Concentration Inequalities on the Multislice and for Sampling Without Replacement

Abstract

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) Erdős–Rényi 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 $$1 - (n / N)$$ 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 the Kolmogorov distance between the empirical measure and the true distribution of the sample.

Private Aggregation of Trajectories

“…Proposition A.1 (Proposition 1 in Sambale & Sinulis (2022)). Let f : S N → R be a real-valued function over S N , such that |f (π) − f (π i↔j )| ≤ c i,j for all π ∈ S N and all 1 ≤ i < j ≤ N for some c i,j ≥ 0.…”