We present a theory of compositionality in stochastic optimal control, showing how task-optimal controllers can be constructed from certain primitives. The primitives are themselves feedback controllers pursuing their own agendas. They are mixed in proportion to how much progress they are making towards their agendas and how compatible their agendas are with the present task. The resulting composite control law is provably optimal when the problem belongs to a certain class. This class is rather general and yet has a number of unique properties - one of which is that the Bellman equation can be made linear even for non-linear or discrete dynamics. This gives rise to the compositionality developed here. In the special case of linear dynamics and Gaussian noise our framework yields analytical solutions (i.e. non-linear mixtures of linear-quadratic regulators) without requiring the final cost to be quadratic. More generally, a natural set of control primitives can be constructed by applying SVD to Greens function of the Bellman equation. We illustrate the theory in the context of human arm movements. The ideas of optimality and compositionality are both very prominent in the field of motor control, yet they are hard to reconcile. Our work makes this possible.

The recently proposed \emph{additive noise model} has advantages over previous structure learning algorithms, when attempting to recover some true data generating mechanism, since it (i) does not assume linearity or Gaussianity and (ii) can recover a unique DAG rather than an equivalence class. However, its original extension to the multivariate case required enumerating all possible DAGs, and for some special distributions, e.g. linear Gaussian, the model is invertible and thus cannot be used for structure learning. We present a new approach which combines a PC style search using recent advances in kernel measures of conditional dependence with local searches for additive noise models in substructures of the equivalence class. This results in a more computationally efficient approach that is useful for arbitrary distributions even when additive noise models are invertible. Experiments with synthetic and real data show that this method is more accurate than previous methods when data are nonlinear and/or non-Gaussian.

Search engines today present results that are often oblivious to recent shifts in intent. For example, the meaning of the query independence day shifts in early July to a US holiday and to a movie around the time of the box office release. While no studies exactly quantify the magnitude of intent-shifting traffic, studies suggest that news events, seasonal topics, pop culture, etc account for 1/2 the search queries. This paper shows that the signals a search engine receives can be used to both determine that a shift in intent happened, as well as find a result that is now more relevant. We present a meta-algorithm that marries a classifier with a bandit algorithm to achieve regret that depends logarithmically on the number of query impressions, under certain assumptions. We provide strong evidence that this regret is close to the best achievable. Finally, via a series of experiments, we demonstrate that our algorithm outperforms prior approaches, particularly as the amount of intent-shifting traffic increases.

We consider the problem of learning probabilistic models for complex relational structures between various types of objects. A model can help us "understand" a dataset of relational facts in at least two ways, by finding interpretable structure in the data, and by supporting predictions, or inferences about whether particular unobserved relations are likely to be true. Often there is a tradeoff between these two aims: cluster-based models yield more easily interpretable representations, while factorization-based approaches have given better predictive performance on large data sets. We introduce the Bayesian Clustered Tensor Factorization (BCTF) model, which embeds a factorized representation of relations in a nonparametric Bayesian clustering framework. Inference is fully Bayesian but scales well to large data sets. The model simultaneously discovers interpretable clusters and yields predictive performance that matches or beats previous probabilistic models for relational data.

We consider the reconstruction of sparse signals in the multiple measurement vector (MMV) model,in which the signal, represented as a matrix, consists of a set of jointly sparse vectors. MMV is an extension of the single measurement vector (SMV) model employed in standard compressive sensing (CS). Recent theoretical studies focus on the convex relaxation of the MMV problem based on the $(2,1)$-norm minimization, which is an extension of the well-known $1$-norm minimization employed in SMV. However, the resulting convex optimization problem in MMV is significantly much more difficult to solve than the one in SMV. Existing algorithms reformulate it as a second-order cone programming (SOCP) or semidefinite programming (SDP), which is computationally expensive to solve for problems of moderate size. In this paper, we propose a new (dual) reformulation of the convex optimization problem in MMV and develop an efficient algorithm based on the prox-method. Interestingly, our theoretical analysis reveals the close connection between the proposed reformulation and multiple kernel learning. Our simulation studies demonstrate the scalability of the proposed algorithm.

