Deterministic and Probabilistic Binary Search in Graphs

We consider the following natural generalization of Binary Search: in a given undirected, positively weighted graph, one vertex is a target. The algorithm's task is to identify the target by adaptively querying vertices. In response to querying a node $q$, the algorithm learns either that $q$ is the target, or is given an edge out of $q$ that lies on a shortest path from $q$ to the target. We study this problem in a general noisy model in which each query independently receives a correct answer with probability $p > \frac{1}{2}$ (a known constant), and an (adversarial) incorrect one with probability $1-p$. Our main positive result is that when $p = 1$ (i.e., all answers are correct), $\log_2 n$ queries are always sufficient. For general $p$, we give an (almost information-theoretically optimal) algorithm that uses, in expectation, no more than $(1 - \delta)\frac{\log_2 n}{1 - H(p)} + o(\log n) + O(\log^2 (1/\delta))$ queries, and identifies the target correctly with probability at leas $1-\delta$. Here, $H(p) = -(p \log p + (1-p) \log(1-p))$ denotes the entropy. The first bound is achieved by the algorithm that iteratively queries a 1-median of the nodes not ruled out yet; the second bound by careful repeated invocations of a multiplicative weights algorithm. Even for $p = 1$, we show several hardness results for the problem of determining whether a target can be found using $K$ queries. Our upper bound of $\log_2 n$ implies a quasipolynomial-time algorithm for undirected connected graphs; we show that this is best-possible under the Strong Exponential Time Hypothesis (SETH). Furthermore, for directed graphs, or for undirected graphs with non-uniform node querying costs, the problem is PSPACE-complete. For a semi-adaptive version, in which one may query $r$ nodes each in $k$ rounds, we show membership in $\Sigma_{2k-1}$ in the polynomial hierarchy, and hardness for $\Sigma_{2k-5}$

Searching a Tree with Permanently Noisy Advice

We consider a search problem on trees using unreliable guiding instructions. Specifically, an agent starts a search at the root of a tree aiming to find a treasure hidden at one of the nodes by an adversary. Each visited node holds information, called advice, regarding the most promising neighbor to continue the search. However, the memory holding this information may be unreliable. Modeling this scenario, we focus on a probabilistic setting. That is, the advice at a node is a pointer to one of its neighbors. With probability q each node is faulty, independently of other nodes, in which case its advice points at an arbitrary neighbor, chosen uniformly at random. Otherwise, the node is sound and points at the correct neighbor. Crucially, the advice is permanent, in the sense that querying a node several times would yield the same answer. We evaluate efficiency by two measures: The move complexity denotes the expected number of edge traversals, and the query complexity denotes the expected number of queries. Let Delta denote the maximal degree. Roughly speaking, the main message of this paper is that a phase transition occurs when the noise parameter q is roughly 1/sqrt{Delta}. More precisely, we prove that above the threshold, every search algorithm has query complexity (and move complexity) which is both exponential in the depth d of the treasure and polynomial in the number of nodes n. Conversely, below the threshold, there exists an algorithm with move complexity O(d sqrt{Delta}), and an algorithm with query complexity O(sqrt{Delta}log Delta log^2 n). Moreover, for the case of regular trees, we obtain an algorithm with query complexity O(sqrt{Delta}log n log log n). For q that is below but close to the threshold, the bound for the move complexity is tight, and the bounds for the query complexity are not far from the lower bound of Omega(sqrt{Delta}log_Delta n). In addition, we also consider a semi-adversarial variant, in which an adversary chooses the direction of advice at faulty nodes. For this variant, the threshold for efficient moving algorithms happens when the noise parameter is roughly 1/Delta. Above this threshold a simple protocol that follows each advice with a fixed probability already achieves optimal move complexity

