A sublinear-time randomized approximation algorithm for matrix games

Grigoriadis, Michael D.; Khachiyan, Leonid

doi:10.1016/0167-6377(95)00032-0

Cited by 96 publications

(107 citation statements)

References 5 publications

Supporting

Mentioning

104

Contrasting

Order By: Relevance

“…In particular, as discussed in the introduction, they gave an -relaxed decision procedure that required O(ρ −2 log m) queries to P , where ρ is the width of the problem instance. A similar result was obtained independently by Grigoriadis and Khachiyan [8]. Many subsequent algorithms (e.g.…”

Section: Fractional Packing and Coveringsupporting

confidence: 64%

On the Number of Iterations for Dantzig-Wolfe Optimization and Packing-Covering Approximation Algorithms

Klein

Young

1999

Integer Programming and Combinatorial Optimization

View full text Add to dashboard Cite

We give a lower bound on the iteration complexity of a natural class of Lagrangeanrelaxation algorithms for approximately solving packing/covering linear programs. We show that, given an input with m random 0/1-constraints on n variables, with high probability, any such algorithm requires Ω(ρ log(m)/ǫ 2 ) iterations to compute a (1 + ǫ)-approximate solution, where ρ is the width of the input. The bound is tight for a range of the parameters (m, n, ρ, ǫ).The algorithms in the class include Dantzig-Wolfe decomposition, Benders' decomposition, Lagrangean relaxation as developed by Held and Karp [1971] for lower-bounding TSP, and many others (e.g. by Plotkin, Shmoys, and Tardos [1988] and Grigoriadis and Khachiyan [1996]). To prove the bound, we use a discrepancy argument to show an analogous lower bound on the support size of (1 + ǫ)-approximate mixed strategies for random two-player zero-sum 0/1-matrix games.

show abstract

Section: Fractional Packing and Coveringsupporting

confidence: 64%

On the Number of Iterations for Dantzig-Wolfe Optimization and Packing-Covering Approximation Algorithms

Klein

Young

1999

Integer Programming and Combinatorial Optimization

View full text Add to dashboard Cite

show abstract

“…This sublinear time behavior [15] was already observed in [4] for the Robust Mirror Descent Stochastic Approximation as applied to a matrix game (the latter problem is the particular case of (52) with p 1 = · · · = p m = 1 and A 0 = 0). Note also that an "ad hoc" sublinear time algorithm for a matrix game, in retrospect close to the one from [4], was discovered in [3] as early as in 1995.…”

Section: 4mentioning

confidence: 99%

“…We quantify the inaccuracy of a candidate solution z ∈ Z by the error Err vi (z) := max u∈Z F (u), z − u ; (3) note that this error is always ≥ 0 and equals zero iff z is a solution to (2).…”

mentioning

confidence: 99%

Solving Variational Inequalities with Stochastic Mirror-Prox Algorithm

2011

View full text Add to dashboard Cite

In this paper we consider iterative methods for stochastic variational inequalities (s.v.i.) with monotone operators. Our basic assumption is that the operator possesses both smooth and nonsmooth components. Further, only noisy observations of the problem data are available. We develop a novel Stochastic Mirror-Prox (SMP) algorithm for solving s.v.i. and show that with the convenient stepsize strategy it attains the optimal rates of convergence with respect to the problem parameters. We apply the SMP algorithm to Stochastic composite minimization and describe particular applications to Stochastic Semidefinite Feasibility problem and deterministic Eigenvalue minimization.1. Introduction. Variational inequalities with monotone operators form a convenient framework for unified treatment (including algorithmic design) of problems with "convex structure", like convex minimization, convexconcave saddle point problems and convex Nash equilibrium problems. In this paper we utilize this framework to develop first order algorithms for stochastic versions of the outlined problems, where the precise first order information is replaced with its unbiased stochastic estimates. This situation arises naturally in convex Stochastic Programming, where the precise first order information is unavailable (see examples in section 4). In some situations, e.g. those considered in [4, Section 3.3] and in Section 4.4, where passing from available, but relatively computationally expensive precise first order information to its cheap stochastic estimates allows to accelerate the solution process, with the gain from randomization growing progressively with problem's sizes.Our "unifying framework" is as follows. Let Z be a convex compact set in Euclidean space E with inner product ·, · , · be a norm on E (not

show abstract

“…We did not consider random nodes here, but they could easily be included as well. We do not write explicitly a proof of the consistency of these algorithms, but we guess that the proof is a consequence of properties in [5,8,2,1]. We'll see the choice of constants below.…”

Section: End While If the Root Is In 1p Or 2p Thenmentioning

confidence: 99%

“…We use the EXP3 algorithm for GSA nodes (variant of the Grigoriadis-Khachiyan algorithm [5,2,1,3]), leading to a probability of choosing an action of the form η + exp(ǫs)/C where η and ǫ are Algorithm 2 Adapting the UCT algorithm for GSA cases.…”

Section: Adapting Uct To the Gsa Acyclic Casementioning

confidence: 99%