CHAPTER 4 S TOCHASTIC A PPROXIMATION FOR R OOT F INDING IN N ONLINEAR M ODELS Organization of chapter in ISSO –Introduction and potpourri of examples Sample mean Quantile and CEP Production function (contrast with maximum likelihood) –Convergence of the SA algorithm –Asymptotic normality of SA and choice of gain sequence –Extensions to standard root-finding SA Joint parameter and state estimation Higher-order methods for algorithm acceleration Iterate averaging Time-varying functions Slides for Introduction to Stochastic Search and Optimization (ISSO) by J. C. Spall

4-2 Stochastic Root-Finding Problem Focus is on finding  (i.e.,   ) such that g(  ) = 0 nonlinear –g(  ) is typically a nonlinear function of  (contrast with Chapter 3 in ISSO) Assume only noisy measurements of g(  ) are available: Y k (  ) = g(  ) + e k (  ), k = 0, 1, 2,…, Above problem arises frequently in practice –Optimization with noisy measurements ( g(  ) represents gradient of loss function) (see Chapter 5 of ISSO) –Quantile-type problems –Equation solving in physics-based models –Machine learning (see Chapter 11 of ISSO)

4-3 Core Algorithm for Stochastic Root-Finding Basic algorithm published in Robbins and Monro (1951) Algorithm is a stochastic analogue to steepest descent when used for optimization –Noisy measurement Y k (  ) replaces exact gradient g(  ) Generally wasteful to average measurements at given value of  across iterations –Average across iterations (changing  ) Core Robbins-Monro algorithm for unconstrained root- finding is Constrained version of algorithm also exists

4-4 Circular Error Probable (CEP): Example of Root-Finding (Example 4.3 in ISSO) Interested in estimating radius of circle about target such that half of impacts lie within circle (  is scalar radius) Define success variable Root-finding algorithm becomes Figure on next slide illustrates results for one study

4-5 True and estimated CEP: 1000 impact points with impact mean differing from target point (Example 4.3 in ISSO)

4-6 Convergence Conditions Central aspect of root-finding SA are conditions for formal convergence of the iterate to a root   –Provides rigorous basis for many popular algorithms (LMS, backpropagation, simulated annealing, etc.) Section 4.3 of ISSO contains two sets of conditions: –“Statistics” conditions based on classical assumptions about g(  ), noise, and gains a k –“Engineering” conditions based on connection to deterministic ordinary differential equation (ODE) Convergence and stability of ODE dZ(  )  /  d  = –g(Z(  )) closely related to convergence of SA algorithm (Z(  ) represents p- dimensional time-varying function and  denotes time) Neither of statistics or engineering conditions is special case of other

4-7 ODE Convergence Paths for Nonlinear Problem in Example 4.6 in ISSO: Satisfies ODE Conditions Due to Asymptotic Stability and Global Domain of Attraction

4-8 Gain Selection Choice of the gain sequence a k is critical to the performance of SA Famous conditions for convergence are =  and A common practical choice of gain sequence is where 1/2 0, and A  0 Strictly positive A (“stability constant”) allows for larger a (possibly faster convergence) without risking unstable behavior in early iterations  and A can usually be pre-specified; critical coefficient a usually chosen by “trial-and-error”

4-9 Extensions to Basic Root-Finding SA (Section 4.5 of ISSO) Joint Parameter and State Evolution –There exists state vector x k related to system being optimized –E.g., state-space model governing evolution of x k, where model depends on values of  Adaptive Estimation and Higher-Order Algorithms –Adaptively estimating gain a k –SA analogues of fast Newton-Raphson search Iterate Averaging –See slides to follow Time-Varying Functions –See slides to follow

4-10 Iterate Averaging Iterate averaging is important and relatively recent development in SA asymptoticProvides means for achieving optimal asymptotic performance without using optimal gains a k Basic iterate average uses following sample mean as final estimate: finite-sampleResults in finite-sample practice are mixed Success relies on large proportion of individual iterates hovering in some balanced way around   –Many practical problems have iterate approaching   in roughly monotonic manner –Monotonicity not consistent with good performance of iterate averaging; see plot on following slide

4-11 Contrasting Search Paths for Typical p = 2 Problem: Ineffective and Effective Uses of Iterate Averaging

4-12 Time-Varying Functions In some problems, the root-finding function varies with iteration: g k (  ) (rather than g(  )) –Adaptive control with time-varying target vector –Experimental design with user-specified input values –Signal processing based on Markov models (Subsection of ISSO) Let denote the root to g k (  ) = 0 Suppose that  for some fixed value (equivalent to the fixed   in conventional root-finding) In such cases, much standard theory continues to apply Plot on following slide shows case when g k (  ) represents a gradient function with scalar 

4-13 Time-Varying g k (  ) =  L k (  )  /   for Loss Functions with Limiting Minimum