Minimizing Convex Functions with Bounded Perturbations

Phú, Hoàng Xuân

doi:10.1137/080741069

Cited by 4 publications

(1 citation statement)

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“…The invariance of the line search oracle under strictly monotone transformations of the objective function already implied that Random Pursuit converges on certain strictly quasiconvex functions. It also seems in reach to derive convergence guarantees for Random Pursuit on the class of globally convex (or δ-convex) functions [12] or on convex functions with bounded perturbations [30] (see right panel of Figure 7.1 for the graph of such an instance). This may be achieved by appropriately adapting line search methods to these function classes.…”

Section: Discussionmentioning

confidence: 99%

Optimization of Convex Functions with Random Pursuit

2013

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We consider unconstrained randomized optimization of convex objective functions. We analyze the Random Pursuit algorithm, which iteratively computes an approximate solution to the optimization problem by repeated optimization over a randomly chosen one-dimensional subspace. This randomized method only uses zeroth-order information about the objective function and does not need any problem-specific parametrization. We prove convergence and give convergence rates for smooth objectives assuming that the one-dimensional optimization can be solved exactly or approximately by an oracle. A convenient property of Random Pursuit is its invariance under strictly monotone transformations of the objective function. It thus enjoys identical convergence behavior on a wider function class. To support the theoretical results we present extensive numerical performance results of Random Pursuit, two gradient-free algorithms recently proposed by Nesterov, and a classical adaptive step-size random search scheme. We also present an accelerated heuristic version of the Random Pursuit algorithm which significantly improves standard Random Pursuit on all numerical benchmark problems. A general comparison of the experimental results reveals that (i) standard Random Pursuit is effective on strongly convex functions with moderate condition number, and (ii) the accelerated scheme is comparable to Nesterov's fast gradient method and outperforms adaptive step-size strategies.

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Section: Discussionmentioning

confidence: 99%