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2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
Approximation Algorithms for Correlated Knapsacks and Nonmartingale Bandits
Palm Springs, California USA
October 22October 25
ISBN: 9780769545714
ASCII Text  x  
Anupam Gupta, Ravishankar Krishnaswamy, Marco Molinaro, R. Ravi, "Approximation Algorithms for Correlated Knapsacks and Nonmartingale Bandits," 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, pp. 827836, 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, 2011.  
BibTex  x  
@article{ 10.1109/FOCS.2011.48, author = {Anupam Gupta and Ravishankar Krishnaswamy and Marco Molinaro and R. Ravi}, title = {Approximation Algorithms for Correlated Knapsacks and Nonmartingale Bandits}, journal ={2013 IEEE 54th Annual Symposium on Foundations of Computer Science}, volume = {0}, year = {2011}, issn = {02725428}, pages = {827836}, doi = {http://doi.ieeecomputersociety.org/10.1109/FOCS.2011.48}, publisher = {IEEE Computer Society}, address = {Los Alamitos, CA, USA}, }  
RefWorks Procite/RefMan/Endnote  x  
TY  CONF JO  2013 IEEE 54th Annual Symposium on Foundations of Computer Science TI  Approximation Algorithms for Correlated Knapsacks and Nonmartingale Bandits SN  02725428 SP827 EP836 A1  Anupam Gupta, A1  Ravishankar Krishnaswamy, A1  Marco Molinaro, A1  R. Ravi, PY  2011 KW  Approximation Algorithms KW  Stochastic Optimization KW  MultiArmed Bandits VL  0 JA  2013 IEEE 54th Annual Symposium on Foundations of Computer Science ER   
DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/FOCS.2011.48
In the stochastic knapsack problem, we are given a knapsack of size B, and a set of items whose sizes and rewards are drawn from a known probability distribution. To know the actual size and reward we have to schedule the item  when it completes, we get to know these values. The goal is to schedule the items (possibly making adaptive decisions based on the sizes seen so far) to maximize the expected total reward of items which successfully pack into the knapsack. We know constantfactor approximations when (i) the rewards and sizes are independent, and (ii) we cannot prematurely cancel items after we schedule them. What if either or both assumptions are relaxed? Related stochastic packing problems are the multiarmed bandit (and budgeted learning) problems, here one is given several arms which evolve in a specified stochastic fashion with each pull, and the goal is to (adaptively) decide which arms to pull, in order to maximize the expected reward obtained after B pulls in total. Much recent work on this problem focuses on the case when the evolution of each arm follows a martingale, i.e., when the expected reward from one pull of an arm is the same as the reward at the current state. What if the rewards do not form a martingale? In this paper, we give O(1)approximation algorithms for the stochastic knapsack problem with correlations and/or cancellations. Extending the ideas developed here, we give O(1)approximations for MAB problems without the martingale assumption. Indeed, we can show that previously proposed linear programming relaxations for these problems have large integrality gaps. So we propose new timeindexed LP relaxations, using a decomposition and "gapfilling" approach, we convert these fractional solutions to distributions over strategies, and then use the LP values and the time ordering information from these strategies to devise randomized adaptive scheduling algorithms.
Index Terms:
Approximation Algorithms, Stochastic Optimization, MultiArmed Bandits
Citation:
Anupam Gupta, Ravishankar Krishnaswamy, Marco Molinaro, R. Ravi, "Approximation Algorithms for Correlated Knapsacks and Nonmartingale Bandits," focs, pp.827836, 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, 2011
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