The <Group Lasso> penalty is
$$
\lambda P(\beta),
\qquad
P(\beta)=\sum_{j=1}^qm_j\|\beta_{G_j}\|_2.
$$
By the <Cauchy-Schwarz inequality> within each group,
$$
|u^Tv|
\leq\sum_j\|u_{G_j}\|_2\|v_{G_j}\|_2
\leq
\max_km_k^{-1}\|v_{G_k}\|_2
\sum_jm_j\|u_{G_j}\|_2.
$$

Put $\delta=\widehat\beta-\beta^0$. Comparing the objective at $\widehat\beta$ and $\beta^0$ gives
$$
\frac1{2n}\|X\delta\|_2^2
\leq\frac1n\epsilon^TX\delta
+\lambda\{P(\beta^0)-P(\widehat\beta)\}.
$$
On $\Omega$, the preceding duality inequality gives
$$
\frac1n|\epsilon^TX\delta|\leq\lambda P(\delta).
$$
The triangle inequality $P(\delta)\leq P(\beta^0)+P(\widehat\beta)$ then yields
$$
\frac1n\|X(\beta^0-\widehat\beta)\|_2^2
\leq4\lambda P(\beta^0).
$$

When $m_j=\sqrt r$ and $X_{G_j}^TX_{G_j}=nI_r$,
$$
\frac{\|X_{G_j}^T\epsilon\|_2^2}{n}\sim\chi_r^2.
$$
Writing $\delta_0=n\lambda^2-1\in(0,1)$, the exponential Markov inequality and the supplied chi-square moment-generating-function bound give
$$
\mathbb P(\chi_r^2>r(1+\delta_0))
\leq\exp(-r\delta_0^2/8).
$$
The prescribed equation makes this $q^{-(A+1)}$. A <union bound> over the $q$ groups therefore gives
$$
\mathbb P(\Omega)\geq1-q^{-A}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 205 / 4 / Solution

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The group Lasso partitions a coefficient vector into groups $G_1,\ldots,G_q$ and replaces the coordinatewise $\ell^1$ penalty with a weighted sum of group norms,
$$
P(\beta)=\sum_{j=1}^q m_j\lVert\beta_{G_j}\rVert_2.
$$
It tends to retain or discard an entire group at once. Its dual norm is
$$
P^*(v)=\max_j m_j^{-1}\lVert v_{G_j}\rVert_2,
$$
so the corresponding <Holder inequality> is $|u^Tv|\leq P(u)P^*(v)$.

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