SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} | \widetilde {\Psi} _ {\Gamma} ^ {\top} \phi_ {x} - \sigma_ {x} ^ {2} | = \| (\widetilde {\Psi} _ {\Gamma} - \Psi^ {*}) ^ {\top} \phi_ {x} \| _ {2} \\ = \left\| \left(\tilde {\Psi} _ {\Gamma} - \Psi^ {*}\right) ^ {\top} \Phi_ {U ^ {*}} ^ {\top} q \right\| _ {2} \\ \leq \left\| \left(\widetilde {\Psi} _ {\Gamma} - \Psi^ {*}\right) ^ {\top} \Phi_ {U ^ {*}} ^ {\top} \right\| _ {2} \| q \| _ {2} \\ = \| \Phi_ {U ^ {*}} \left(\tilde {\Psi} _ {\Gamma} - \Psi^ {*}\right) \| _ {2} \| \left(\Phi_ {U ^ {*}} ^ {- 1}\right) ^ {\top} \phi_ {x} \| _ {2} \\ \leq \left\| \Phi_ {U ^ {*}} \left(\widetilde {\Psi} _ {\Gamma} - \Psi^ {*}\right) \right\| _ {2} \left\| \left(\Phi_ {U ^ {*}} ^ {- 1}\right) ^ {\top} \right\| _ {2} \left\| \phi_ {x} \right\| _ {2} \\ \stackrel {(a)} {=} \frac {2}{\zeta_ {\operatorname* {m i n}} \left(\Phi_ {U ^ {*}}\right)} \| \Phi_ {U ^ {*}} \big (\widetilde {\Psi} _ {\Gamma} - \Psi^ {*} \big) \| _ {2} \| \phi_ {x} \| _ {2}, \\ \end{array}
$$
where  $(a)$  follows from the definition of the matrix norm. We now leverage the nature of the duplication matrix,  $G$ , and the fact that  $\|x\|_2 \leq B$  to bound  $\|\phi_x\|_2^2$ .
$$
\left\| \phi_ {x} \right\| _ {2} ^ {2} = \operatorname {v e c} \left(x x ^ {\top}\right) ^ {\top} G G ^ {\top} \operatorname {v e c} \left(x x ^ {\top}\right)
$$
$$
\begin{array}{l} = \operatorname {v e c} \left(x x ^ {\top}\right) ^ {\top} G \left(G ^ {\top} G\right) \left(G ^ {\top} G\right) ^ {- 1} G ^ {\top} \operatorname {v e c} \left(x x ^ {\top}\right) \\ = \operatorname {v e c} (x x ^ {\top}) ^ {\top} G (G ^ {\top} G) ^ {- 1 / 2} (G ^ {\top} G) (G ^ {\top} G) ^ {- 1 / 2} G ^ {\top} \operatorname {v e c} (x x ^ {\top}) \\ \end{array}
$$
$$
\begin{array}{l} \stackrel {(a)} {\leq} 2 \operatorname {v e c} (x x ^ {\top}) ^ {\top} G (G ^ {\top} G) ^ {- 1} G ^ {\top} \operatorname {v e c} (x x ^ {\top}) \\ \stackrel {(b)} {=} 2 \operatorname {v e c} (x x ^ {\top}) ^ {\top} \operatorname {v e c} (x x ^ {\top}) \\ = 2 x ^ {\top} x x ^ {\top} x \\ \leq 2 B ^ {4}, \\ \end{array}
$$
where  $(a)$  follows from the fact that  $G^{\top}G\preceq 2I$  as shown in Equation (61) of [40], and  $(b)$  uses the fact that  $G(G^{\top}G)^{-1}G^{\top}\mathrm{vec}(xx^{\top}) = \mathrm{vec}(xx^{\top})$  as shown in Equations (54) and (52) of [40]. Consequently, with probability greater than  $1 - \delta /2$ , we have that for any  $x\in \mathcal{X}$
$$
