Sublinear time algorithms

Rubinfeld, Ronitt

doi:10.4171/022-3/53

Cited by 60 publications

(64 citation statements)

References 6 publications

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“…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: 4supporting

confidence: 72%

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Solving Variational Inequalities with Stochastic Mirror-Prox Algorithm

2011

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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

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Section: 4supporting

confidence: 72%

“…1 0 . We start with the following simple observation: if r e is a solution to (15), then ∂ Z ω(r e ) contains −e and thus is nonempty, so that r e ∈ Z o . Moreover, one has (68) ω ′ (r e ) + e, u − r e ≥ 0 ∀u ∈ Z.…”

Section: Preliminaries We Need the Following Technical Results About mentioning

confidence: 99%

Solving Variational Inequalities with Stochastic Mirror-Prox Algorithm

2011

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“…For a collection of vectors A, we write wt 2 (A) = α∈A f (α) 2 and wt 4 (A) = α∈A f (α) 4 . This notation suppresses the dependence on f , but it will always be clear from the context.…”

Section: Preliminariesmentioning

confidence: 99%

“…Since then, the field of property testing has been very active. For an overview of recent developments, we refer the reader to the surveys [3,4] and the book [5].…”

Section: Introductionmentioning

confidence: 99%

Testing Linear-Invariant Function Isomorphism

Wimmer

Yoshida

2013

Automata, Languages, and Programming

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Abstract. A function f : F n 2 → {−1, 1} is called linear-isomorphic to g if f = g • A for some non-singular matrix A. In the g-isomorphism problem, we want a randomized algorithm that distinguishes whether an input function f is linear-isomorphic to g or far from being so. We show that the query complexity to test g-isomorphism is essentially determined by the spectral norm of g. That is, if g is close to having spectral norm s, then we can test g-isomorphism with poly(s) queries, and if g is far from having spectral norm s, then we cannot test g-isomorphism with o(log s) queries. The upper bound is almost tight since there is indeed a function g close to having spectral norm s whereas testing gisomorphism requires Ω(s) queries. As far as we know, our result is the first characterization of this type for functions. Our upper bound is essentially the Kushilevitz-Mansour learning algorithm, modified for use in the implicit setting. Exploiting our upper bound, we show that any property is testable if it can be well-approximated by functions with small spectral norm. We also extend our algorithm to the setting where A is allowed to be singular.

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“…Sampling is perhaps the simplest example of a computational problem where randomization enables algorithms with running time sublinear in the size of the input. Such sublinear-time algorithms are now known for a wide variety of interesting computational problems; see the surveys [337,339].…”

Section: Chapter Notes and Referencesmentioning

confidence: 99%

Chapter 8 Notes and References

2011

Feynman-Kac-Type Theorems and Gibbs Measures on Path Space

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Sublinear time algorithms

Cited by 60 publications

References 6 publications

Solving Variational Inequalities with Stochastic Mirror-Prox Algorithm

Solving Variational Inequalities with Stochastic Mirror-Prox Algorithm

Testing Linear-Invariant Function Isomorphism

Chapter 8 Notes and References

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