Gradient-free proximal methods with inexact oracle for convex stochastic nonsmooth optimization problems on the simplex

Gasnikov, Alexander; Lagunovskaya, Anastasia; Usmanova, Ilnura; Fedorenko, Fedor

doi:10.1134/s0005117916110114

Cited by 25 publications

(16 citation statements)

References 8 publications

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“…We consider stochastic mirror descent (MD) with inexact oracle [34,44,54] 3 . For a proxfunction d(x) and the corresponding Bregman divergence B d (x, x 1 ), the proximal mirror descent step is…”

Section: The Sa Approach: Stochastic Mirror Descentmentioning

confidence: 99%

Stochastic approximation versus sample average approximation for Wasserstein barycenters

Dvinskikh

2021

Optimization Methods and Software

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In the machine learning and optimization community, there are two main approaches for the convex risk minimization problem, namely the Stochastic Approximation (SA) and the Sample Average Approximation (SAA). In terms of the oracle complexity (required number of stochastic gradient evaluations), both approaches are considered equivalent on average (up to a logarithmic factor). The total complexity depends on a specific problem, however, starting from the work [A. Nemirovski, A. Juditsky, G. Lan, and A. Shapiro, Robust stochastic approximation approach to stochastic programming, SIAM. J. Opt. 19 (2009), pp. 1574-1609] it was generally accepted that the SA is better than the SAA. We show that for the Wasserstein barycenter problem, this superiority can be inverted. We provide a detailed comparison by stating the complexity bounds for the SA and SAA implementations calculating barycenters defined with respect to optimal transport distances and entropy-regularized optimal transport distances. As a byproduct, we also construct confidence intervals for the barycenter defined with respect to entropyregularized optimal transport distances in the 2 -norm. The preliminary results are derived for a general convex optimization problem given by the expectation to have other applications besides the Wasserstein barycenter problem.

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Section: The Sa Approach: Stochastic Mirror Descentmentioning

confidence: 99%

Stochastic approximation versus sample average approximation for Wasserstein barycenters

Dvinskikh

2021

Optimization Methods and Software

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“…We use the technique, developed in [7,8] for stochastic gradient-free nonsmooth convex optimization problems (gradient-free version of mirror descent [2]) to propose a stochastic gradient-free version of saddle-point variant of mirror descent [2] for non-smooth convex-concave saddle-point problems.…”

Section: Introductionmentioning

confidence: 99%

Gradient-Free Methods with Inexact Oracle for Convex-Concave Stochastic Saddle-Point Problem

Beznosikov¹,

Sadiev²,

Gasnikov³

2020

Mathematical Optimization Theory and Operations Research

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In the paper, we generalize the approach Gasnikov et. al, 2017, which allows to solve (stochastic) convex optimization problems with an inexact gradient-free oracle, to the convex-concave saddle-point problem. The proposed approach works, at least, like the best existing approaches. But for a special set-up (simplex type constraints and closeness of Lipschitz constants in 1 and 2 norms) our approach reduces n /log n times the required number of oracle calls (function calculations). Our method uses a stochastic approximation of the gradient via finite differences. In this case, the function must be specified not only on the optimization set itself, but in a certain neighbourhood of it. In the second part of the paper, we analyze the case when such an assumption cannot be made, we propose a general approach on how to modernize the method to solve this problem, and also we apply this approach to particular cases of some classical sets.

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“…Oracle complexity, O (•) p = 1 Stoch. Noise MD [Duchi et al (2015); Gasnikov et al (2016aGasnikov et al ( , 2016b]…”

Section: Assumptionsmentioning

confidence: 99%

“…Derivative-free or zeroth-order optimization Rosenbrock (1960); Brent (1973); Spall (2003) is one of the oldest areas in optimization, which constantly attracts attention of the learning community, mostly in connection to the online learning in the bandit setup Bubeck and Cesa-Bianchi (2012). We study stochastic derivativefree optimization problems in a two-point feedback situation, considered by Agarwal et al (2010); Duchi et al (2015); Shamir (2017) in the learning community and by Nesterov and Spokoiny (2017); Stich et al (2011); Ghadimi and Lan (2013); Ghadimi et al (2016); Gasnikov et al (2016a) in the optimization community. Two-point setup allows to prove complexity bounds, which typically coincide with the complexity bounds for gradient-based algorithms up to a small-degree polynomial of n, where n is the dimension of the decision variable.…”

Section: Introductionmentioning

confidence: 99%

An Accelerated Method for Derivative-Free Smooth Stochastic Convex Optimization

Gorbunov¹,

Dvurechensky²,

Gasnikov³

2018

Preprint

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We consider an unconstrained problem of minimization of a smooth convex function which is only available through noisy observations of its values, the noise consisting of two parts. Similar to stochastic optimization problems, the first part is of a stochastic nature. On the opposite, the second part is an additive noise of an unknown nature, but bounded in the absolute value. In the two-point feedback setting, i.e. when pairs of function values are available, we propose an accelerated derivative-free algorithm together with its complexity analysis. The complexity bound of our derivative-free algorithm is only by a factor of √ n larger than the bound for accelerated gradient-based algorithms, where n is the dimension of the decision variable. We also propose a non-accelerated derivative-free algorithm with a complexity bound similar to the stochastic-gradient-based algorithm, that is, our bound does not have any dimension-dependent factor. Interestingly, if the difference between the starting point and the solution is a sparse vector, for both our algorithms, we obtain better complexity bound if the algorithm uses a 1 -norm proximal setup, rather than the Euclidean proximal setup, which is a standard choice for unconstrained problems

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Gradient-free proximal methods with inexact oracle for convex stochastic nonsmooth optimization problems on the simplex

Cited by 25 publications

References 8 publications

Stochastic approximation versus sample average approximation for Wasserstein barycenters

Stochastic approximation versus sample average approximation for Wasserstein barycenters

Gradient-Free Methods with Inexact Oracle for Convex-Concave Stochastic Saddle-Point Problem

An Accelerated Method for Derivative-Free Smooth Stochastic Convex Optimization

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