\begin{array}{l} | \widetilde {\sigma} _ {x} ^ {2} - \sigma_ {x} ^ {2} | \leq \frac {5 B ^ {2} \sqrt {2}}{\zeta_ {\mathrm {m i n}} (\Phi_ {U ^ {*}})} \max \left\{\frac {4 \sqrt {2} \sigma_ {\mathrm {m a x}} ^ {2} M \sqrt {M} \log (8 M / \delta)}{\Gamma}, \sqrt {\frac {3 2 \sigma_ {\mathrm {m a x}} ^ {4} M ^ {2} \log (8 M / \delta)}{\Gamma}} \right\} \\ \leq \frac {5 B ^ {2} \sqrt {2}}{\zeta_ {\mathrm {m i n}} (\Phi_ {U ^ {*}})} \max \left\{\frac {4 \sqrt {2} \sigma_ {\mathrm {m a x}} ^ {2} d ^ {3} \log (8 M / \delta)}{\Gamma}, \sqrt {\frac {3 2 \sigma_ {\mathrm {m a x}} ^ {4} d ^ {4} \log (8 M / \delta)}{\Gamma}} \right\}. \\ \end{array}
$$
D.2 Analysis of the HEAD Estimator
We now present the maximum absolute error guarantees of the HEAD procedure, Alg. 1.
Proof of Theorem 3.1. In this proof, we bound the estimation error  $|\hat{\sigma}_x^2 - \sigma_x^2|, \forall x \in \mathcal{X}$ . Define  $\sigma_{\max}^2 = \max_{x \in \mathcal{X}} \sigma_x^2$ ,  $\Psi^* = \mathrm{vech}(\Sigma^*)$ ,  $N_1 = \{1, \dots, \Gamma/2\}$ ,  $N_2 = \{\Gamma/2 + 1, \dots, \Gamma\}$  and
$$
\widehat {\Psi} _ {\Gamma} = \arg \min  _ {\boldsymbol {s} \in \mathbb {R} ^ {M}} \sum_ {t \in N _ {2}} \left\{\boldsymbol {s} ^ {\top} \phi_ {x _ {t}} - (y _ {t} - x _ {t} ^ {\top} \widehat {\theta} _ {\Gamma / 2}) ^ {2} \right\} ^ {2},
$$
where  $\widehat{\Psi}_{\Gamma} = \mathrm{vech}(\widehat{\Sigma}_{\Gamma})$ . Furthermore, define vector  $e$  with entries  $e_t = (y_t - x_t^\top \widehat{\theta}_{\Gamma/2})^2$  and  $\Phi$  such that the rows of  $\Phi$  are composed of  $\{\phi_{x_t}\}_{t \in N_2}$ . We find that
$$
\widehat {\Psi} _ {\Gamma} = \left(\Phi^ {\top} \Phi\right) ^ {- 1} \Phi^ {\top} e.
$$
For any  $\phi_x\in \mathcal{W}$ , we are interested in bounding  $\left|\phi_x^\top (\widehat{\Psi}_\Gamma -\Psi^*)\right|$
$$
\begin{array}{l} \left| \phi_ {x} ^ {\top} \left(\widehat {\Psi} _ {\Gamma} - \Psi^ {*}\right) \right| = \left| \phi_ {x} ^ {\top} \left\{\left(\Phi^ {\top} \Phi\right) ^ {- 1} \Phi^ {\top} e - \Psi^ {*} \right\} \right| \\ = | \underbrace {\phi_ {x} ^ {\top} (\Phi^ {\top} \Phi) ^ {- 1} \Phi^ {\top}} _ {z ^ {\top}} (e - \Phi \Psi^ {*}) |. \\ \end{array}
$$
Define  $z^{\top} = \phi_{x}^{\top}(\Phi^{\top}\Phi)^{-1}\Phi^{\top}$  and  $X$  such that the rows of  $X$  are composed of  $\{x_{t}\}_{t\in N_{2}}$ . We can bound this expression with the triangle inequality as
$$
