Talagrand's concentration inequality explained

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

t away, is bounded from above by the exponential factor e ^{- t ^{2/4}} . 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]

Statement

The inequality states that if

\Omega = \Omega _{1} x \Omega _{2} x … x \Omega _{n} is a product space endowed with a product probability measure and A is a subset in this space, then for any t \ge 0 \Pr[A] ⋅ \Pr\left[{A ^{c} _{t}\right]} \le e ^{- t ^{2/4}} ,

where

{A ^{c} _{t}} is the complement of A _{t} where this is defined byA _{t} = \{ x \in \Omega\colon~ \rho(A,x) \le t \} and where \rho is Talagrand's convex distance defined as\rho(A,x) = max _{\alpha, \|\alpha\| _{2} \le 1} min _{y} \sum _{i\colon~x _{i} ≠ y _{i}} \alpha _{i}

where

\alpha \in R ^{n} , x,y \in \Omega are n -dimensional vectors with entries\alpha _{i,} x _{i,} y _{i} respectively and \| ⋅ \| _{2} is the \ell ^{2} -norm. That is,\|\alpha\| _{2=\left(\sum} _{i\alpha} ^{2 \right)} ^{i} ^{1/2}

Notes and References

1. Book: Alon . Noga . Spencer . Joel H. . The Probabilistic Method . 2nd . 2000 . John Wiley & Sons, Inc. . 0-471-37046-0.
2. Book: Ledoux, Michel . The Concentration of Measure Phenomenon . American Mathematical Society . 2001 . 0-8218-2864-9.
3. Talagrand . Michel . Concentration of measure and isoperimetric inequalities in product spaces . Publications Mathématiques de l'IHÉS. 1995 . 81 . 73–205 . Springer-Verlag . 0073-8301. 10.1007/BF02699376. math/9406212 . 119668709 .
4. Web site: Castelvecchi . Davide . Mathematician who tamed randomness wins Abel Prize . 21 March 2024 . Nature . 10.1038/d41586-024-00839-6.