随机扰动梯度近似中的矩收敛率

Convergence rate of moments for stochastic perturbation gradient approximation

采用递归估计器的序列求函数的最小值 ,并扩展了Spall的同步扰动随机近似方法 ,进而提出随机扰动梯度近似算法 .对于任意的 1≤q<∞ ,可用估计误差的范数Lq 来度量收敛率 ,序列的收敛速度为O(n- 2 ) ,( >0 ) .在最小点上 ,若代价函数的Hessian矩阵的所有本征值都在 1 2的右面 ,则误差指数 2可任意地接近 1 2 ,并且代价函数足够光滑 。

The sequence of recursive estimators is adopted for function minimization, and Spall's simultaneous perturbation stochastic approximation method is extended, and then an algorithm for stochastic perturbation gradient approximation is proposed. It is proved that this sequence converges under certain conditions with rate O(n -/2 ) for some >0, where the rate is measured by the L q-norm of the estimation error for any 1≤q<∞. It is shown that the error exponent /2 can be arbitrarily close to 1/2 if the Hessian matrix of the cost function at the minimizing point has all its eigenvalues to the right of 1/2, the cost function is sufficiently smooth, and a sufficiently high-order approximation of the derivative is used.