Zero-Order Stochastic Optimization: Keifer-Wolfowitz

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 finite difference has some impact on the rate of convergence.

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Continuous Probability Distributions

We consider distributions that have a continuous range of values. Discrete probability distributions where defined by a probability mass function. Analogously continuous probability distributions are defined by a probability density function.

(This is a section in the notes here.)

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Discrete Probability Distributions

There are some probability distributions that occur frequently. This is because they either have a particularly natural or simple construction. Or they arise as the limit of some simpler distribution. Here we cover

  • Bernoulli random variables
  • Binomial distribution
  • Geometric distribution
  • Poisson distribution.

(This is a section in the notes here.)

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Probability and Set Operations

(This is a section in the notes here.)

We want to calculate probabilities for different events. Events are sets of outcomes, and we recall that there are various ways of combining sets. The current section is a bit abstract but will become more useful for concrete calculations later.

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