Concentration inequalities are mathematical inequalities that provide bounds on how a random variable deviates from a certain value (typically its mean). These inequalities are essential in probability theory and statistics, particularly in the fields of machine learning, information theory, and statistical learning, because they help analyze the behavior of sums of random variables, as well as the performance of estimators and algorithms. There are several well-known concentration inequalities, each suitable for different types of random variables and different settings.
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A concentration inequality bounds the probability that a random variable differs substantially from a typical value such as its expected value or median.