Sharp high-dimensional central limit theorems for log-concave distributions

Let X1, ... , Xn be i.i.d. log-concave random vectors in IPd with mean 0 and covariance matrix E. We study the problem of quantifying the normal approximation error for W = n-1/2 Sigma ni=1 Xi with explicit dependence on the dimension d. Specifically, without any restriction on E, we show that the approximation error over rectangles in IPd is bounded by C(log13(dn)/n)1/2 for some universal constant C. Moreover, if the Kannan-Lovasz-Simonovits (KLS) spectral gap conjecture is true, this bound can be improved to C(log3(dn)/n)1/2. This improved bound is optimal in terms of both n and d in the regime log n = O(logd). We also give pWasserstein bounds with all p >= 1 and a Cramer type moderate deviation result for this normal approximation error, and they are all optimal under the KLS conjecture. To prove these bounds, we develop a new Gaussian coupling inequality that gives almost dimensionfree bounds for projected versions of p-Wasserstein distance for every p >= 1. We prove this coupling inequality by combining Stein's method and Eldan's stochastic localization procedure.