The purpose of stochastic approximation (SA) is to find the roots of f(·) or the maximiz-er (minimizer) of L(·) when the unknown function f(·) or L(·) can be observed but with noise. SA is an impor...The purpose of stochastic approximation (SA) is to find the roots of f(·) or the maximiz-er (minimizer) of L(·) when the unknown function f(·) or L(·) can be observed but with noise. SA is an important tool in dealing with many problems arising from systems and control, whose solutions often rely on convergence of the SA algorithm applied. Here the pathwise convergence of SA algorithms is considered, when the observation noise may depend on state by which we mean those x at which f( x) or L( x) are observed. The conditions imposed on the observation noise are the weakest in comparison with the existing ones. When the algorithm is to find the roots of f(·), the superiority of the condition given in the paper over those used in literature consists in the fact that the present condition is directly verifiable, needless to see the behaviour of the algorithm. When the algorithm is to find the maximizer (minimizer) of L(·), the present conditioin allows the observation noise to depend on the state. The conditions imposed on f(·) and L(·) are truly general: f(·) is required to be measurable and locally bounded if the roots of f(·) are sought, and the gradient of L(·) is required to be locally Lipschitz continuous if the maximizer (minimizer) of L(·) is searched.展开更多
文摘The purpose of stochastic approximation (SA) is to find the roots of f(·) or the maximiz-er (minimizer) of L(·) when the unknown function f(·) or L(·) can be observed but with noise. SA is an important tool in dealing with many problems arising from systems and control, whose solutions often rely on convergence of the SA algorithm applied. Here the pathwise convergence of SA algorithms is considered, when the observation noise may depend on state by which we mean those x at which f( x) or L( x) are observed. The conditions imposed on the observation noise are the weakest in comparison with the existing ones. When the algorithm is to find the roots of f(·), the superiority of the condition given in the paper over those used in literature consists in the fact that the present condition is directly verifiable, needless to see the behaviour of the algorithm. When the algorithm is to find the maximizer (minimizer) of L(·), the present conditioin allows the observation noise to depend on the state. The conditions imposed on f(·) and L(·) are truly general: f(·) is required to be measurable and locally bounded if the roots of f(·) are sought, and the gradient of L(·) is required to be locally Lipschitz continuous if the maximizer (minimizer) of L(·) is searched.
