We go beyond the notion of pairwise similarity and look into search problems with $k$-way similarity functions. In this paper, we focus on problems related to \emph{3-way Jaccard} similarity: $\mathcal{R}^{3way}= \frac{|S_1 \cap S_2 \cap S_3|}{|S_1 \cup S_2 \cup S_3|}$, $S_1, S_2, S_3 \in \mathcal{C}$, where $\mathcal{C}$ is a size $n$ collection of sets (or binary vectors). We show that approximate $\mathcal{R}^{3way}$ similarity search problems admit fast algorithms with provable guarantees, analogous to the pairwise case. Our analysis and speedup guarantees naturally extend to $k$-way resemblance. In the process, we extend traditional framework of \emph{locality sensitive hashing (LSH)} to handle higher order similarities, which could be of independent theoretical interest. The applicability of $\mathcal{R}^{3way}$ search is shown on the Google sets" application. In addition, we demonstrate the advantage of $\mathcal{R}^{3way}$ resemblance over the pairwise case in improving retrieval quality."

A probabilistic model based on the horseshoe prior is proposed for learning dependencies in the process of identifying relevant features for prediction. Exact inference is intractable in this model. However, expectation propagation offers an approximate alternative. Because the process of estimating feature selection dependencies may suffer from over-fitting in the model proposed, additional data from a multi-task learning scenario are considered for induction. The same model can be used in this setting with few modifications. Furthermore, the assumptions made are less restrictive than in other multi-task methods: The different tasks must share feature selection dependencies, but can have different relevant features and model coefficients. Experiments with real and synthetic data show that this model performs better than other multi-task alternatives from the literature. The experiments also show that the model is able to induce suitable feature selection dependencies for the problems considered, only from the training data.

We consider the problem of learning good trajectories for manipulation tasks. This is challenging because the criterion defining a good trajectory varies with users, tasks and environments. In this paper, we propose a co-active online learning framework for teaching robots the preferences of its users for object manipulation tasks. The key novelty of our approach lies in the type of feedback expected from the user: the human user does not need to demonstrate optimal trajectories as training data, but merely needs to iteratively provide trajectories that slightly improve over the trajectory currently proposed by the system. We argue that this co-active preference feedback can be more easily elicited from the user than demonstrations of optimal trajectories, which are often challenging and non-intuitive to provide on high degrees of freedom manipulators. Nevertheless, theoretical regret bounds of our algorithm match the asymptotic rates of optimal trajectory algorithms. We also formulate a score function to capture the contextual information and demonstrate the generalizability of our algorithm on a variety of household tasks, for whom, the preferences were not only influenced by the object being manipulated but also by the surrounding environment.

Given a Markov Decision Process (MDP) with $n$ states and $m$ actions per state, we study the number of iterations needed by Policy Iteration (PI) algorithms to converge to the optimal $\gamma$-discounted optimal policy. We consider two variations of PI: Howard's PI that changes the actions in all states with a positive advantage, and Simplex-PI that only changes the action in the state with maximal advantage. We show that Howard's PI terminates after at most $ O \left( \frac{ n m}{1-\gamma} \log \left( \frac{1}{1-\gamma} \right)\right) $ iterations, improving by a factor $O(\log n)$ a result by Hansen et al. (2013), while Simplex-PI terminates after at most $ O \left( \frac{n^2 m}{1-\gamma} \log \left( \frac{1}{1-\gamma} \right)\right) $ iterations, improving by a factor $O(\log n)$ a result by Ye (2011). Under some structural assumptions of the MDP, we then consider bounds that are independent of the discount factor~$\gamma$: given a measure of the maximal transient time $\tau_t$ and the maximal time $\tau_r$ to revisit states in recurrent classes under all policies, we show that Simplex-PI terminates after at most $ \tilde O \left( n^3 m^2 \tau_t \tau_r \right) $ iterations. This generalizes a recent result for deterministic MDPs by Post & Ye (2012), in which $\tau_t \le n$ and $\tau_r \le n$. We explain why similar results seem hard to derive for Howard's PI. Finally, under the additional (restrictive) assumption that the state space is partitioned in two sets, respectively states that are transient and recurrent for all policies, we show that Simplex-PI and Howard's PI terminate after at most $ \tilde O(nm (\tau_t+\tau_r))$ iterations.

