In Lemma 9
$$(1-2\mu\eta_r)^{(K-1)/2}=(1-\frac{2\gamma_r}{R(K-1)})^{(K-1)/2}\leq 1-\gamma_r/R$$ only holds for $K\leq 3$. Otherwise, LHS $\geq$ RHS rather than $\leq$.
In Lemma 14
- For $\tau_3$, haven’t found a clear reason to support why it is $c-c_i+\nabla f_i(x)$ rather than $c-c_i-\nabla f_i(x)$. If this is a typo, then the proof for $\tau_3$ doesn’t work.
- In the line of equation below (25), there is an extra $\beta$ in the term $7\beta\eta_l^2\sigma^2$, so that in the final result, the coefficient of $\sigma^2$ is missing by a factor of $63\tilde{\eta}^2$.