The Influence of Shape Constraints on the Thresholding Bandit Problem

We investigate the stochastic Thresholding Bandit problem (TBP) under several shape constraints. On top of (i) the vanilla, unstructured TBP, we consider the case where (ii) the sequence of arm's means $(\mu_k)_k$ is monotonically increasing MTBP, (iii) the case where $(\mu_k)_k$ is unimodal UTBP and (iv) the case where $(\mu_k)_k$ is concave CTBP. In the TBP problem the aim is to output, at the end of the sequential game, the set of arms whose means are above a given threshold. The regret is the highest gap between a misclassified arm and the threshold. In the fixed budget setting, we provide problem independent minimax rates for the expected regret in all settings, as well as associated algorithms. We prove that the minimax rates for the regret are (i) $\sqrt{\log(K)K/T}$ for TBP, (ii) $\sqrt{\log(K)/T}$ for MTBP, (iii) $\sqrt{K/T}$ for UTBP and (iv) $\sqrt{\log\log K/T}$ for CTBP, where $K$ is the number of arms and $T$ is the budget. These rates demonstrate that the dependence on $K$ of the minimax regret varies significantly depending on the shape constraint. This highlights the fact that the shape constraints modify fundamentally the nature of the TBP

Driven by Compression Progress: A Simple Principle Explains Essential Aspects of Subjective Beauty, Novelty, Surprise, Interestingness, Attention, Curiosity, Creativity, Art, Science, Music, Jokes

I argue that data becomes temporarily interesting by itself to some self-improving, but computationally limited, subjective observer once he learns to predict or compress the data in a better way, thus making it subjectively simpler and more beautiful. Curiosity is the desire to create or discover more non-random, non-arbitrary, regular data that is novel and surprising not in the traditional sense of Boltzmann and Shannon but in the sense that it allows for compression progress because its regularity was not yet known. This drive maximizes interestingness, the first derivative of subjective beauty or compressibility, that is, the steepness of the learning curve. It motivates exploring infants, pure mathematicians, composers, artists, dancers, comedians, yourself, and (since 1990) artificial systems.Comment: 35 pages, 3 figures, based on KES 2008 keynote and ALT 2007 / DS 2007 joint invited lectur

A Finite-Horizon Approach to Active Level Set Estimation

We consider the problem of active learning in the context of spatial sampling for level set estimation (LSE), where the goal is to localize all regions where a function of interest lies above/below a given threshold as quickly as possible. We present a finite-horizon search procedure to perform LSE in one dimension while optimally balancing both the final estimation error and the distance traveled for a fixed number of samples. A tuning parameter is used to trade off between the estimation accuracy and distance traveled. We show that the resulting optimization problem can be solved in closed form and that the resulting policy generalizes existing approaches to this problem. We then show how this approach can be used to perform level set estimation in higher dimensions under the popular Gaussian process model. Empirical results on synthetic data indicate that as the cost of travel increases, our method's ability to treat distance nonmyopically allows it to significantly improve on the state of the art. On real air quality data, our approach achieves roughly one fifth the estimation error at less than half the cost of competing algorithms

The Dependent Doors Problem: An Investigation into Sequential Decisions without Feedback

We introduce the dependent doors problem as an abstraction for situations in which one must perform a sequence of possibly dependent decisions, without receiving feedback information on the effectiveness of previously made actions. Informally, the problem considers a set of d doors that are initially closed, and the aim is to open all of them as fast as possible. To open a door, the algorithm knocks on it and it might open or not according to some probability distribution. This distribution may depend on which other doors are currently open, as well as on which other doors were open during each of the previous knocks on that door. The algorithm aims to minimize the expected time until all doors open. Crucially, it must act at any time without knowing whether or which other doors have already opened. In this work, we focus on scenarios where dependencies between doors are both positively correlated and acyclic. The fundamental distribution of a door describes the probability it opens in the best of conditions (with respect to other doors being open or closed). We show that if in two configurations of d doors corresponding doors share the same fundamental distribution, then these configurations have the same optimal running time up to a universal constant, no matter what are the dependencies between doors and what are the distributions. We also identify algorithms that are optimal up to a universal constant factor. For the case in which all doors share the same fundamental distribution we additionally provide a simpler algorithm, and a formula to calculate its running time. We furthermore analyse the price of lacking feedback for several configurations governed by standard fundamental distributions. In particular, we show that the price is logarithmic in d for memoryless doors, but can potentially grow to be linear in d for other distributions. We then turn our attention to investigate precise bounds. Even for the case of two doors, identifying the optimal sequence is an intriguing combinatorial question. Here, we study the case of two cascading memoryless doors. That is, the first door opens on each knock independently with probability p_1. The second door can only open if the first door is open, in which case it will open on each knock independently with probability p_2. We solve this problem almost completely by identifying algorithms that are optimal up to an additive term of 1