In the probability theory field of mathematics, Talagrand's concentration inequality is an isoperimetric-type inequality for product probability spaces.
[1][2] It was first proved by the French mathematician Michel Talagrand.
[3] The inequality is one of the manifestations of the concentration of measure phenomenon.
[2] Roughly, the product of the probability to be in some subset of a product space (e.g. to be in one of some collection of states described by a vector) multiplied by the probability to be outside of a neighbourhood of that subspace at least a distance
away, is bounded from above by the exponential factor
It becomes rapidly more unlikely to be outside of a larger neighbourhood of a region in a product space, implying a highly concentrated probability density for states described by independent variables, generically.
The inequality can be used to streamline optimisation protocols by sampling a limited subset of the full distribution and being able to bound the probability to find a value far from the average of the samples.
[4] The inequality states that if
is a product space endowed with a product probability measure and
is a subset in this space, then for any
is Talagrand's convex distance defined as where
-dimensional vectors with entries
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