Modified log-Sobolev inequalities and two-level concentration

We consider a generic modified logarithmic Sobolev inequality (mLSI) of the form $\mathrm{Ent}_{\mu}(e^f) \le \tfrac{\rho}{2} \mathbb{E}_\mu e^f \Gamma(f)^2$ for some difference operator $\Gamma$, and show how it implies two-level concentration inequalities akin to the Hanson--Wright or Bernstein inequality. This can be applied to the continuous (e.\,g. the sphere or bounded perturbations of product measures) as well as discrete setting (the symmetric group, finite measures satisfying an approximate tensorization property, \ldots). Moreover, we use modified logarithmic Sobolev inequalities on the symmetric group $S_n$ and for slices of the hypercube to prove Talagrand's convex distance inequality, and provide concentration inequalities for locally Lipschitz functions on $S_n$. Some examples of known statistics are worked out, for which we obtain the correct order of fluctuations, which is consistent with central limit theorems

Higher order concentration of measure and applications

Sinulis A. Higher order concentration of measure and applications. Bielefeld: Universität Bielefeld; 2019

Concentration Inequalities for Bounded Functionals via Log-Sobolev-Type Inequalities

Götze F, Sambale H, Sinulis A. Concentration Inequalities for Bounded Functionals via Log-Sobolev-Type Inequalities. JOURNAL OF THEORETICAL PROBABILITY. 2020.In this paper, we prove multilevel concentration inequalities for bounded functionals f = f ( X-1,..., X-n) of random variables X-1,..., X-n that are either independent or satisfy certain logarithmic Sobolev inequalities. The constants in the tail estimates depend on the operator norms of k-tensors of higher order differences of f. We provide applications for both dependent and independent random variables. This includes deviation inequalities for empirical processes f (X) = supg is an element of F vertical bar g(X)vertical bar and suprema of homogeneous chaos in bounded random variables in theBanach space case f (X) = sup(t)parallel to Sigma(i1 not equal)...not equal(id) t(i1)... (i)d X-i1 ... X-id parallel to(B). The latter application is comparable to earlier results of Boucheron, Bousquet, Lugosi, and Massart and provides the upper tail bounds of Talagrand. In the case of Rademacher random variables, we give an interpretation of the results in terms of quantities familiar in Boolean analysis. Further applications are concentration inequalities forU-statistics with bounded kernels h and for the number of triangles in an exponential random graph model

Fluctuation Results for General Block Spin Ising Models

Knopfel H, Lowe M, Schubert K, Sinulis A. Fluctuation Results for General Block Spin Ising Models. JOURNAL OF STATISTICAL PHYSICS. 2020;178(5):1175-1200.We study a block spin mean-field Ising model, i.e. a model of spins in which the vertices are divided into a finite number of blocks with each block having a fixed proportion of vertices, and where pair interactions are given according to their blocks. For the vector of block magnetizations we prove Large Deviation Principles and Central Limit Theorems under general assumptions for the block interaction matrix. Using the exchangeable pair approach of Stein's method we establish a rate of convergence in the Central Limit Theorem for the block magnetization vector in the high temperature regime

Mixing times of Glauber dynamics via entropy methods

In this work we prove sufficient conditions for the Glauber dynamics corresponding to a sequence of (non-product) measures on finite product spaces to be rapidly mixing, i.e. that the mixing time with respect to the total variation distance satisfies $t_{mix} = O(N \log N)$, where $N$ is the system size. The proofs do not rely on coupling arguments, but instead use functional inequalities. As a byproduct, we obtain exponential decay of the relative entropy along the Glauber semigroup. These conditions can be checked in various examples, which include the exponential random graph models with sufficiently small parameters (which does not require any monotonicity in the system and thus also applies to negative parameters, as long the associated monotone system is in the high temperature phase), the vertex-weighted exponential random graph models, as well as models with hard constraints such as the random coloring and the hard-core model.Comment: due to various similarities in the proofs the article has been merged with arXiv:1807.07765 - see the second version thereo

Concentration Inequalities on the Multislice and for Sampling Without Replacement

Sambale H, Sinulis A. Concentration Inequalities on the Multislice and for Sampling Without Replacement. Journal of Theoretical Probability . 2021.We present concentration inequalities on the multislice which are based on (modified) log-Sobolev inequalities. This includes bounds for convex functions and multilinear polynomials. As an application, we show concentration results for the triangle count in the G(n, M) Erdos-Renyi model resembling known bounds in the G(n, p) case. Moreover, we give a proof of Talagrand's convex distance inequality for the multislice. Interpreting the multislice in a sampling without replacement context, we furthermore present concentration results for n out of N sampling without replacement. Based on a bounded difference inequality involving the finite-sampling correction factor 1 - (n/ N), we present an easy proof of Serfling's inequality with a slightly worse factor in the exponent, as well as a sub-Gaussian right tail for theKolmogorov distance between the empirical measure and the true distribution of the sample

Logarithmic Sobolev inequalities for finite spin systems and applications

Sambale H, Sinulis A. Logarithmic Sobolev inequalities for finite spin systems and applications. BERNOULLI. 2020;26(3):1863-1890.We derive sufficient conditions for a probability measure on a finite product space (a spin system) to satisfy a (modified) logarithmic Sobolev inequality. We establish these conditions for various examples, such as the (vertex-weighted) exponential random graph model, the random coloring and the hard-core model with fugacity. This leads to two separate branches of applications. The first branch is given by mixing time estimates of the Glauber dynamics. The proofs do not rely on coupling arguments, but instead use functional inequalities. As a byproduct, this also yields exponential decay of the relative entropy along the Glauber semigroup. Secondly, we investigate the concentration of measure phenomenon (particularly of higher order) for these spin systems. We show the effect of better concentration properties by centering not around the mean, but around a stochastic term in the exponential random graph model. From there, one can deduce a central limit theorem for the number of triangles from the CLT of the edge count. In the Erdos-Renyi model the first-order approximation leads to a quantification and a proof of a central limit theorem for subgraph counts

Concentration inequalities for polynomials in alpha-sub-exponential random variables

Götze F, Sambale H, Sinulis A. Concentration inequalities for polynomials in alpha-sub-exponential random variables. Electronic Journal of Probability. 2021;26: 48.We derive multi-level concentration inequalities for polynomials in independent random variables with an alpha-sub-exponential tail decay. A particularly interesting case is given by quadratic forms f (X-1, ..., X-n) = , for which we prove Hanson-Wright-type inequalities with explicit dependence on various norms of the matrix A. A consequence of these inequalities is a two-level concentration inequality for quadratic forms in alpha-sub-exponential random variables, such as quadratic Poisson chaos. We provide various applications of these inequalities. Among them are generalizations of some results proven by Rudelson and Vershynin from sub-Gaussian to alpha-sub-exponential random variables, i. e. concentration of the Euclidean norm of the linear image of a random vector and concentration inequalities for the distance between a random vector and a fixed subspace