\begin{array}{l} \left| \phi_ {x} ^ {\top} \left(\widehat {\Psi} _ {\Gamma} - \Psi^ {*}\right) \right| = \left| \sum_ {t \in N _ {2}} z _ {t} \left\{x _ {t} ^ {\top} \left(\widehat {\theta} _ {\Gamma / 2} - \theta^ {*}\right) \right\} ^ {2} + \sum_ {t \in N _ {2}} z _ {t} \left(\eta_ {t} ^ {2} - \phi_ {t} ^ {\top} \Psi^ {*}\right) + \sum_ {t \in N _ {2}} 2 z _ {t} x _ {t} ^ {\top} \left(\widehat {\theta} _ {\Gamma / 2} - \theta^ {*}\right) \eta_ {t} \right| \\ \leq \left| \sum_ {t \in N _ {2}} z _ {t} \left\{x _ {t} ^ {\top} (\widehat {\theta} _ {\Gamma / 2} - \theta^ {*}) \right\} ^ {2} \right| + \left| \sum_ {t \in N _ {2}} z _ {t} \left(\eta_ {t} ^ {2} - \phi_ {t} ^ {\top} \Psi^ {*}\right) \right| + \left| \sum_ {t \in N _ {2}} 2 z _ {t} x _ {t} ^ {\top} (\widehat {\theta} _ {\Gamma / 2} - \theta^ {*}) \eta_ {t} \right|. \\ \end{array}
$$
We can simplify the first term as,
$$
\begin{array}{l} \left| \sum_ {t \in N _ {2}} z _ {t} \left\{x _ {t} ^ {\top} \left(\widehat {\theta} _ {\Gamma / 2} - \theta^ {*}\right) \right\} ^ {2} \right| = \left| \sum_ {t \in N _ {2}} z _ {t} \phi_ {x _ {t}} ^ {\top} \operatorname {v e c h} \left\{\left(\widehat {\theta} _ {\Gamma / 2} - \theta^ {*}\right) \left(\widehat {\theta} _ {\Gamma / 2} - \theta^ {*}\right) ^ {\top} \right\} \right| \\ = \left| \phi_ {x} ^ {\top} (\Phi^ {\top} \Phi) ^ {- 1} \Phi^ {\top} \Phi \times \operatorname {v e c h} \left\{(\widehat {\theta} _ {\Gamma / 2} - \theta^ {*}) (\widehat {\theta} _ {\Gamma / 2} - \theta^ {*}) ^ {\top} \right\} \right| \\ = \left| x ^ {\top} \left(\widehat {\theta} _ {\Gamma / 2} - \theta^ {*}\right) \left(\widehat {\theta} _ {\Gamma / 2} - \theta^ {*}\right) ^ {\top} x \right| \\ = \left\{x ^ {\top} (\widehat {\theta} _ {\Gamma / 2} - \theta^ {*}) \right\} ^ {2}. \\ \end{array}
$$
Additionally, the third term is equivalent to  $2|(\eta \circ z)^{\top}X(\widehat{\theta}_{\Gamma /2} - \theta^{*})|$ , where “ $\circ$ ” is element-wise multiplication. Consequently, the absolute error is equal to
$$
| \phi_ {x} ^ {\top} (\widehat {\Psi} _ {\Gamma} - \Psi^ {*}) | = \underbrace {\left\{x ^ {\top} (\widehat {\theta} _ {\Gamma / 2} - \theta^ {*}) \right\} ^ {2}} _ {\mathbf {A}} + \underbrace {\left| \sum_ {t \in N _ {2}} z _ {t} (\eta_ {t} ^ {2} - \phi_ {t} ^ {\top} \Psi^ {*}) \right|} _ {\mathbf {B}} + \underbrace {2 \left| (\eta \circ z) ^ {\top} X (\widehat {\theta} _ {\Gamma / 2} - \theta^ {*}) \right|} _ {\mathbf {C}}.
$$
Note that to bound  $|\phi_x^\top (\widehat{\Psi}_\Gamma - \Psi^*)|$  with probability  $1 - \delta / 2$ , it is necessary to bound quantities  $\mathbf{A}$ ,  $\mathbf{B}$  and  $\mathbf{C}$  with probability  $1 - \delta / 6$  in anticipation of a union bound.
