Stochastic gradient algorithm with random truncations |
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| Affiliation: | 1. Institut de Mathématiques de Toulouse, UMR 5219, Université de Toulouse, UPS IMT, F-31062 Toulouse Cedex 9, France;2. Center for Mathematical Modeling (CNRS UMI 2807), University of Chile, Chile;3. Université de Lorraine, CNRS, Inria, IECL, F-54000 Nancy, France;1. Université de Lorraine, CNRS, Inria, IECL, F-54000 Nancy, France;2. Inria, 2 rue Simone Iff, 75012 Paris, France;3. Université Paris-Est, CERMICS (ENPC), 77455 Marne-la-Vallée 2, France;1. School of Information Technology, Jiangxi University of Finance and Economics, Nanchang, Jiangxi 330032, China;2. Institute of Computer Science and Technology, Peking University, Beijing 100080, China;3. School of Computer Science and Engineering, Nanyang Technological University, 639798, Singapore;1. Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK;2. Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France |
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| Abstract: | Let f : Rd × Rd → R be a Borel-measurable function which satisfies ∫Rd′|f(θ, x) < ∞, ∨θ ϵ Rd, where q0(·) is a probability measure on (Rd′, Bd′). The problem of minimization of the function f0(θ) = ∫Rd′(θ, x)q0(d), θ ϵ Rd, is considered for the case when the probability measure q0(·) is unknown, but a realization of a non-stationary random process {Xn}n⩾1 whose single probability measures in a certain sense tend to q0(·), is available. The random process {Xn}n⩾1 is defined on a common probability space, R′-valued, correlated and satisfies certain uniform mix conditions. The function f(·, ·) is completely known. A stochastic gradient algorithm with random truncations is used for the minimization of f0(·), and its almost sure convergence is proved. |
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