  Leveraging algorithmic stability to derive sharp generalization bounds is a
classic and powerful approach in learning theory. Since Vapnik and Chervonenkis
[1974] first formalized the idea for analyzing SVMs, it has been utilized to
study many fundamental learning algorithms (e.g., $k$-nearest neighbors [Rogers
and Wagner, 1978], stochastic gradient method [Hardt et al., 2016], linear
regression [Maurer, 2017], etc). In a recent line of great works by Feldman and
Vondrak [2018, 2019] as well as Bousquet et al. [2020b], they prove a high
probability generalization upper bound of order $\tilde{\mathcal{O}}(\gamma
+\frac{L}{\sqrt{n}})$ for any uniformly $\gamma$-stable algorithm and
$L$-bounded loss function. Although much progress was achieved in proving
generalization upper bounds for stable algorithms, our knowledge of lower
bounds is rather limited. In fact, there is no nontrivial lower bound known
ever since the study of uniform stability [Bousquet and Elisseeff, 2002], to
the best of our knowledge. In this paper we fill the gap by proving a tight
generalization lower bound of order $\Omega(\gamma+\frac{L}{\sqrt{n}})$, which
matches the best known upper bound up to logarithmic factors
