=- + * *np random rand( , =- + * *np random rand( , = ( , )** * / ( ,yestcmc np mean(zres np hstack ((res estcmc )plt plot(lams resplt xlabel ( '$lambda'plt ylabel ( '$\hat{ }(lambda )$'plt ticklabel_format style ='sci 'axis=' 'scilimits =( , )plt show ( figure the stochastic counterpart method replaces the unknown ( (that isthe scaled variance of the importance sampling estimatorwith its sample averageb (lthe minimum value of is attained around third method for stochastic optimization is the cross-entropy method in particularalgorithm can easily be modified to minimize noisy functions (xese(xx)as defined in ( the only change required in the algorithm is that every function value (xbe replaced by its estimate (xdepending on the level of noise in the functionthe sample size might have to be increased considerably example (cross-entropy method for noisy optimizationto explore the use of the ce method for noisy optimizationtake the following noisy discrete optimization problem suppose there is "black boxthat contains an unknown binary sequence of bits if one feeds the black box any input vectorit will first scramble the input by independently flipping the bits (changing to and to with probability th and then return the number of bits that do not match the true (unknownbinary sequence this is illustrated in figure for