Results 1  10
of
88
Online Linear Optimization and Adaptive Routing
, 2006
"... This paper studies an online linear optimization problem generalizing the multiarmed bandit problem. Motivated primarily by the task of designing adaptive routing algorithms for overlay networks, we present two randomized online algorithms for selecting a sequence of routing paths in a network with ..."
Abstract

Cited by 37 (5 self)
 Add to MetaCart
element of this algorithm is the notion of a barycentric spanner, a special type of basis for the vector space of strategies which allows any feasible strategy to be expressed as a linear combination of basis vectors using bounded coefficients. We also present a second algorithm for the online shortest
Playing games with approximation algorithms
 In Proceedings of the 39 th annual ACM Symposium on Theory of Computing
, 2007
"... Abstract. In an online linear optimization problem, on each period t, an online algorithm chooses st ∈ S from a fixed (possibly infinite) set S of feasible decisions. Nature (who may be adversarial) chooses a weight vector wt ∈ R n, and the algorithm incurs cost c(st, wt), where c is a fixed cost fu ..."
Abstract

Cited by 27 (2 self)
 Add to MetaCart
algorithm. Standard techniques generalize the above result to the bandit setting, except that a “Barycentric Spanner ” for the problem is also (provably) necessary as input. Our algorithm can also be viewed as a method for playing large repeated games, where one can only compute approximate best
ABSTRACT Playing Games with Approximation Algorithms
"... In an online linear optimization problem, on each period t, an online algorithm chooses st ∈ S from a fixed (possibly infinite) set S of feasible decisions. Nature (who may be adversarial) chooses a weight vector wt ∈ R n, and the algorithm incurs cost c(st, wt), where c is a fixed cost function tha ..."
Abstract
 Add to MetaCart
. Standard techniques generalize the above result to the bandit setting, except that a “Barycentric Spanner ” for the problem is also (provably) necessary as input. Our algorithm can also be viewed as a method for playing
Yahoo Labs
"... 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 bases. We show how efficient volumetric span ..."
Abstract
 Add to MetaCart
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 bases. We show how efficient volumetric
Online Learning for Network Optimization under Unknown Models
"... Abstract—We consider the shortest path problem in a communication network with random link costs drawn from unknown distributions. A realization of the total endtoend cost is obtained when a path is selected for communication. The objective is an online learning algorithm that minimizes the total ..."
Abstract
 Add to MetaCart
Abstract—We consider the shortest path problem in a communication network with random link costs drawn from unknown distributions. A realization of the total endtoend cost is obtained when a path is selected for communication. The objective is an online learning algorithm that minimizes the total expected communication cost in the long run. The problem is formulated as a multiarmed bandit problem with dependent arms, and an algorithm based on basisbased learning integrated with a Best Linear Unbiased Estimator (BLUE) is developed. Index Terms—Bandit problem, shortest path, best linear unbiased estimator. I.
SAMPLING ALGORITHMS AND CORESETS FOR ℓp REGRESSION
 SIAM J. COMPUT. VOL. 38, NO. 5, PP. 2060–2078
, 2009
"... The ℓp regression problem takes as input a matrix A ∈ Rn×d, a vector b ∈ Rn, and a number p ∈ [1, ∞), and it returns as output a number Z and a vector xopt ∈ Rd such that Z =minx∈Rd‖Ax − b‖p = ‖Axopt − b‖p. In this paper, we construct coresets and obtain an efficient twostage samplingbased approx ..."
Abstract

Cited by 19 (6 self)
 Add to MetaCart
The ℓp regression problem takes as input a matrix A ∈ Rn×d, a vector b ∈ Rn, and a number p ∈ [1, ∞), and it returns as output a number Z and a vector xopt ∈ Rd such that Z =minx∈Rd‖Ax − b‖p = ‖Axopt − b‖p. In this paper, we construct coresets and obtain an efficient twostage samplingbased approximation algorithm for the very overconstrained (n ≫ d) version of this classical problem, for all p ∈ [1, ∞). The first stage of our algorithm nonuniformly samples ˆr1 = O(36pdmax{p/2+1,p}+1)rowsofAand the corresponding elements of b, and then it solves the ℓp regression problem on the sample; we prove this is an 8approximation. The second stage of our algorithm uses the output of the first stage to resample ˆr1/ɛ2 constraints, and then it solves the ℓp regression problem on the new sample; we prove this is a (1 + ɛ)approximation. Our algorithm unifies, improves upon, and extends the existing algorithms for special cases of ℓp regression, namely,
Adaptive Routing with EndtoEnd feedback: Distributed Learning and Geometric Approaches
"... Minimal delay routing is a fundamental task in networks. Since delays depend on the (potentially unpredictable) traffic distribution, online delay optimization can be quite challenging. While uncertainty about the current network delays may make the current routing choices suboptimal, the algorithm ..."
Abstract
 Add to MetaCart
Minimal delay routing is a fundamental task in networks. Since delays depend on the (potentially unpredictable) traffic distribution, online delay optimization can be quite challenging. While uncertainty about the current network delays may make the current routing choices suboptimal, the algorithm can nevertheless try to learn the traffic patterns and keep adapting its choice of routing paths so as to perform nearly as well as the best static path. This online shortest path problem is a special case of online linear optimization, a problem in which an online algorithm must choose, in each round, a strategy from some compact set S ⊆ R d so as to try to minimize a linear cost function which is only revealed at the end of the round. Kalai and Vempala [4] gave an algorithm for such problems in the transparent feedback model, where the entire cost function is revealed at the end of the round. Here we present an algorithm for online linear optimization in the more challenging opaque feedback model, in which only the cost of the chosen strategy is revealed at the end of the round. In the special case of shortest paths, opaque feedback corresponds to the notion that in each round the algorithm learns only the endtoend cost of the chosen path, not the cost of every edge in the network. We also present a second algorithm for online shortest paths, which solves the shortestpath problem using a chain of online decision oracles, one at each node of the graph. This has several advantages over the online linear optimization approach. First, it is effective against an adaptive adversary, whereas our linear optimization algorithm assumes an oblivious adversary. Second, even in the case of an oblivious adversary, the second algorithm performs better than the first, as measured by their additive regret.
Stochastic Online Learning for Network Optimization under Random Unknown Weights 1
"... We consider network optimization problems under random weights with unknown distributions. We first consider the shortest path problem that aims to optimize the quality of communication between a source and a destination through adaptive path selection. Due to the randomness and uncertainties in the ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
We consider network optimization problems under random weights with unknown distributions. We first consider the shortest path problem that aims to optimize the quality of communication between a source and a destination through adaptive path selection. Due to the randomness and uncertainties in the network dynamics, the state of each communication link varies over time according to a stochastic process with unknown distributions. The link states are not directly observable. The aggregated endtoend cost of a path from the source to the destination is revealed after the path is chosen for communication. The objective is an adaptive path selection algorithm that minimizes regret defined as the additional cost over the ideal scenario where the best path is known a priori. This problem can be cast as a variation of the classic multiarmed bandit (MAB) problem with each path as an arm and arms dependent through common links. We show that by exploiting arm dependencies, a regret polynomial with the network size can be achieved while maintaining the optimal logarithmic order with time. This is in sharp contrast with the exponential regret order with the network size offered by a direct application of the classic MAB policies that ignores arm dependencies. Furthermore, these results are obtained under a general model of link state distributions (including heavytailed distributions). We then extend the results to general network optimization problems (e.g., minimum spanning tree and dominating set) and stochastic online linear optimization problems. These results find applications in cognitive radio and ad hoc networks with unknown and dynamic communication environments.
Results 1  10
of
88