latent factors with a Kalman filter and the actions with Poisson distributions. We derive a scalable variational inference algorithm to infer the latent factors. Finally, we demonstrate dPF on 10 years of user click data from arXiv.org, one of the largest repository of scientific papers and a formidable source of information about the behavior of scientists. Empirically we show performance improvement over both static and, more recently proposed, dynamic recommendation models. We also provide a thorough exploration of the inferred posteriors over the latent variables.