On the Convergence of Random Search Algorithms In Continuous Time with Applications to Adaptive Control
On the Convergence of Random Search Algorithms In Continuous Time with Applications to Adaptive Control
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DOI：
10.1109/tsmc.1973.5408578
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发表时间：
1970-12
期刊：
影响因子：
--
通讯作者：
R. Gran
中科院分区：
文献类型：
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作者：
R. Gran
使用梯度搜索算法来计算函数的最小值是很好理解的。特别地，在连续时间中，该算法可以被公式化为微分方程，其平衡解是函数的局部极小值。Khas'Minskii提出了使用具有白噪声驱动项的连续梯度算法。他表明，使用一个参数的收敛性的概率密度函数，即平衡解决这一微分方程是全球最低的功能。本文从扩散过程理论的角度对这一结果进行了评述。证明了Khas'minskii的结论是不正确的，而且实际上随机梯度搜索的运算与无噪声梯度搜索的运算是相同的。事实上，有噪声的搜索以概率1收敛到任何局部最小值，与无噪声搜索收敛到局部最小值的方式相同。一些随机自适应控制系统使用随机搜索算法进行操作。其中之一，由于巴伦，分析表明，它满足收敛的条件所施加的结果已经在这里推导出来。
The use of a gradient search algorithm for the computation of the minimum of a function is well understood. In particular, in continuous time, the algorithm can be formulated as a differential equation whose equilibrium solution is a local minimum of the function. Khas'minskii has proposed using the continuous gradient algorithm with a white-noise driving term. He shows, using an argument on the convergence of the probability density function, that the equilibrium solution of this differential equation is the global minimum of the function. This paper reviews that result from the point of view of the theory of diffusion processes. It is shown that the conclusion of Khas'minskii is not correct, and that in fact the operation of the stochastic gradient search is the same as the operation of the noise-free gradient search. In fact, the search with noise converges with probability one to any of the local minima in the same way as the noise-free search converges to a local minimum. Several stochastic adaptive control systems use random search algorithms for their operation. One of these, due to Barron, is analyzed to show that it meets the conditions for convergence imposed by the results which have been derived here.