A high-dimensional CLT in $\mathcal{W}_2$ distance with near optimal convergence rate

Let $X_1, \ldots , X_n$ be i.i.d. random vectors in $\mathbb{R}^d$ with $\|X_1\| \le \beta$. Then, we show that $\frac{1}{\sqrt{n}}(X_1 + \ldots + X_n)$ converges to a Gaussian in quadratic transportation (also known as "Kantorovich" or "Wasserstein") distance at a rate of $O\left( \frac{\sqrt{d} \beta \log n}{\sqrt{n}} \right)$, improving a result of Valiant and Valiant. The main feature of our theorem is that the rate of convergence is within $\log n$ of optimal for $n, d \rightarrow \infty$.