Online convex programming has recently emerged as a powerful primitive for designing machine learning algorithms. For example, OCP can be used for learning a linear classifier, dynamically rebalancing a binary search tree, finding the shortest path in a graph with unknown edge lengths, solving a structured classification problem, or finding a good strategy in an extensive-form game. Several researchers have designed no-regret algorithms for OCP. But, compared to algorithms for special cases of OCP such as learning from expert advice, these algorithms are not very numerous or flexible. In learning from expert advice, one tool which has proved particularly valuable is the correspondence between no-regret algorithms and convex potential functions: by reasoning about these potential functions, researchers have designed algorithms with a wide variety of useful guarantees such as good performance when the target hypothesis is sparse. Until now, there has been no such recipe for the more general OCP problem, and therefore no ability to tune OCP algorithms to take advantage of properties of the problem or data. In this paper we derive a new class of no-regret learning algorithms for OCP. These Lagrangian Hedging algorithms are based on a general class of potential functions, and are a direct generalization of known learning rules like weighted majority and external-regret matching. In addition to proving regret bounds, we demonstrate our algorithms learning to play one-card poker.

Planning in partially observable domains is a notoriously difficult problem. However, in many real-world scenarios, planning can be simplified by decomposing the task into a hierarchy of smaller planning problems. Several approaches have been proposed to optimize a policy that decomposes according to a hierarchy specified a priori. In this paper, we investigate the problem of automatically discovering the hierarchy. More precisely, we frame the optimization of a hierarchical policy as a non-convex optimization problem that can be solved with general nonlinear solvers, a mixed-integer nonlinear approximation or a form of bounded hierarchical policy iteration. By encoding the hierarchical structure as variables of the optimization problem, we can automatically discover a hierarchy. Our method is flexible enough to allow any parts of the hierarchy to be specified based on prior knowledge while letting the optimization discover the unknown parts. It can also discover hierarchical policies, including recursive policies, that are more compact (potentially infinitely fewer parameters) and often easier to understand given the decomposition induced by the hierarchy.

Efficient coding models predict that the optimal code for natural images is a population of oriented Gabor receptive fields. These results match response properties of neurons in primary visual cortex, but not those in the retina. Does the retina use an optimal code, and if so, what is it optimized for? Previous theories of retinal coding have assumed that the goal is to encode the maximal amount of information about the sensory signal. However, the image sampled by retinal photoreceptors is degraded both by the optics of the eye and by the photoreceptor noise. Therefore, de-blurring and de-noising of the retinal signal should be important aspects of retinal coding.

Linear Gaussian State-Space Models are widely used and a Bayesian treatment of parameters is therefore of considerable interest. The approximate Variational Bayesian method applied to these models is an attractive approach, used successfully in applications ranging from acoustics to bioinformatics. The most challenging aspect of implementing the method is in performing inference on the hidden state sequence of the model. We show how to convert the inference problem so that standard Kalman Filtering/Smoothing recursions from the literature may be applied. This is in contrast to previously published approaches based on Belief Propagation. Our framework both simplifies and unifies the inference problem, so that future applications may be more easily developed. We demonstrate the elegance of the approach on Bayesian temporal ICA, with an application to finding independent dynamical processes underlying noisy EEG signals.

Markov networks are commonly used in a wide variety of applications, ranging from computer vision, to natural language, to computational biology. In most current applications, even those that rely heavily on learned models, the structure of the Markov network is constructed by hand, due to the lack of effective algorithms for learning Markov network structure from data. In this paper, we provide a computationally efficient method for learning Markov network structure from data.

Kernel-based regularized learning seeks a model in a hypothesis space by minimizing the empirical error and the model's complexity. Based on the representer theorem, the solution consists of a linear combination of translates of a kernel. This paper investigates a generalized form of representer theorem for kernel-based learning. After mapping predefined features and translates of a kernel simultaneously onto a hypothesis space by a specific way of constructing kernels, we proposed a new algorithm by utilizing a generalized regularizer which leaves part of the space unregularized. Using a squared-loss function in calculating the empirical error, a simple convex solution is obtained which combines predefined features with translates of the kernel. Empirical evaluations have confirmed the effectiveness of the algorithm for supervised learning tasks.

Dirichlet Process (DP) mixture models are promising candidates for clustering applications where the number of clusters is unknown a priori. Due to computational considerations these models are unfortunately unsuitable for large scale data-mining applications. We propose a class of deterministic accelerated DP mixture models that can routinely handle millions of data-cases. The speedup is achieved by incorporating kd-trees into a variational Bayesian algorithm for DP mixtures in the stick-breaking representation, similar to that of Blei and Jordan (2005). Our algorithm differs in the use of kd-trees and in the way we handle truncation: we only assume that the variational distributions are fixed at their priors after a certain level. Experiments show that speedups relative to the standard variational algorithm can be significant.

A number of exact and approximate methods are available for inference calculations in graphical models. Many recent approximate methods for graphs with cycles are based on tractable algorithms for tree structured graphs. Here we base the approximation on a different tractable model, planar graphs with binary variables and pure interaction potentials (no external field). The partition function for such models can be calculated exactly using an algorithm introduced by Fisher and Kasteleyn in the 1960s. We show how such tractable planar models can be used in a decomposition to derive upper bounds on the partition function of non-planar models. The resulting algorithm also allows for the estimation of marginals. We compare our planar decomposition to the tree decomposition method of Wainwright et.

In this paper, we generalize the previous iLSTD algorithm and present three new results: (1) the first convergence proof for an iLSTD algorithm; (2) an extension to incorporate eligibility traces without changing the asymptotic computational complexity; and (3) the first empirical results with an iLSTD algorithm for a problem (mountain car) with feature vectors large enough (n 10, 000) to show substantial computational advantages over LSTD.

We propose new PAC-Bayes bounds for the risk of the weighted majority vote that depend on the mean and variance of the error of its associated Gibbs classifier. We show that these bounds can be smaller than the risk of the Gibbs classifier and can be arbitrarily close to zero even if the risk of the Gibbs classifier is close to 1/2. Moreover, we show that these bounds can be uniformly estimated on the training data for all possible posteriors Q. Moreover, they can be improved by using a large sample of unlabelled data.