At NIPS yesterday, James Spall gave a nice overview of stochastic optimization. Stochastic optimization is the process of finding the minimum of a function $f(x)$ where are measurements or samples of the function are noisy. He stressed that the no free lunch theorems (Wolpert & Macready 1995) limit the efficiency of any global minimization problem if there are no restrictions on $f$. He described in detail the Simultaneous Perturbation Stochastic Approximation (SPSA) method which appears to be a great method for optimizing with noisy measurements. The basic is idea is that you don’t need to approximate the gradient by making $p+1$ measurements in a $p$ dimensional domain. Instead, you sample $f$ at two nearby randomly generated points and make a nearly unbiased estimate of the gradient from those two measurements. There is also way to form estimates of the Hessian with just 4 samples which leads to a stochastic algorithm similar to Newton-Raphson. None of these methods can converge faster than $O(1/\sqrt{n})$ due to the noise, but they may be very useful as robust semi-global optimizers for functions with lots of local minima or high frequencies.

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