We want to optimize the expected value of some random function. This is the problem we solved with Stochastic Gradient Descent. However, we assume that we no longer have access to unbiased estimate of the gradient. We only can obtain estimates of the function itself. In this case we can apply the Kiefer-Wolfowitz procedure. The idea here is to replace the random gradient estimate used in stochastic gradient descent with a finite difference. If the increments used for these finite differences are sufficiently small, then over time convergence can be achieved. The approximation error for the fin ..read more

Let’s explain why the normal distribution is so important. (This is a section in the notes here.) Suppose that I throw a coin times and count the number of heads The proportion of heads should be close to its mean and for it should be even closer. This can be shown mathematically (not just for coin throws but for quite general random variables) Theorem [Weak Law of Large Numbers] For independent random variables , , with mean and variance bounded above by , if we define then for all We will prove this result a little later. But, continuing the discussion, suppose are independent iden ..read more

Next semester, I will teach a short course on Probability for university students who have not taken probability before, who know some basic mathematics, but who are not necessarily going to be studying mathematic. The notes for this are here: Probability_notesDownload ..read more