We consider the task of nearest-neighbor search with the class of binary-space-partitioning trees, which includes kd-trees, principal axis trees and random projection trees, and try to rigorously answer the question which tree to use for nearest-neighbor search?'' To this end, we present the theoretical results which imply that trees with better vector quantization performance have better search performance guarantees. We also explore another factor affecting the search performance -- margins of the partitions in these trees. We demonstrate, both theoretically and empirically, that large margin partitions can improve the search performance of a space-partitioning tree. "

Many real-world problems have complicated objective functions. To optimize such functions, humans utilize sophisticated sequential decision-making strategies. Many optimization algorithms have also been developed for this same purpose, but how do they compare to humans in terms of both performance and behavior? We try to unravel the general underlying algorithm people may be using while searching for the maximum of an invisible 1D function. Subjects click on a blank screen and are shown the ordinate of the function at each clicked abscissa location. Their task is to find the function’s maximum in as few clicks as possible. Subjects win if they get close enough to the maximum location. Analysis over 23 non-maths undergraduates, optimizing 25 functions from different families, shows that humans outperform 24 well-known optimization algorithms. Bayesian Optimization based on Gaussian Processes, which exploit all the x values tried and all the f(x) values obtained so far to pick the next x, predicts human performance and searched locations better. In 6 follow-up controlled experiments over 76 subjects, covering interpolation, extrapolation, and optimization tasks, we further confirm that Gaussian Processes provide a general and unified theoretical account to explain passive and active function learning and search in humans.

A Constraint Satisfaction Problem (CSP) is a framework used for modeling and solving constrained problems. Tree-search algorithms like backtracking try to construct a solution to a CSP by selecting the variables of the problem one after another. The order in which these algorithm select the variables potentially have significant impact on the search performance. Various heuristics have been proposed for choosing good variable ordering. Many powerful variable ordering heuristics weigh the constraints first and then utilize the weights for selecting good order of the variables. Constraint weighting are basically employed to identify global bottlenecks in a CSP. In this paper, we propose a new approach for learning weights for the constraints using competitive coevolutionary Genetic Algorithm (GA). Weights learned by the coevolutionary GA later help to make better choices for the first few variables in a search. In the competitive coevolutionary GA, constraints and candidate solutions for a CSP evolve together through an inverse fitness interaction process. We have conducted experiments on several random, quasi-random and patterned instances to measure the efficiency of the proposed approach. The results and analysis show that the proposed approach is good at learning weights to distinguish the hard constraints for quasi-random instances and forced satisfiable random instances generated with the Model RB. For other type of instances, RNDI still seems to be the best approach as our experiments show.

This paper defines a generalized column subset selection problem which is concerned with the selection of a few columns from a source matrix A that best approximate the span of a target matrix B. The paper then proposes a fast greedy algorithm for solving this problem and draws connections to different problems that can be efficiently solved using the proposed algorithm.

Bayesian model-based reinforcement learning is a formally elegant approach to learning optimal behaviour under model uncertainty, trading off exploration and exploitation in an ideal way. Unfortunately, finding the resulting Bayes-optimal policies is notoriously taxing, since the search space becomes enormous. In this paper we introduce a tractable, sample-based method for approximate Bayes-optimal planning which exploits Monte-Carlo tree search. Our approach outperformed prior Bayesian model-based RL algorithms by a significant margin on several well-known benchmark problems -- because it avoids expensive applications of Bayes rule within the search tree by lazily sampling models from the current beliefs. We illustrate the advantages of our approach by showing it working in an infinite state space domain which is qualitatively out of reach of almost all previous work in Bayesian exploration.

We introduce a new local search algorithm for satisfiability problems. Usual approaches focus uniformly on unsatisfied clauses. The new method works by picking uniformly random variables in unsatisfied clauses. A Variable-based Focused Metropolis Search (V-FMS) is then applied to random 3-SAT. We show that it is quite comparable in performance to the clause-based FMS. Consequences for algorithmic design are discussed.

Let G=(V,E) be an undirected graph, lambda_k be the k-th smallest eigenvalue of the normalized laplacian matrix of G. There is a basic fact in algebraic graph theory that lambda_k > 0 if and only if G has at most k-1 connected components. We prove a robust version of this fact. If lambda_k>0, then for some 1\leq \ell\leq k-1, V can be {\em partitioned} into l sets P_1,\ldots,P_l such that each P_i is a low-conductance set in G and induces a high conductance induced subgraph. In particular, \phi(P_i)=O(l^3\sqrt{\lambda_l}) and \phi(G[P_i]) >= \lambda_k/k^2). We make our results algorithmic by designing a simple polynomial time spectral algorithm to find such partitioning of G with a quadratic loss in the inside conductance of P_i's. Unlike the recent results on higher order Cheeger's inequality [LOT12,LRTV12], our algorithmic results do not use higher order eigenfunctions of G. If there is a sufficiently large gap between lambda_k and lambda_{k+1}, more precisely, if \lambda_{k+1} >= \poly(k) lambda_{k}^{1/4} then our algorithm finds a k partitioning of V into sets P_1,...,P_k such that the induced subgraph G[P_i] has a significantly larger conductance than the conductance of P_i in G. Such a partitioning may represent the best k clustering of G. Our algorithm is a simple local search that only uses the Spectral Partitioning algorithm as a subroutine. We expect to see further applications of this simple algorithm in clustering applications.

Chebyshev Greedy Algorithm is a generalization of the well known Orthogonal Matching Pursuit defined in a Hilbert space to the case of Banach spaces. We apply this algorithm for constructing sparse approximate solutions (with respect to a given dictionary) to convex optimization problems. Rate of convergence results in a style of the Lebesgue-type inequalities are proved.

Reconstructing transcriptional regulatory networks is an important task in functional genomics. Data obtained from experiments that perturb genes by knockouts or RNA interference contain useful information for addressing this reconstruction problem. However, such data can be limited in size and/or are expensive to acquire. On the other hand, observational data of the organism in steady state (e.g. wild-type) are more readily available, but their informational content is inadequate for the task at hand. We develop a computational approach to appropriately utilize both data sources for estimating a regulatory network. The proposed approach is based on a three-step algorithm to estimate the underlying directed but cyclic network, that uses as input both perturbation screens and steady state gene expression data. In the first step, the algorithm determines causal orderings of the genes that are consistent with the perturbation data, by combining an exhaustive search method with a fast heuristic that in turn couples a Monte Carlo technique with a fast search algorithm. In the second step, for each obtained causal ordering, a regulatory network is estimated using a penalized likelihood based method, while in the third step a consensus network is constructed from the highest scored ones. Extensive computational experiments show that the algorithm performs well in reconstructing the underlying network and clearly outperforms competing approaches that rely only on a single data source. Further, it is established that the algorithm produces a consistent estimate of the regulatory network.

Bayesian planning is a formally elegant approach to learning optimal behaviour under model uncertainty, trading off exploration and exploitation in an ideal way. Unfortunately, planning optimally in the face of uncertainty is notoriously taxing, since the search space is enormous. In this paper we introduce a tractable, sample-based method for approximate Bayes-optimal planning which exploits Monte-Carlo tree search. Our approach avoids expensive applications of Bayes rule within the search tree by sampling models from current beliefs, and furthermore performs this sampling in a lazy manner. This enables it to outperform previous Bayesian model-based reinforcement learning algorithms by a significant margin on several well-known benchmark problems. As we show, our approach can even work in problems with an infinite state space that lie qualitatively out of reach of almost all previous work in Bayesian exploration.