Submodular meets Spectral: Greedy Algorithms for Subset Selection, Sparse Approximation and Dictionary Selection

We study the problem of selecting a subset of k random variables from a large set, in order to obtain the best linear prediction of another variable of interest. This problem can be viewed in the context of both feature selection and sparse approximation. We analyze the performance of widely used greedy heuristics, using insights from the maximization of submodular functions and spectral analysis. We introduce the submodularity ratio as a key quantity to help understand why greedy algorithms perform well even when the variables are highly correlated. Using our techniques, we obtain the strongest known approximation guarantees for this problem, both in terms of the submodularity ratio and the smallest k-sparse eigenvalue of the covariance matrix. We further demonstrate the wide applicability of our techniques by analyzing greedy algorithms for the dictionary selection problem, and significantly improve the previously known guarantees. Our theoretical analysis is complemented by experiments on real-world and synthetic data sets; the experiments show that the submodularity ratio is a stronger predictor of the performance of greedy algorithms than other spectral parameters

Cognitive hierarchies in adaptive play

Inspired by the behavior in repeated guessing game experiments, we study adaptive play bypopulations containing individuals that reason with different levels of cognition. Individualsplay a higher order best response to samples from the empirical data on the history of play, wherethe order of best response is determined by their exogenously given level of cognition. As inYoung''s model of adaptive play, (unperturbed) play still converges to a minimal curb set. However,with the random perturbations of this (higher order) best response dynamic, the stochasticallystable states obtained may now be different, but in a deterministic manner. Perhapscounter-intuitively, higher cognition may actually be bad for both the individual with highercognition and his parent population.microeconomics ;