The first part of the paper presents a reinforcement learning (adaptive filtering) algorithm that facilitates learning an equilibrium by resorting to diffusion cooperation strategies in a social network. Agents form homophilic social groups, within which they exchange past experiences over an undirected graph. It is shown that, if all agents follow the proposed algorithm, their global behavior is attracted to the correlated equilibria set of the game. The second part of the paper provides a test to detect if the actions of agents are consistent with play from the equilibrium of a concave potential game. The theory of revealed preference from microeconomics is used to construct a nonparametric decision test and statistical test which only require the probe and associated actions of agents. A stochastic gradient algorithm is given to optimize the probe in real time to minimize the Type-II error probabilities of the detection test subject to specified Type-I error probability. We provide a real-world example using the energy market, and a numerical example to detect malicious agents in an online social network. Index Terms--Multi-agent signal processing, non-cooperative games, social networks, correlated equilibrium, diffusion cooperation, homophily behavior, revealed preferences, Afriat's theorem, stochastic approximation algorithm.

We present a new approach for transferring knowledge from groups to individuals that comprise them. We evaluate our method in text, by inferring the ratings of individual sentences using full-review ratings. This approach, which combines ideas from transfer learning, deep learning and multi-instance learning, reduces the need for laborious human labelling of fine-grained data when abundant labels are available at the group level.

This work presents a technique for statistically modeling errors introduced by reduced-order models. The method employs Gaussian-process regression to construct a mapping from a small number of computationally inexpensive `error indicators' to a distribution over the true error. The variance of this distribution can be interpreted as the (epistemic) uncertainty introduced by the reduced-order model. To model normed errors, the method employs existing rigorous error bounds and residual norms as indicators; numerical experiments show that the method leads to a near-optimal expected effectivity in contrast to typical error bounds. To model errors in general outputs, the method uses dual-weighted residuals---which are amenable to uncertainty control---as indicators. Experiments illustrate that correcting the reduced-order-model output with this surrogate can improve prediction accuracy by an order of magnitude; this contrasts with existing `multifidelity correction' approaches, which often fail for reduced-order models and suffer from the curse of dimensionality. The proposed error surrogates also lead to a notion of `probabilistic rigor', i.e., the surrogate bounds the error with specified probability.

In many domains, scientists build complex simulators of natural phenomena that encode their hypotheses about the underlying processes. These simulators can be deterministic or stochastic, fast or slow, constrained or unconstrained, and so on. Optimizing the simulators with respect to a set of parameter values is common practice, resulting in a single parameter setting that minimizes an objective subject to constraints. We propose a post optimization posterior analysis that computes and visualizes all the models that can generate equally good or better simulation results, subject to constraints. These optimization posteriors are desirable for a number of reasons among which easy interpretability, automatic parameter sensitivity and correlation analysis and posterior predictive analysis. We develop a new sampling framework based on approximate Bayesian computation (ABC) with one-sided kernels. In collaboration with two groups of scientists we applied POPE to two important biological simulators: a fast and stochastic simulator of stem-cell cycling and a slow and deterministic simulator of tumor growth patterns.

Predictive rate-distortion analysis suffers from the curse of dimensionality: clustering arbitrarily long pasts to retain information about arbitrarily long futures requires resources that typically grow exponentially with length. The challenge is compounded for infinite-order Markov processes, since conditioning on finite sequences cannot capture all of their past dependencies. Spectral arguments show that algorithms which cluster finite-length sequences fail dramatically when the underlying process has long-range temporal correlations and can fail even for processes generated by finite-memory hidden Markov models. We circumvent the curse of dimensionality in rate-distortion analysis of infinite-order processes by casting predictive rate-distortion objective functions in terms of the forward- and reverse-time causal states of computational mechanics. Examples demonstrate that the resulting causal rate-distortion theory substantially improves current predictive rate-distortion analyses.