When individuals independently recollect events or retrieve facts from memory, how can we aggregate these retrieved memories to reconstruct the actual set of events or facts? In this research, we report the performance of individuals in a series of general knowledge tasks, where the goal is to reconstruct from memory the order of historic events, or the order of items along some physical dimension. We introduce two Bayesian models for aggregating order information based on a Thurstonian approach and Mallows model. Both models assume that each individuals reconstruction is based on either a random permutation of the unobserved ground truth, or by a pure guessing strategy. We apply MCMC to make inferences about the underlying truth and the strategies employed by individuals. The models demonstrate a wisdom of crowds" effect, where the aggregated orderings are closer to the true ordering than the orderings of the best individual."

In many domains, humans appear to combine perceptual cues in a near-optimal, probabilistic fashion: two noisy pieces of information tend to be combined linearly with weights proportional to the precision of each cue. Here we present a case where structural information plays an important role. The presence of a background cue gives rise to the possibility of occlusion, and places a soft constraint on the location of a target – in effect propelling it forward. We present an ideal observer model of depth estimation for this situation where structural or ordinal information is important and then fit the model to human data from a stereo-matching task. To test whether subjects are truly using ordinal cues in a probabilistic manner we then vary the uncertainty of the task. We find that the model accurately predicts shifts in subject’s behavior. Our results indicate that the nervous system estimates depth ordering in a probabilistic fashion and estimates the structure of the visual scene during depth perception.

Embeddings of probability measures into reproducing kernel Hilbert spaces have been proposed as a straightforward and practical means of representing and comparing probabilities.In particular, the distance between embeddings (the maximum mean discrepancy, or MMD) has several key advantages over many classical metrics on distributions, namely easy computability, fast convergence and low bias of finite sample estimates. An important requirement of the embedding RKHS is that it be characteristic: in this case, the MMD between two distributions is zero if and only if the distributions coincide. Three new results on the MMD are introduced inthe present study. First, it is established that MMD corresponds to the optimal risk of a kernel classifier, thus forming a natural link between the distance between distributions and their ease of classification. An important consequence is that a kernel must be characteristic to guarantee classifiability between distributions inthe RKHS.

Directed graphical models such as Bayesian networks are a favored formalism to model the dependency structures in complex multivariate systems such as those encountered in biology and neural sciences. When the system is undergoing dynamic transformation, often a temporally rewiring network is needed for capturing the dynamic causal influences between covariates. In this paper, we propose a time-varying dynamic Bayesian network (TV-DBN) for modeling the structurally varying directed dependency structures underlying non-stationary biological/neural time series. This is a challenging problem due the non-stationarity and sample scarcity of the time series. We present a kernel reweighted $\ell_1$ regularized auto-regressive procedure for learning the TV-DBN model. Our method enjoys nice properties such as computational efficiency and provable asymptotic consistency. Applying TV-DBN to time series measurements during yeast cell cycle and brain response to visual stimuli reveals interesting dynamics underlying the respective biological systems.

We investigate how well Gaussian process regression can learn functions defined on graphs, using large regular random graphs as a paradigmatic example. Random-walk based kernels are shown to have some surprising properties: within the standard approximation of a locally tree-like graph structure, the kernel does not become constant, i.e.neighbouring function values do not become fully correlated, when the lengthscale $\sigma$ of the kernel is made large. Instead the kernel attains a non-trivial limiting form, which we calculate. The fully correlated limit is reached only once loops become relevant, and we estimate where the crossover to this regime occurs. Our main subject are learning curves of Bayes error versus training set size. We show that these are qualitatively well predicted by a simple approximation using only the spectrum of a large tree as input, and generically scale with $n/V$, the number of training examples per vertex. We also explore how this behaviour changes once kernel lengthscales are large enough for loops to become important.