DOI link for Stochastic Approximation
Stochastic Approximation book
Robbins and Monro (1951) introduce the first stochastic approximation method to address the problem of finding the root of a regression function M(x). Precisely, let Y =Y (x) denote a random outcome of interest at the stimulus level x with expectation E(Y ) = M(x). The objective is to sequentially approach the root x∗ of the equation
M(x∗) = θ (14.1)
for a given θ . In the special case when Y is a binary indicator for toxicity and the stimulus level is dose, the regression function is equal to the toxicity probability at dose x, that is, M(x) = pi(x) = Pr(Y = 1 | x), and the root x∗ is the θ th percentile on the dose-toxicity curve pi . Using dose finding terminology, solving (14.1) amounts to estimating the MTD under the surrogacy perspective (Definition 2.1). As such, the Robbins-Monro stochastic approximation is a natural method for dose finding in clinical trials. In this chapter we explore the clinical relevance of the stochastic approximation method via its connections to the CRM. Section 14.2 reviews the Robbins-Monro method and some of its refinements, and discusses their implications on the CRM. Section 14.3 presents some adaptations of stochastic approximation for dose finding trials. Section 14.4 discuss some future directions in dose finding methodology. Section 14.5 ends this chapter with some technical details regarding the regularity conditions for M(x) and Y (x).