Inverse problems and regularization theory is a central theme in contemporary signal processing, where the goal is to reconstruct an unknown signal from partial indirect, and possibly noisy, measurements of it. A now standard method for recovering the unknown signal is to solve a convex optimization problem that enforces some prior knowledge about its structure. This has proved efficient in many problems routinely encountered in imaging sciences, statistics and machine learning. This chapter delivers a review of recent advances in the field where the regularization prior promotes solutions conforming to some notion of simplicity/low-complexity. These priors encompass as popular examples sparsity and group sparsity (to capture the compressibility of natural signals and images), total variation and analysis sparsity (to promote piecewise regularity), and low-rank (as natural extension of sparsity to matrix-valued data). Our aim is to provide a unified treatment of all these regularizations under a single umbrella, namely the theory of partial smoothness. This framework is very general and accommodates all low-complexity regularizers just mentioned, as well as many others. Partial smoothness turns out to be the canonical way to encode low-dimensional models that can be linear spaces or more general smooth manifolds. This review is intended to serve as a one stop shop toward the understanding of the theoretical properties of the so-regularized solutions. It covers a large spectrum including: (i) recovery guarantees and stability to noise, both in terms of $\ell^2$-stability and model (manifold) identification; (ii) sensitivity analysis to perturbations of the parameters involved (in particular the observations), with applications to unbiased risk estimation ; (iii) convergence properties of the forward-backward proximal splitting scheme, that is particularly well suited to solve the corresponding large-scale regularized optimization problem.

Robert Gagne (1968) defined a learning hierarchy as a set of specified intellectual capabilities or intellectual skills. The capabilities in the hierarchy have an ordered relationship to each other and the hierarchy, as a whole, bears some relation to a plan for effective instruction. The hierarchy is built in a manner to reflect that a lower level skill must be acquired or mastered before an upper-level one, that is, lower level capabilities are prerequisites for upper level ones. Intellectual capabilities or skills are the nodes of the hierarchy. Gagne (1968) defines them as cognitive strategies that denote capabilities for action. Additionally, they also depict a learning route, a path, from simple skills to a final complex capability. Learning hierarchies not only serve to represent effective instruction plans in terms of skills or capabilities, but also, they serve as diagnosis instruments for providing individual or personalized remediation to students. However, for classrooms with a large number of students, the application of learning hierarchies for individualized (remedial) instruction is a highly time consuming task. Learning hierarchies belong to the behaviorist view on cognition and www.ijera.com

This work presents a general framework for solving the low rank and/or sparse matrix minimization problems, which may involve multiple non-smooth terms. The Iteratively Reweighted Least Squares (IRLS) method is a fast solver, which smooths the objective function and minimizes it by alternately updating the variables and their weights. However, the traditional IRLS can only solve a sparse only or low rank only minimization problem with squared loss or an affine constraint. This work generalizes IRLS to solve joint/mixed low rank and sparse minimization problems, which are essential formulations for many tasks. As a concrete example, we solve the Schatten-$p$ norm and $\ell_{2,q}$-norm regularized Low-Rank Representation (LRR) problem by IRLS, and theoretically prove that the derived solution is a stationary point (globally optimal if $p,q\geq1$). Our convergence proof of IRLS is more general than previous one which depends on the special properties of the Schatten-$p$ norm and $\ell_{2,q}$-norm. Extensive experiments on both synthetic and real data sets demonstrate that our IRLS is much more efficient.

Stein showed that the multivariate sample mean is outperformed by "shrinking" to a constant target vector. Ledoit and Wolf extended this approach to the sample covariance matrix and proposed a multiple of the identity as shrinkage target. In a general framework, independent of a specific estimator, we extend the shrinkage concept by allowing simultaneous shrinkage to a set of targets. Application scenarios include settings with (A) additional data sets from potentially similar distributions, (B) non-stationarity, (C) a natural grouping of the data or (D) multiple alternative estimators which could serve as targets. We show that this Multi-Target Shrinkage can be translated into a quadratic program and derive conditions under which the estimation of the shrinkage intensities yields optimal expected squared error in the limit. For the sample mean and the sample covariance as specific instances, we derive conditions under which the optimality of MTS is applicable. We consider two asymptotic settings: the large dimensional limit (LDL), where the dimensionality and the number of observations go to infinity at the same rate, and the finite observations large dimensional limit (FOLDL), where only the dimensionality goes to infinity while the number of observations remains constant. We then show the effectiveness in extensive simulations and on real world data.

To estimate a sparse linear model from data with Gaussian noise, consilience from lasso and compressed sensing literatures is that thresholding estimators like lasso and the Dantzig selector have the ability in some situations to identify with high probability part of the significant covariates asymptotically, and are numerically tractable thanks to convexity. Yet, the selection of a threshold parameter $\lambda$ remains crucial in practice. To that aim we propose Quantile Universal Thresholding, a selection of $\lambda$ at the detection edge. We show with extensive simulations and real data that an excellent compromise between high true positive rate and low false discovery rate is achieved, leading also to good predictive risk.