The need for appropriate ways to measure the distance or similarity between data is ubiquitous in machine learning, pattern recognition and data mining, but handcrafting such good metrics for specific problems is generally difficult. This has led to the emergence of metric learning, which aims at automatically learning a metric from data and has attracted a lot of interest in machine learning and related fields for the past ten years. This survey paper proposes a systematic review of the metric learning literature, highlighting the pros and cons of each approach. We pay particular attention to Mahalanobis distance metric learning, a well-studied and successful framework, but additionally present a wide range of methods that have recently emerged as powerful alternatives, including nonlinear metric learning, similarity learning and local metric learning. Recent trends and extensions, such as semi-supervised metric learning, metric learning for histogram data and the derivation of generalization guarantees, are also covered. Finally, this survey addresses metric learning for structured data, in particular edit distance learning, and attempts to give an overview of the remaining challenges in metric learning for the years to come.

Least squares fitting is in general not useful for high-dimensional linear models, in which the number of predictors is of the same or even larger order of magnitude than the number of samples. Theory developed in recent years has coined a paradigm according to which sparsity-promoting regularization is regarded as a necessity in such setting. Deviating from this paradigm, we show that non-negativity constraints on the regression coefficients may be similarly effective as explicit regularization if the design matrix has additional properties, which are met in several applications of non-negative least squares (NNLS). We show that for these designs, the performance of NNLS with regard to prediction and estimation is comparable to that of the lasso. We argue further that in specific cases, NNLS may have a better $\ell_{\infty}$-rate in estimation and hence also advantages with respect to support recovery when combined with thresholding. From a practical point of view, NNLS does not depend on a regularization parameter and is hence easier to use.

We present a numerical algorithm for nonnegative matrix factorization (NMF) problems under noisy separability. An NMF problem under separability can be stated as one of finding all vertices of the convex hull of data points. The research interest of this paper is to find the vectors as close to the vertices as possible in a situation in which noise is added to the data points. Our algorithm is designed to capture the shape of the convex hull of data points by using its enclosing ellipsoid. We show that the algorithm has correctness and robustness properties from theoretical and practical perspectives; correctness here means that if the data points do not contain any noise, the algorithm can find the vertices of their convex hull; robustness means that if the data points contain noise, the algorithm can find the near-vertices. Finally, we apply the algorithm to document clustering, and report the experimental results.

This chapter overviews a recently introduced network-based model of combinatorial landscapes: Local Optima Networks (LON). The model compresses the information given by the whole search space into a smaller mathematical object that is a graph having as vertices the local optima and as edges the possible weighted transitions between them. Two definitions of edges have been proposed: basin-transition and escape-edges, which capture relevant topological features of the underlying search spaces. This network model brings a new set of metrics to characterize the structure of combinatorial landscapes, those associated with the science of complex networks. These metrics are described, and results are presented of local optima network extraction and analysis for two selected combinatorial landscapes: NK landscapes and the quadratic assignment problem. Network features are found to correlate with and even predict the performance of heuristic search algorithms operating on these problems.

Learning policies that generalize across multiple tasks is an important and challenging research topic in reinforcement learning and robotics. Training individual policies for every single potential task is often impractical, especially for continuous task variations, requiring more principled approaches to share and transfer knowledge among similar tasks. We present a novel approach for learning a nonlinear feedback policy that generalizes across multiple tasks. The key idea is to define a parametrized policy as a function of both the state and the task, which allows learning a single policy that generalizes across multiple known and unknown tasks. Applications of our novel approach to reinforcement and imitation learning in real-robot experiments are shown.

Existing Bayesian models, especially nonparametric Bayesian methods, rely on specially conceived priors to incorporate domain knowledge for discovering improved latent representations. While priors can affect posterior distributions through Bayes' rule, imposing posterior regularization is arguably more direct and in some cases more natural and general. In this paper, we present regularized Bayesian inference (RegBayes), a novel computational framework that performs posterior inference with a regularization term on the desired post-data posterior distribution under an information theoretical formulation. RegBayes is more flexible than the procedure that elicits expert knowledge via priors, and it covers both directed Bayesian networks and undirected Markov networks whose Bayesian formulation results in hybrid chain graph models. When the regularization is induced from a linear operator on the posterior distributions, such as the expectation operator, we present a general convex-analysis theorem to characterize the solution of RegBayes. Furthermore, we present two concrete examples of RegBayes, infinite latent support vector machines (iLSVM) and multi-task infinite latent support vector machines (MT-iLSVM), which explore the large-margin idea in combination with a nonparametric Bayesian model for discovering predictive latent features for classification and multi-task learning, respectively. We present efficient inference methods and report empirical studies on several benchmark datasets, which appear to demonstrate the merits inherited from both large-margin learning and Bayesian nonparametrics. Such results were not available until now, and contribute to push forward the interface between these two important subfields, which have been largely treated as isolated in the community.

We show how game-theoretic solution concepts such as Nash equilibrium, correlated equilibrium, rationalizability, and sequential equilibrium can be given a uniform definition in terms of a knowledge-based program with counterfactual semantics. In a precise sense, this program can be viewed as providing a procedural characterization of rationality.

Information Geometric Causal Inference (IGCI) is a new approach to distinguish between cause and effect for two variables. It is based on an independence assumption between input distribution and causal mechanism that can be phrased in terms of orthogonality in information space. We describe two intuitive reinterpretations of this approach that makes IGCI more accessible to a broader audience. Moreover, we show that the described independence is related to the hypothesis that unsupervised learning and semi-supervised learning only works for predicting the cause from the effect and not vice versa.

Kernel adaptive filters (KAF) are a class of powerful nonlinear filters developed in Reproducing Kernel Hilbert Space (RKHS). The Gaussian kernel is usually the default kernel in KAF algorithms, but selecting the proper kernel size (bandwidth) is still an open important issue especially for learning with small sample sizes. In previous research, the kernel size was set manually or estimated in advance by Silvermans rule based on the sample distribution. This study aims to develop an online technique for optimizing the kernel size of the kernel least mean square (KLMS) algorithm. A sequential optimization strategy is proposed, and a new algorithm is developed, in which the filter weights and the kernel size are both sequentially updated by stochastic gradient algorithms that minimize the mean square error (MSE). Theoretical results on convergence are also presented. The excellent performance of the new algorithm is confirmed by simulations on static function estimation and short term chaotic time series prediction. Keywords: Kernel methods, kernel adaptive filtering, kernel least mean square, kernel selection.

We consider the problem of learning in repeated games against arbitrary associates. Specifically, we study the ability of expert algorithms to quickly learn effective strategies in repeated games, towards the ultimate goal of learning near-optimal behavior against any arbitrary associate within only a handful of interactions. Our contribution is three-fold. First, we advocate a new metric, called disappointment, for evaluating expert algorithms in repeated games. Unlike minimizing traditional notions of regret, minimizing disappointment in repeated games is equivalent to maximizing payoffs. Unfortunately, eliminating disappointment is impossible to guarantee in general. However, it is possible for an expert algorithm to quickly achieve low disappointment against many known classes of algorithms in many games. Second, we show that popular existing expert algorithms often fail to achieve low disappointment against a variety of associates, particularly in early rounds of the game. Finally, we describe a new meta-algorithm that can be applied to existing expert algorithms to substantially reduce disappointment in many two-player repeated games when associates follow various static, reinforcement learning, and expert algorithms.