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Online EXP3 Learning in Adversarial Bandits with Delayed Feedback
Ilai Bistritz, Zhengyuan Zhou, Xi Chen, Nicholas Bambos, Jose Blanchet
Consider a player that in each of T rounds chooses one of K arms. An adversary chooses the cost of each arm in a bounded interval, and a sequence of feedback delays {dt} that are unknown to the player. After picking arm at at round t, the player receives the cost of playing this arm dt rounds later. In cases where t + dt > T, this feedback is simply missing.
Teen brothers build a Disney-inspired ride in family basement
Nico (right) and Matteo Mucchetti pose with their homemade dark ride vehicle. We may earn revenue from the products available on this page and participate in affiliate programs. When 12-year-old Matteo Mucchetti mapped out an amusement-style attraction that he wanted to create in his family's basement and then showed it to his older brother Nico, the high-school sophomore was immediately sold. "This is amazing," said Nico. "Let's make it!" Matteo had sketched on paper a top-down view of the multi-room space in Bear, Delaware, where they live.
Benefits of over-parameterization with EM
Ji Xu, Daniel J. Hsu, Arian Maleki
Expectation Maximization (EM) is among the most popular algorithms for maximum likelihood estimation, but it is generally only guaranteed to find its stationary points of the log-likelihood objective. The goal of this article is to present theoretical and empirical evidence that over-parameterization can help EM avoid spurious local optima in the log-likelihood. We consider the problem of estimating the mean vectors of a Gaussian mixture model in a scenario where the mixing weights are known. Our study shows that the global behavior of EM, when one uses an over-parameterized model in which the mixing weights are treated as unknown, is better than that when one uses the (correct) model with the mixing weights fixed to the known values. For symmetric Gaussians mixtures with two components, we prove that introducing the (statistically redundant) weight parameters enables EM to find the global maximizer of the log-likelihood starting from almost any initial mean parameters, whereas EM without this over-parameterization may very often fail. For other Gaussian mixtures, we provide empirical evidence that shows similar behavior. Our results corroborate the value of over-parameterization in solving non-convex optimization problems, previously observed in other domains.