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Optimal stopping problems can be found in areas of statistics, economics, and mathematical finance (related to the pricing of American options). A key example of an optimal stopping problem is the secretary problem .
Graphs of probabilities of getting the best candidate (red circles) from n applications, and k/n (blue crosses) where k is the sample size. The secretary problem demonstrates a scenario involving optimal stopping theory [1] [2] that is studied extensively in the fields of applied probability, statistics, and decision theory.
A Bellman equation, named after Richard E. Bellman, is a necessary condition for optimality associated with the mathematical optimization method known as dynamic programming. [1] It writes the "value" of a decision problem at a certain point in time in terms of the payoff from some initial choices and the "value" of the remaining decision ...
In machine learning, early stopping is a form of regularization used to avoid overfitting when training a model with an iterative method, such as gradient descent. Such methods update the model to make it better fit the training data with each iteration. Up to a point, this improves the model's performance on data outside of the training set (e ...
Another attempt proposed to make progress on the problem is a continuous time version of the problem where the observations follow a Poisson arrival process of homogeneous rate 1. Under some mild assumptions, the corresponding value function w ( t ) {\displaystyle w(t)} is bounded and Lipschitz continuous, and the differential equation for this ...
An equation (or set of equations) stating that the first derivative(s) equal(s) zero at an interior optimum is called a 'first-order condition' or a set of first-order conditions. Optima of equality-constrained problems can be found by the Lagrange multiplier method.
The primary application of the Levenberg–Marquardt algorithm is in the least-squares curve fitting problem: given a set of empirical pairs (,) of independent and dependent variables, find the parameters of the model curve (,) so that the sum of the squares of the deviations () is minimized:
In probability theory, the optional stopping theorem (or sometimes Doob's optional sampling theorem, for American probabilist Joseph Doob) says that, under certain conditions, the expected value of a martingale at a stopping time is equal to its initial expected value. Since martingales can be used to model the wealth of a gambler participating ...