Gaussian processes are typically used for smoothing and interpolation on small datasets. We introduce a new Bayesian nonparametric framework -- GPatt -- enabling automatic pattern extrapolation with Gaussian processes on large multidimensional datasets. GPatt unifies and extends highly expressive kernels and fast exact inference techniques. Without human intervention -- no hand crafting of kernel features, and no sophisticated initialisation procedures -- we show that GPatt can solve large scale pattern extrapolation, inpainting, and kernel discovery problems, including a problem with 383400 training points. We find that GPatt significantly outperforms popular alternative scalable Gaussian process methods in speed and accuracy. Moreover, we discover profound differences between each of these methods, suggesting expressive kernels, nonparametric representations, and exact inference are useful for modelling large scale multidimensional patterns.

We consider the partial observability model for multi-armed bandits, introduced by Mannor and Shamir (2011). Our main result is a characterization of regret in the directed observability model in terms of the dominating and independence numbers of the observability graph. We also show that in the undirected case, the learner can achieve optimal regret without even accessing the observability graph before selecting an action. Both results are shown using variants of the Exp3 algorithm operating on the observability graph in a time-efficient manner.

Recently, it has become evident that submodularity naturally captures widely occurring concepts in machine learning, signal processing and computer vision. In consequence, there is need for efficient optimization procedures for submodular functions, in particular for minimization problems. While general submodular minimization is challenging, we propose a new approach that exploits existing decomposability of submodular functions. In contrast to previous approaches, our method is neither approximate, nor impractical, nor does it need any cumbersome parameter tuning. Moreover, it is easy to implement and parallelize. A key component of our approach is a formulation of the discrete submodular minimization problem as a continuous best approximation problem. It is solved through a sequence of reflections and its solution can be automatically thresholded to obtain an optimal discrete solution. Our method solves both the continuous and discrete formulations of the problem, and therefore has applications in learning, inference, and reconstruction. In our experiments, we show the benefits of our new algorithms for two image segmentation tasks.

Recent extensions of the Perceptron, as e.g. the Tempotron, suggest that this theoretical concept is highly relevant also for understanding networks of spiking neurons in the brain. It is not known, however, how the computational power of the Perceptron and of its variants might be accomplished by the plasticity mechanisms of real synapses. Here we prove that spike-timing-dependent plasticity having an anti-Hebbian form for excitatory synapses as well as a spike-timing-dependent plasticity of Hebbian shape for inhibitory synapses are sufficient for realizing the original Perceptron Learning Rule if the respective plasticity mechanisms act in concert with the hyperpolarisation of the post-synaptic neurons. We also show that with these simple yet biologically realistic dynamics Tempotrons are efficiently learned. The proposed mechanism might underly the acquisition of mappings of spatio-temporal activity patterns in one area of the brain onto other spatio-temporal spike patterns in another region and of long term memories in cortex. Our results underline that learning processes in realistic networks of spiking neurons depend crucially on the interactions of synaptic plasticity mechanisms with the dynamics of participating neurons.

We propose a novel class of algorithms for low rank matrix completion. Our approach builds on novel penalty functions on the singular values of the low rank matrix. By exploiting a mixture model representation of this penalty, we show that a suitably chosen set of latent variables enables to derive an Expectation-Maximization algorithm to obtain a Maximum A Posteriori estimate of the completed low rank matrix. The resulting algorithm is an iterative soft-thresholded algorithm which iteratively adapts the shrinkage coefficients associated to the singular values. The algorithm is simple to implement and can scale to large matrices. We provide numerical comparisons between our approach and recent alternatives showing the interest of the proposed approach for low rank matrix completion.

State-space models are successfully used in many areas of science, engineering and economics to model time series and dynamical systems. We present a fully Bayesian approach to inference and learning in nonlinear nonparametric state-space models. We place a Gaussian process prior over the transition dynamics, resulting in a flexible model able to capture complex dynamical phenomena. However, to enable efficient inference, we marginalize over the dynamics of the model and instead infer directly the joint smoothing distribution through the use of specially tailored Particle Markov Chain Monte Carlo samplers. Once an approximation of the smoothing distribution is computed, the state transition predictive distribution can be formulated analytically. We make use of sparse Gaussian process models to greatly reduce the computational complexity of the approach.