Talagrand’s Bernoulli Conjecture, resolved.

Apparently, Bednorz and Latała have solved Talagrand’s $5,000 Bernoulli Conjecture. The conjecture concerns the supremum of a Bernoulli process.

Consider a finite subset {T \subseteq \ell^2} and define the value

\displaystyle  b(T) = \mathbb{E} \max_{t \in T} \sum_{i \geq 1} \varepsilon_i t_i\,,

where {\varepsilon_1, \varepsilon_2, \ldots} are i.i.d. random {\pm 1} . This looks somewhat similar to the corresponding value

\displaystyle  g(T) = \mathbb{E} \max_{t \in T} \sum_{i \geq 1} g_i t_i\,,

where {g_1, g_2, \ldots} are i.i.d. standard normal random variables. But while {g(T)} can be characterized (up to universal constant factors) by the Fernique-Talagrand majorizing measures theory, no similar control was known for {b(T)} . One stark difference between the two cases is that {g(T)} depends only on the distance geometry of {T} , i.e. {g(A(T))=g(T)} whenever {A} is an affine isometry. On the other hand, {b(T)} can depend heavily on the coordinate structure of {T} .

There are two basic ways to prove an upper bound on {b(T)} . One is via the trivial bound

\displaystyle  b(T) \leq \max_{t \in T} \|t\|_1\,. \ \ \ \ \ (1)

The other uses the fact that the tails of Gaussians are “fatter” than those of Bernoullis.

\displaystyle  b(T) \leq \sqrt{\frac{\pi}{2}} g(T)\,. \ \ \ \ \ (2)

This can be proved easily using Jensen’s inequality.

Talagrand’s Bernoulli conjecture is that {b(T)} can be characterized by these two upper bounds.

Bernoulli conjecture: There exists a constant {C > 0} such that for every {T \subseteq \ell^2} , there are two subsets {T_1, T_2 \subseteq \ell^2} such that

\displaystyle  T \subseteq T_1 + T_2 = \{ t_1 + t_2 : t_1 \in T_1, t_2 \in T_2 \}\,,

and

\displaystyle  g(T_1) + \sup_{t \in T_2} \|t\|_1 \leq C b(T)\,.

Note that this is a “characterization” because given such sets {T_1} and {T_2} , equations (1) and (2) imply

\displaystyle  b(T) \leq \sqrt{\frac{\pi}{2}} g(T_1) + \sup_{t \in T_2} \|t\|_1\,.

The set {T_1} controls the “Gaussian” part of the Bernoulli process, while the set {T_2} controls the part that is heavily dependent on the coordinate structure.

This beautiful problem finally appears to have met a solution. While I don’t know of any applications yet in TCS, it does feel like something powerful and relevant.