Robbins-Monro 随机逼近算法和序列学习(Sequential Learning)

本文介绍了Robbins-Monro随机逼近算法,用于解决当函数未知且存在随机误差的情况下寻找方程根的问题。通过迭代公式,即使在数据具有随机性时,算法依然能通过控制收敛速度找到近似解。此外,文章还探讨了该算法与序列学习的关系,特别是在动态估计函数和寻找函数零点的应用中,如何利用新数据不断修正估计结果。

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随机逼近的问题背景:

设因素x的值可由试验者控制,x的“响应”的指标值为Y,当取x之值x进行试验时,响应Y可表为Y=h(x)+ε,式中h

The Robbins-Monro setting is a mathematical framework used to study stochastic approximation and optimization problems. It is named after Herbert Robbins and Sutton Monro, who introduced this framework in the 1950s. In the Robbins-Monro setting, we consider a sequence of random variables (Xn, Yn) where Xn is a sequence of random variables taking values in a measurable space and Yn is a sequence of random variables taking values in the real line. The goal is to find a sequence of real numbers (θn) that converges to a solution of the optimization problem minimize f(θ) where f is an unknown function that we want to minimize. The key assumption in the Robbins-Monro setting is that the function f is not known, but we can evaluate it at any point θ. Moreover, we assume that the random variables (Xn, Yn) are independent and identically distributed, and that the conditional distribution of Yn given Xn and θn is a probability distribution that depends on θn. The Robbins-Monro algorithm is a way to solve the optimization problem in this setting. It is an iterative algorithm that updates the estimate θn based on the observed value of Yn: θn+1 = θn - an(Xn, θn)(Yn - f(θn)) where an(Xn, θn) is a sequence of positive numbers that satisfies certain conditions. The idea behind this update rule is to move θn in the direction that reduces the expected value of the objective function f. The Robbins-Monro algorithm has been widely used in statistics, machine learning, and control theory to solve various optimization and estimation problems. It is a powerful tool for dealing with complex and noisy data, and it can be used to find the optimal solution even when the objective function is non-convex or non-smooth.
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