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Erscheinungsjahr: 
2023
Quellenangabe: 
[Journal:] Computational Optimization and Applications [ISSN:] 1573-2894 [Volume:] 87 [Issue:] 3 [Publisher:] Springer US [Place:] New York, NY [Year:] 2023 [Pages:] 935-976
Verlag: 
Springer US, New York, NY
Zusammenfassung: 
In this contribution, we present a full overview of the continuous stochastic gradient (CSG) method, including convergence results, step size rules and algorithmic insights. We consider optimization problems in which the objective function requires some form of integration, e.g., expected values. Since approximating the integration by a fixed quadrature rule can introduce artificial local solutions into the problem while simultaneously raising the computational effort, stochastic optimization schemes have become increasingly popular in such contexts. However, known stochastic gradient type methods are typically limited to expected risk functions and inherently require many iterations. The latter is particularly problematic, if the evaluation of the cost function involves solving multiple state equations, given, e.g., in form of partial differential equations. To overcome these drawbacks, a recent article introduced the CSG method, which reuses old gradient sample information via the calculation of design dependent integration weights to obtain a better approximation to the full gradient. While in the original CSG paper convergence of a subsequence was established for a diminishing step size, here, we provide a complete convergence analysis of CSG for constant step sizes and an Armijo-type line search. Moreover, new methods to obtain the integration weights are presented, extending the application range of CSG to problems involving higher dimensional integrals and distributed data.
Schlagwörter: 
Stochastic gradient scheme
Convergence analysis
Step size rule
Backtracking line search
Constant step size
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