Quantity A. We sample  $\{x_{t}, t \in N_{1}\}$  according to the rounded G-optimal design over  $\mathcal{X}$  constructed in Stage 1 of Alg. 1. We use the concentration bounds of Proposition B.1, to state that with probability  $1 - \delta / 6$ ,
$$
\mathbf {A} \leq 2 \underbrace {x ^ {\top} \left(\sum_ {t \in N _ {1}} x _ {t} x _ {t} ^ {\top}\right) ^ {- 1}} _ {\mathbf {G}} \sigma_ {\max } ^ {2} \log (1 2 | \mathcal {X} | / \delta).
$$
We use Proposition B.5 in combination with the rounding procedure guarantees of Proposition A.1 to bound quantity  $\mathbf{G}$ . For  $\epsilon \in (0, 1/3]$ , with probability  $1 - \delta/6$ ,
$$
\mathbf {A} \leq \frac {4 d (1 + 6 \epsilon) \sigma_ {\max } ^ {2} \log (1 2 | \mathcal {X} | / \delta)}{\Gamma}.
$$
Quantity B. According to Proposition B.3,  $\eta_t^2$  is sub-exponential with parameters  $(4\sqrt{2}\sigma_{\max}^2, 4\sigma_{\max}^2)$ . Consequently, by Proposition B.4, we know that  $\mathbf{B}$  is sub-exponential with parameters
$$
\left(\sqrt {\sum_ {t \in N _ {2}} z _ {t} ^ {2} 3 2 \sigma_ {\mathrm {m a x}} ^ {4}}, \max _ {t \in N _ {2}} | z _ {t} | 4 \sigma_ {\mathrm {m a x}} ^ {2}\right).
$$
We now invoke the G-optimal design over  $\mathcal{W}$  space conducted in Stage 2 of Alg. 1 and Propositions B.5 and A.1 to find that for  $\epsilon \in (0,1/3]$ ,
$$
\begin{array}{l} \sum_ {t \in N _ {2}} z _ {t} ^ {2} = \sum_ {t \in N _ {2}} \phi_ {x} ^ {\top} \left(\Phi^ {\top} \Phi\right) ^ {- 1} \phi_ {t} \phi_ {t} ^ {\top} \left(\Phi^ {\top} \Phi\right) ^ {- 1} \phi_ {x} \\ = \phi_ {x} ^ {\top} \left(\Phi^ {\top} \Phi\right) ^ {- 1} \Phi^ {\top} \Phi \left(\Phi^ {\top} \Phi\right) ^ {- 1} \phi_ {x} \\ \leq \phi_ {x} ^ {\top} (\Phi^ {\top} \Phi) ^ {- 1} \phi_ {x} \leq \frac {2 d ^ {2} (1 + 6 \epsilon)}{\Gamma}. \\ \end{array}
$$
$$
\sum_ {t \in N _ {2}} z _ {t} ^ {2} \leq \frac {2 d ^ {2} (1 + 6 \epsilon)}{\Gamma}, \tag {7}
$$
$$
\begin{array}{l} \left| z _ {t} \right| = \left| \phi_ {x} ^ {\top} \left(\Phi^ {\top} \Phi\right) ^ {- 1} \phi_ {t} \right| \\ \leq \sqrt {\phi_ {x} ^ {\top} (\Phi^ {\top} \Phi) ^ {- 1} \phi_ {x}} \sqrt {\phi_ {t} ^ {\top} (\Phi^ {\top} \Phi) ^ {- 1} \phi_ {t}} \\ \leq \frac {2 d ^ {2} (1 + 6 \epsilon)}{\Gamma}. \\ \end{array}
$$
Therefore,  $\mathbf{B}$  has sub-exponential parameters  $(\sqrt{64\sigma_{\mathrm{max}}^4d^2(1 + 6\epsilon) / \Gamma},8\sigma_{\mathrm{max}}^2 d^2 (1 + 6\epsilon) / \Gamma)$ . Using Proposition B.2, with probability  $1 - \delta /6$
