We present a novel feature selection algorithm for the $k$-means clustering problem. Our algorithm is randomized and, assuming an accuracy parameter $\epsilon \in (0,1)$, selects and appropriately rescales in an unsupervised manner $\Theta(k \log(k / \epsilon) / \epsilon^2)$ features from a dataset of arbitrary dimensions. We prove that, if we run any $\gamma$-approximate $k$-means algorithm ($\gamma \geq 1$) on the features selected using our method, we can find a $(1+(1+\epsilon)\gamma)$-approximate partition with high probability.

We present a novel linear program for the approximation of the dynamic programming cost-to-go function in high-dimensional stochastic control problems. LP approaches to approximate DP naturally restrict attention to approximations that are lower bounds to the optimal cost-to-go function. Our program -- the `smoothed approximate linear program -- relaxes this restriction in an appropriate fashion while remaining computationally tractable. Doing so appears to have several advantages: First, we demonstrate superior bounds on the quality of approximation to the optimal cost-to-go function afforded by our approach. Second, experiments with our approach on a challenging problem (the game of Tetris) show that the approach outperforms the existing LP approach (which has previously been shown to be competitive with several ADP algorithms) by an order of magnitude.

We consider the problem of using nearest neighbor methods to provide a conditional probability estimate, P(y|a), when the number of labels y is large and the labels share some underlying structure. We propose a method for learning error-correcting output codes (ECOCs) to model the similarity between labels within a nearest neighbor framework. The learned ECOCs and nearest neighbor information are used to provide conditional probability estimates. We apply these estimates to the problem of acoustic modeling for speech recognition. We demonstrate an absolute reduction in word error rate (WER) of 0.9% (a 2.5% relative reduction in WER) on a lecture recognition task over a state-of-the-art baseline GMM model.

We study unsupervised learning in a probabilistic generative model for occlusion. The model uses two types of latent variables: one indicates which objects are present in the image, and the other how they are ordered in depth. This depth order then determines how the positions and appearances of the objects present, specified in the model parameters, combine to form the image. We show that the object parameters can be learnt from an unlabelled set of images in which objects occlude one another. Exact maximum-likelihood learning is intractable. However, we show that tractable approximations to Expectation Maximization (EM) can be found if the training images each contain only a small number of objects on average. In numerical experiments it is shown that these approximations recover the correct set of object parameters. Experiments on a novel version of the bars test using colored bars, and experiments on more realistic data, show that the algorithm performs well in extracting the generating causes. Experiments based on the standard bars benchmark test for object learning show that the algorithm performs well in comparison to other recent component extraction approaches. The model and the learning algorithm thus connect research on occlusion with the research field of multiple-cause component extraction methods.

A visual attention system should respond placidly when common stimuli are presented, while at the same time keep alert to anomalous visual inputs. In this paper, a dynamic visual attention model based on the rarity of features is proposed. We introduce the Incremental Coding Length (ICL) to measure the perspective entropy gain of each feature. The objective of our model is to maximize the entropy of the sampled visual features. In order to optimize energy consumption, the limit amount of energy of the system is re-distributed amongst features according to their Incremental Coding Length. By selecting features with large coding length increments, the computational system can achieve attention selectivity in both static and dynamic scenes. We demonstrate that the proposed model achieves superior accuracy in comparison to mainstream approaches in static saliency map generation. Moreover, we also show that our model captures several less-reported dynamic visual search behaviors, such as attentional swing and inhibition of return.

This paper tackles the problem of selecting among several linear estimators in non-parametric regression; this includes model selection for linear regression, the choice of a regularization parameter in kernel ridge regression or spline smoothing, and the choice of a kernel in multiple kernel learning. We propose a new algorithm which first estimates consistently the variance of the noise, based upon the concept of minimal penalty which was previously introduced in the context of model selection. Then, plugging our variance estimate in Mallows $C_L$ penalty is proved to lead to an algorithm satisfying an oracle inequality. Simulation experiments with kernel ridge regression and multiple kernel learning show that the proposed algorithm often improves significantly existing calibration procedures such as 10-fold cross-validation or generalized cross-validation.

Sequential decision-making with multiple agents and imperfect information is commonly modeled as an extensive game. One efficient method for computing Nash equilibria in large, zero-sum, imperfect information games is counterfactual regret minimization (CFR). In the domain of poker, CFR has proven effective, particularly when using a domain-specific augmentation involving chance outcome sampling. In this paper, we describe a general family of domain independent CFR sample-based algorithms called Monte Carlo counterfactual regret minimization (MCCFR) of which the original and poker-specific versions are special cases. We start by showing that MCCFR performs the same regret updates as CFR on expectation. Then, we introduce two sampling schemes: {\it outcome sampling} and {\it external sampling}, showing that both have bounded overall regret with high probability. Thus, they can compute an approximate equilibrium using self-play. Finally, we prove a new tighter bound on the regret for the original CFR algorithm and relate this new bound to MCCFRs bounds. We show empirically that, although the sample-based algorithms require more iterations, their lower cost per iteration can lead to dramatically faster convergence in various games.

A series of corrections is developed for the fixed points of Expectation Propagation (EP),which is one of the most popular methods for approximate probabilistic inference. These corrections can lead to improvements of the inference approximation orserve as a sanity check, indicating when EP yields unrealiable results.

Solving multi-agent reinforcement learning problems has proven difficult because of the lack of tractable algorithms. We provide the first approximation algorithm which solves stochastic games to within $\epsilon$ relative error of the optimal game-theoretic solution, in time polynomial in $1/\epsilon$. Our algorithm extends Murrays and Gordon’s (2007) modified Bellman equation which determines the \emph{set} of all possible achievable utilities; this provides us a truly general framework for multi-agent learning. Further, we empirically validate our algorithm and find the computational cost to be orders of magnitude less than what the theory predicts.

We propose a Bayesian nonparametric approach to relating multiple time series via a set of latent, dynamical behaviors. Using a beta process prior, we allow data-driven selection of the size of this set, as well as the pattern with which behaviors are shared among time series. Via the Indian buffet process representation of the beta process predictive distributions, we develop an exact Markov chain Monte Carlo inference method. In particular, our approach uses the sum-product algorithm to efficiently compute Metropolis-Hastings acceptance probabilities, and explores new dynamical behaviors via birth/death proposals. We validate our sampling algorithm using several synthetic datasets, and also demonstrate promising unsupervised segmentation of visual motion capture data.