TY - CHAP
T1 - Open-Loop Control
T2 - The Stochastic Gradient Method
AU - Carpentier, Pierre
AU - Chancelier, Jean Philippe
AU - Cohen, Guy
AU - De Lara, Michel
N1 - Publisher Copyright:
© 2015, Springer International Publishing Switzerland.
PY - 2015/1/1
Y1 - 2015/1/1
N2 - The stochastic gradient method has a rather long history. The method foundations were given by (Robbins and Monro, Annals of Mathematical Statistics, 22:400-407, (1951) [129]) on the one hand, and by (Kiefer and Wolfowitz, Annals of Mathematical Statistics, 23:462-466, (1952) [93]) on the other. Later on, (Polyak, Automation and Remote Control, 37(12):1858-1868, (1976) [120], Polyak and Tsypkin, Automation and Remote Control, 40(3):378-389, (1979) [123]) gave results about the convergence rate. Based on this work, (Dodu et al., Bulletin de la Direction des Études et Recherches EDF, (1981) [57]) studied the optimality of the stochastic gradient algorithm, that is, the asymptotic efficiency of the associated estimator. An important contribution by (Polyak, Automation and Remote Control, 51(7):937-946, (1990) [121], Polyak and Juditsky, SIAM Journal on Control and Optimization, 30(4):838-855, (1992) [122]) has been to combine stochastic gradient method and averaging techniques in order to reach the optimal efficiency.
AB - The stochastic gradient method has a rather long history. The method foundations were given by (Robbins and Monro, Annals of Mathematical Statistics, 22:400-407, (1951) [129]) on the one hand, and by (Kiefer and Wolfowitz, Annals of Mathematical Statistics, 23:462-466, (1952) [93]) on the other. Later on, (Polyak, Automation and Remote Control, 37(12):1858-1868, (1976) [120], Polyak and Tsypkin, Automation and Remote Control, 40(3):378-389, (1979) [123]) gave results about the convergence rate. Based on this work, (Dodu et al., Bulletin de la Direction des Études et Recherches EDF, (1981) [57]) studied the optimality of the stochastic gradient algorithm, that is, the asymptotic efficiency of the associated estimator. An important contribution by (Polyak, Automation and Remote Control, 51(7):937-946, (1990) [121], Polyak and Juditsky, SIAM Journal on Control and Optimization, 30(4):838-855, (1992) [122]) has been to combine stochastic gradient method and averaging techniques in order to reach the optimal efficiency.
KW - Auxiliary Problem Principle
KW - Open-loop Optimization Problem
KW - Sample Average Approximation (SAA)
KW - Stochastic Gradient Algorithm
KW - Surely Convergent
UR - https://www.scopus.com/pages/publications/85130809460
U2 - 10.1007/978-3-319-18138-7_2
DO - 10.1007/978-3-319-18138-7_2
M3 - Chapter
AN - SCOPUS:85130809460
T3 - Probability Theory and Stochastic Modelling
SP - 27
EP - 62
BT - Probability Theory and Stochastic Modelling
PB - Springer Nature
ER -