Random gradient-free minimization of convex functions
AbstractIn this paper, we prove the complexity bounds for methods of Convex Optimization based only on computation of the function value. The search directions of our schemes are normally distributed random Gaussian vectors. It appears that such methods usually need at most n times more iterations than the standard gradient methods, where n is the dimension of the space of variables. This conclusion is true both for nonsmooth and smooth problems. For the later class, we present also an accelerated scheme with the expected rate of convergence O(n[ exp ]2 /k[ exp ]2), where k is the iteration counter. For Stochastic Optimization, we propose a zero-order scheme and justify its expected rate of convergence O(n/k[ exp ]1/2). We give also some bounds for the rate of convergence of the random gradient-free methods to stationary points of nonconvex functions, both for smooth and nonsmooth cases. Our theoretical results are supported by preliminary computational experiments.
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Bibliographic InfoPaper provided by Université catholique de Louvain, Center for Operations Research and Econometrics (CORE) in its series CORE Discussion Papers with number 2011001.
Date of creation: 01 Jan 2011
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convex optimization; stochastic optimization; derivative-free methods; random methods; complexity bounds;
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