Volumetric Spanners: an Efficient Exploration Basis for Learning

Numerous machine learning problems require an exploration basis - a mechanism to explore the action space. We define a novel geometric notion of exploration basis with low variance, called volumetric spanners, and give efficient algorithms to construct such a basis. We show how efficient volumetric spanners give rise to the first efficient and optimal regret algorithm for bandit linear optimization over general convex sets. Previously such results were known only for specific convex sets, or under special conditions such as the existence of an efficient self-concordant barrier for the underlying set

On Strong Diameter Padded Decompositions

Given a weighted graph G=(V,E,w), a partition of V is Delta-bounded if the diameter of each cluster is bounded by Delta. A distribution over Delta-bounded partitions is a beta-padded decomposition if every ball of radius gamma Delta is contained in a single cluster with probability at least e^{-beta * gamma}. The weak diameter of a cluster C is measured w.r.t. distances in G, while the strong diameter is measured w.r.t. distances in the induced graph G[C]. The decomposition is weak/strong according to the diameter guarantee. Formerly, it was proven that K_r free graphs admit weak decompositions with padding parameter O(r), while for strong decompositions only O(r^2) padding parameter was known. Furthermore, for the case of a graph G, for which the induced shortest path metric d_G has doubling dimension ddim, a weak O(ddim)-padded decomposition was constructed, which is also known to be tight. For the case of strong diameter, nothing was known. We construct strong O(r)-padded decompositions for K_r free graphs, matching the state of the art for weak decompositions. Similarly, for graphs with doubling dimension ddim we construct a strong O(ddim)-padded decomposition, which is also tight. We use this decomposition to construct (O(ddim),O~(ddim))-sparse cover scheme for such graphs. Our new decompositions and cover have implications to approximating unique games, the construction of light and sparse spanners, and for path reporting distance oracles

Minimax Policies for Combinatorial Prediction Games

We address the online linear optimization problem when the actions of the forecaster are represented by binary vectors. Our goal is to understand the magnitude of the minimax regret for the worst possible set of actions. We study the problem under three different assumptions for the feedback: full information, and the partial information models of the so-called "semi-bandit", and "bandit" problems. We consider both $L_\infty$-, and $L_2$-type of restrictions for the losses assigned by the adversary. We formulate a general strategy using Bregman projections on top of a potential-based gradient descent, which generalizes the ones studied in the series of papers Gyorgy et al. (2007), Dani et al. (2008), Abernethy et al. (2008), Cesa-Bianchi and Lugosi (2009), Helmbold and Warmuth (2009), Koolen et al. (2010), Uchiya et al. (2010), Kale et al. (2010) and Audibert and Bubeck (2010). We provide simple proofs that recover most of the previous results. We propose new upper bounds for the semi-bandit game. Moreover we derive lower bounds for all three feedback assumptions. With the only exception of the bandit game, the upper and lower bounds are tight, up to a constant factor. Finally, we answer a question asked by Koolen et al. (2010) by showing that the exponentially weighted average forecaster is suboptimal against $L_{\infty}$ adversaries

The on-line shortest path problem under partial monitoring

The on-line shortest path problem is considered under various models of partial monitoring. Given a weighted directed acyclic graph whose edge weights can change in an arbitrary (adversarial) way, a decision maker has to choose in each round of a game a path between two distinguished vertices such that the loss of the chosen path (defined as the sum of the weights of its composing edges) be as small as possible. In a setting generalizing the multi-armed bandit problem, after choosing a path, the decision maker learns only the weights of those edges that belong to the chosen path. For this problem, an algorithm is given whose average cumulative loss in n rounds exceeds that of the best path, matched off-line to the entire sequence of the edge weights, by a quantity that is proportional to 1/\sqrt{n} and depends only polynomially on the number of edges of the graph. The algorithm can be implemented with linear complexity in the number of rounds n and in the number of edges. An extension to the so-called label efficient setting is also given, in which the decision maker is informed about the weights of the edges corresponding to the chosen path at a total of m << n time instances. Another extension is shown where the decision maker competes against a time-varying path, a generalization of the problem of tracking the best expert. A version of the multi-armed bandit setting for shortest path is also discussed where the decision maker learns only the total weight of the chosen path but not the weights of the individual edges on the path. Applications to routing in packet switched networks along with simulation results are also presented.Comment: 35 page

09451 Abstracts Collection -- Geometric Networks, Metric Space Embeddings and Spatial Data Mining

From November 1 to 6, 2009, the Dagstuhl Seminar 09451 ``Geometric Networks, Metric Space Embeddings and Spatial Data Mining\u27\u27 was held in Schloss Dagstuhl~--~Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

FLOC-SPANNER: An O(1) time, locally self-stabilizing algorithm for geometric spanner construction in a wireless sensor network

Geometric spanners are a popular form of topology control in wireless networks because they yield an efficient, reduced interference subgraph for both unicast and broadcast routing.;In this thesis work a distributed algorithm for creation of geometric spanners in a wireless sensor network is presented. Given any connected network, we show that the algorithm terminates in O(1) time, irrespective of network size. Our algorithm uses an underlying clustering algorithm as a foundation for creating spanners, and only relies on the periodic heartbeat messages associated with cluster maintenance for the creation of the spanners. The algorithm is also shown to stabilize locally in the presence of node additions and deletions. The performance of our algorithm is verified using large scale simulations. The average path length ratio for routing along the spanner for large networks is shown to be less than 2.;Geometric Spanners is a well-researched topic. The algorithm presented in this thesis differs from other spanner algorithms in the following ways: 1. It is a distributed locally self-stabilizing algorithm. 2. It does not require location information for its operation. 3. Creates spanner network in constant time irrespective of network size and network density

FLOC-SPANNER: An Time, Locally Self-Stabilizing Algorithm for Geometric Spanner Construction in a Wireless Sensor Network

We present a distributed algorithm for creation of geometric spanners in a wireless sensor network. Given any connected network, we show that the algorithm terminates in time, irrespective of network size. Our algorithm uses an underlying clustering algorithm as a foundation for creating spanners and only relies on the periodic heartbeat messages associated with cluster maintenance for the creation of the spanners. The algorithm is also shown to stabilize locally in the presence of node additions and deletions. The performance of our algorithm is verified using large scale simulations. The average path length ratio for routing along the spanner for large networks is shown to be less than 2