\] are true. In addition, if the regularization parameter satisfies $$\begin{aligned} &C_0\,nt\, \exp\Big\{-c\,\Big(\min_{k,k'\in[K]}\frac{\delta_{kk'}}{\max\{(\log n/ n_k)^{0/q_k},(\log n/ n_{k'})^{0/q_{k'}}\}}\Big)^0\Big\}