Simulated annealing is a popular method for approaching the solution of a global optimization problem. Existing results on its performance apply to discrete combinatorial optimization where the optimization variables can assume only a finite set of possible values. We introduce a new general formulation of simulated annealing which allows one to guarantee finite-time performance in the optimization of functions of continuous variables. The results hold universally for any optimization problem on a bounded domain and establish a connection between simulated annealing and up-to-date theory of convergence of Markov chain Monte Carlo methods on continuous domains. This work is inspired by the concept of finite-time learning with known accuracy and confidence developed in statistical learning theory.

Semi-supervised methods use unlabeled data in addition to labeled data to construct predictors. While existing semi-supervised methods have shown some promising empirical performance, their development has been based largely based on heuristics. In this paper we study semi-supervised learning from the viewpoint of minimax theory. Our first result shows that some common methods based on regularization using graph Laplacians do not lead to faster minimax rates of convergence. Thus, the estimators that use the unlabeled data do not have smaller risk than the estimators that use only labeled data. We then develop several new approaches that provably lead to improved performance. The statistical tools of minimax analysis are thus used to offer some new perspective on the problem of semi-supervised learning.

In many applications, one has to actively select among a set of expensive observations before making an informed decision. Often, we want to select observations which perform well when evaluated with an objective function chosen by an adversary. Examples include minimizing the maximum posterior variance in Gaussian Process regression, robust experimental design, and sensor placement for outbreak detection. In this paper, we present the Submodular Saturation algorithm, a simple and efficient algorithm with strong theoretical approximation guarantees for the case where the possible objective functions exhibit submodularity, an intuitive diminishing returns property. Moreover, we prove that better approximation algorithms do not exist unless NPcomplete problems admit efficient algorithms. We evaluate our algorithm on several real-world problems. For Gaussian Process regression, our algorithm compares favorably with state-of-the-art heuristics described in the geostatistics literature, while being simpler, faster and providing theoretical guarantees. For robust experimental design, our algorithm performs favorably compared to SDP-based algorithms.

Point-based algorithms have been surprisingly successful in computing approximately optimal solutions for partially observable Markov decision processes (POMDPs) in high dimensional belief spaces. In this work, we seek to understand the belief-space properties that allow some POMDP problems to be approximated efficiently and thus help to explain the point-based algorithms' success often observed in the experiments. We show that an approximately optimal POMDP solution can be computed in time polynomial in the covering number of a reachable belief space, which is the subset of the belief space reachable from a given belief point. We also show that under the weaker condition of having a small covering number for an optimal reachable space, which is the subset of the belief space reachable under an optimal policy, computing an approximately optimal solution is NPhard. However, given a suitable set of points that "cover" an optimal reachable space well, an approximate solution can be computed in polynomial time. The covering number highlights several interesting properties that reduce the complexity of POMDP planning in practice, e.g., fully observed state variables, beliefs with sparse support, smooth beliefs, and circulant state-transition matrices.

Although kernel measures of independence have been widely applied in machine learning (notably in kernel ICA), there is as yet no method to determine whether they have detected statistically significant dependence. We provide a novel test of the independence hypothesis for one particular kernel independence measure, the Hilbert-Schmidt independence criterion (HSIC).

Machine learning techniques are increasingly being used to produce a wide-range of classifiers for complex real-world applications that involve nonuniform testing costs and misclassification costs. As the complexity of these applications grows, the management of resources during the learning and classification processes becomes a challenging task. In this work we introduce ACT (Anytime Cost-sensitive Trees), a novel framework for operating in such environments. ACT is an anytime algorithm that allows trading computation time for lower classification costs. It builds a tree top-down and exploits additional time resources to obtain better estimations for the utility of the different candidate splits.

Although theorists have interpreted classical conditioning as a laboratory model of Bayesian belief updating, a recent reanalysis showed that the key features that theoretical models capture about learning are artifacts of averaging over subjects. Rather than learning smoothly to asymptote (reflecting, according to Bayesian models, the gradual tradeoff from prior to posterior as data accumulate), subjects learn suddenly and their predictions fluctuate perpetually. We suggest that abrupt and unstable learning can be modeled by assuming subjects are conducting inference using sequential Monte Carlo sampling with a small number of samples -- one, in our simulations. Ensemble behavior resembles exact Bayesian models since, as in particle filters, it averages over many samples. Further, the model is capable of exhibiting sophisticated behaviors like retrospective revaluation at the ensemble level, even given minimally sophisticated individuals that do not track uncertainty in their beliefs over trials.

A novel approach to measure the interdependence of two time series is proposed, referred to as "stochastic event synchrony" (SES); it quantifies the alignment of two point processes by means of the following parameters: time delay, variance of the timing jitter, fraction of "spurious" events, and average similarity of events. SES may be applied to generic one-dimensional and multidimensional point processes, however, the paper mainly focusses on point processes in time-frequency domain. The average event similarity is in that case described by two parameters: the average frequency offset between events in the time-frequency plane, and the variance of the frequency offset ("frequency jitter"); SES then consists of five parameters in total. Those parameters quantify the synchrony of oscillatory events, and hence, they provide an alternative to existing synchrony measures that quantify amplitude or phase synchrony. The pairwise alignment of point processes is cast as a statistical inference problem, which is solved by applying the maxproduct algorithm on a graphical model. The SES parameters are determined from the resulting pairwise alignment by maximum a posteriori (MAP) estimation. The proposed interdependence measure is applied to the problem of detecting anomalies in EEG synchrony of Mild Cognitive Impairment (MCI) patients; the results indicate that SES significantly improves the sensitivity of EEG in detecting MCI.

Neural spike trains present challenges to analytical efforts due to their noisy, spiking nature. Many studies of neuroscientific and neural prosthetic importance rely on a smoothed, denoised estimate of the spike train's underlying firing rate. Current techniques to find time-varying firing rates require ad hoc choices of parameters, offer no confidence intervals on their estimates, and can obscure potentially important single trial variability. We present a new method, based on a Gaussian Process prior, for inferring probabilistically optimal estimates of firing rate functions underlying single or multiple neural spike trains. We test the performance of the method on simulated data and experimentally gathered neural spike trains, and we demonstrate improvements over conventional estimators.

In this paper we formulate a novel AND/OR graph representation capable of describing the different configurations of deformable articulated objects such as horses. The representation makes use of the summarization principle so that lower level nodes in the graph only pass on summary statistics to the higher level nodes. The probability distributions are invariant to position, orientation, and scale. We develop a novel inference algorithm that combined a bottom-up process for proposing configurations for horses together with a top-down process for refining and validating these proposals. The strategy of surround suppression is applied to ensure that the inference time is polynomial in the size of input data. The algorithm was applied to the tasks of detecting, segmenting and parsing horses. We demonstrate that the algorithm is fast and comparable with the state of the art approaches.