Slides for Introduction to Stochastic Search and Optimization (ISSO) by J. C. Spall CHAPTER 15 SIMULATION-BASED OPTIMIZATION II: STOCHASTIC GRADIENT AND SAMPLE PATH METHODS Organization of chapter in ISSO Introduction to gradient estimation Interchange of derivative and integral Gradient estimation techniques Likelihood ratio/score function (LR/SF) Infinitesimal perturbation analysis (IPA) Optimization with gradient estimates Sample path method

Issues in Gradient Estimation Estimate the gradient of the loss function with respect to parameters for optimization from simulation outputs where L(q) is a scalar-valued loss function to minimize and q is a p-dimensional vector of parameters Essential properties of gradient estimates Unbiased: Small variance

Two Types of Parameters where V is the random effect in the system, is the probability density function of V Distributional parameters qD: Elements of q that enter via their effect on probability distribution of V. For example, if scalar V has distribution N(m,s2), then m and s2 are distributional parameters Structural parameters qS: Elements of q that have effects directly on the loss function (via Q) Distinction not always obvious

Interchange of Derivative and Integral Unbiased gradient estimations using only one simulation require the interchange of derivative and integral: Above generally not true. Technical conditions needed for validity: Q ·pV and are continuous Above has implications in practical applications

A General Form of Gradient Estimate Assume that all the conditions required for the exchange of derivative and integral are satisfied, Hence, an unbiased gradient estimate can be obtained as Output from one simulation!

Two Gradient Estimates: LR/SF and IPA pure LR/SF pure IPA Likelihood Ratio/ Score Function (LR/SF): only distributional parameters Infinitestimal Perturbation Analysis (IPA): only structural parameters

Comparison of Pure LR/SF and IPA In practice, neither extreme (LR/SF or IPA) may provide a framework for reasonable implementation: LR/SF may require deriving a complex distribution function starting from U(0,1) IPA may lead to intractable Q/q with a complex Q(q,V) Pure LR/SF gradient estimate tend to suffer from large variance (variance can grow with the number of components in V) Pure IPA may result in a Q(q,V) that fails to meet the conditions for valid interchange of derivative and integral. Hence can lead to biased gradient estimate. In many cases where IPA is feasible, it leads to low variance gradient estimate

A Simple Example: Exponential Distribution Let Z be exponential random variable with mean q. That is . Define L = E(Z) = q. Then L/q = 1. LR/SF estimate: V = Z; Q(q,V) = V. IPA estimate: V = U(0,1); Q(q,V) = -qlogV (Z = -qlogV). Both of LR/SF and IPA estimators are unbiased

Stochastic Optimization with Gradient Estimate Use the gradient estimates in the root-finding stochastic approximation (SA) algorithm to minimize the loss function L(q) = E[Q(q,V)]: Find q* such that g(q*) = 0 based on simulation outputs A general root-finding SA algorithm: where ak is the step size with If Yk is unbiased and has bounded variance (and other appropriate assumptions hold), then (a.s.) an estimate of

Simulation-Based Optimization Use gradient estimate derived from one simulation run in the iteration of SA: where Vk is the realization of V from a simulation run with parameter q set at run one simulation with q = to obtain Vk derive gradient estimate from Vk iterate SA with the gradient estimate

Example: Experimental Response (Examples 15.4 and 15.5 in ISSO) Let {Vk} be i.i.d. randomly generated binary (on-off) stimuli with “on” probability l. Assume Q(l,b,Vk) represents negative of specimen response, where b is design parameter. Objective is to design experiment to maximize the response (i.e., minimize Q) by selecting values for l and b. Gradient estimate: q = [l, b]T; where and denotes derivative w.r.t. x

Experimental Response (continued) Specific response function: where b is a structural parameter, but l is both a distributional and structural parameter. Then:

Search Path in Experimental Response Problem

Sample Path Method Sample path method based on reusing a fixed set of simulation runs Method based on minimizing rather than L() represents sample mean of N simulation runs If N is large, then minimum of is close to minimum of L() (under conditions) Optimization problem with is effectively deterministic Can use standard nonlinear programming IPA and/or LR/SF methods of gradient estimation still relevant Generally need to choose a fixed value of  (reference value) to produce the N simulation runs Choice of reference value has impact on for finite N

Accuracy of Sample Path Method Interested in accuracy of sample path method in seeking true optimal  (minimum of L()) Let represent minimum of surrogate loss Let denote final solution from nonlinear programming method Hence, error in estimate is due to two sources: Error in nonlinear programming solution to finding Difference in  and Triangle inequality can be used to provide bound to overall error: Sometimes numerical values can be assigned to two right-hand terms in triangle inequality