57,695 research outputs found
Improved Bounds in Stochastic Matching and Optimization
We consider two fundamental problems in stochastic optimization: approximation algorithms for stochastic matching, and sampling bounds in the black-box model. For the former, we improve the current-best bound of 3.709 due to Adamczyk et al. (2015), to 3.224; we also present improvements on Bansal et al. (2012) for hypergraph matching and for relaxed versions of the problem. In the context of stochastic optimization, we improve upon the sampling bounds of Charikar et al. (2005)
Reduced Complexity Filtering with Stochastic Dominance Bounds: A Convex Optimization Approach
This paper uses stochastic dominance principles to construct upper and lower
sample path bounds for Hidden Markov Model (HMM) filters. Given a HMM, by using
convex optimization methods for nuclear norm minimization with copositive
constraints, we construct low rank stochastic marices so that the optimal
filters using these matrices provably lower and upper bound (with respect to a
partially ordered set) the true filtered distribution at each time instant.
Since these matrices are low rank (say R), the computational cost of evaluating
the filtering bounds is O(XR) instead of O(X2). A Monte-Carlo importance
sampling filter is presented that exploits these upper and lower bounds to
estimate the optimal posterior. Finally, using the Dobrushin coefficient,
explicit bounds are given on the variational norm between the true posterior
and the upper and lower bounds
On Kernelized Multi-armed Bandits
We consider the stochastic bandit problem with a continuous set of arms, with
the expected reward function over the arms assumed to be fixed but unknown. We
provide two new Gaussian process-based algorithms for continuous bandit
optimization-Improved GP-UCB (IGP-UCB) and GP-Thomson sampling (GP-TS), and
derive corresponding regret bounds. Specifically, the bounds hold when the
expected reward function belongs to the reproducing kernel Hilbert space (RKHS)
that naturally corresponds to a Gaussian process kernel used as input by the
algorithms. Along the way, we derive a new self-normalized concentration
inequality for vector- valued martingales of arbitrary, possibly infinite,
dimension. Finally, experimental evaluation and comparisons to existing
algorithms on synthetic and real-world environments are carried out that
highlight the favorable gains of the proposed strategies in many cases
Confidence level solutions for stochastic programming
We propose an alternative approach to stochastic programming based on Monte-Carlo sampling and stochastic gradient optimization. The procedure is by essence probabilistic and the computed solution is a random variable. The associated objective value is doubly random, since it depends on two outcomes: the event in the stochastic program and the randomized algorithm. We propose a solution concept in which the probability that the randomized algorithm produces a solution with an expected objective value departing from the optimal one by more than is small enough. We derive complexity bounds for this process. We show that by repeating the basic process on independent sample, one can significantly sharpen the complexity bounds
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