I am reading this paper and Theorem 3.3, says that under some conditions,
$$ \inf_{M \in \bf{M}} \sup_{P\in \bf{P}} E|M(X)-\mu|_2^2 \gtrsim \sigma^2\left(\frac{s^* \log d}{n}+\frac{\left(s^* \log d\right)^2}{n^2 \varepsilon^2}\right) . $$
(Here $P$ is the law of each observation in a sample of $n$ points $X$ and $M$ is an estimator of $\mu$ in some class.)
Later, in Theorem 3.4, its given that under some conditions, with probability $1-f(n,d)$, it holds that
$$|M(X)-\mu|_2^2\lesssim \sigma^2\left(\frac{s^* \log d}{n}+\frac{\left(s^* \log d\right)^2 \log (1 / \delta) \log n}{n^2 \varepsilon^2}\right).$$
Then there is the statement "the convergence rate of Algorithm 3 attains the lower bound, Theorem 3.3, up to a gap." But Theorem 3.3 is about expected deviations and Theorem 3.4 is a high probability deviations bound. I am not sure how the lower bound on the expected risk implies the lower bound on the deviations. I think it has to do with the fact that the data and, in this case the estimator, are subGaussian. My question is that in general, when can lower bounds on $\ell_2$ minimax risk be used to obtain lower bounds on deviations (with high probability)?