$$
\begin{array}{l} \mathbf {B} \leq \max  \left\{\sqrt {\frac {1 2 8 \log (1 2 / \delta) \sigma_ {\operatorname* {m a x}} ^ {4} d ^ {2} (1 + 6 \epsilon)}{\Gamma}}, \frac {1 6 \log (1 2 / \delta) \sigma_ {\operatorname* {m a x}} ^ {2} d ^ {2} (1 + 6 \epsilon)}{\Gamma} \right\} \\ = \max \left\{\sqrt {8} \sigma_ {\mathrm {m a x}} \sqrt {\frac {1 6 \log (1 2 / \delta) \sigma_ {\mathrm {m a x}} ^ {2} d ^ {2} (1 + 6 \epsilon)}{\Gamma}}, \frac {1 6 \log (1 2 / \delta) \sigma_ {\mathrm {m a x}} ^ {2} d ^ {2} (1 + 6 \epsilon)}{\Gamma} \right\}. \\ \end{array}
$$
Quantity C. Construct  $V \in \mathbb{R}^{\Gamma/2 \times d}$  such that the rows of  $V$  correspond to  $\{x_t : t \in N_1\}$ . We can upper bound quantity C using the Cauchy-Schwartz inequality in the following manner,
$$
\begin{array}{l} | (\eta \circ z) ^ {\top} X (\widehat {\theta} _ {\Gamma / 2} - \theta^ {*}) | = | (\eta \circ z) ^ {\top} X (V ^ {\top} V) ^ {- 1 / 2} (V ^ {\top} V) ^ {1 / 2} (\widehat {\theta} _ {\Gamma / 2} - \theta^ {*}) | \\ \leq \| (\eta \circ z) ^ {\top} X (V ^ {\top} V) ^ {- 1 / 2} \| _ {2} \| (V ^ {\top} V) ^ {1 / 2} (\widehat {\theta} _ {\Gamma / 2} - \theta^ {*}) \| _ {2}. \\ \end{array}
$$
By Proposition B.8, with probability  $1 - \delta / 12$ ,
$$
\left\| \left(V ^ {\top} V\right) ^ {1 / 2} \left(\widehat {\theta} _ {\Gamma / 2} - \theta^ {*}\right) \right\| _ {2} \leq 2 \sqrt {2 \sigma_ {\max } ^ {2} \left\{d \log (6) + \log (1 2 / \delta) \right\}}.
$$
Now construct  $D_z = \mathrm{diag}(\{z_t : t \in N_2\})$  and express,
$$
\left\| (\eta \circ z) ^ {\top} X (V ^ {\top} V) ^ {- 1 / 2} \right\| _ {2} = \sqrt {\eta^ {\top} D _ {z} X (V ^ {\top} V) ^ {- 1} X ^ {\top} D _ {z} \eta}.
$$
We first establish that by the G-optimal designs in both  $\mathcal{X}$  and  $\mathcal{W}$  space along with Propositions B.5 and A.1,
$$
\begin{array}{l} \operatorname {t r} \left\{D _ {z} X (V ^ {\top} V) ^ {- 1} X ^ {\top} D _ {z} \right\} = \operatorname {t r} \left\{D _ {z} ^ {2} X (V ^ {\top} V) ^ {- 1} X ^ {\top} \right\} \\ = \sum_ {t \in N _ {2}} z _ {t} ^ {2} x _ {t} ^ {\top} (V ^ {\top} V) ^ {- 1} x _ {t} \\ \leq \frac {2 d (1 + 6 \epsilon)}{\Gamma} \sum_ {t \in N _ {2}} z _ {t} ^ {2} \\ \stackrel {(a)} {\leq} \frac {4 d ^ {3} (1 + 6 \epsilon) ^ {2}}{\Gamma^ {2}}, \\ \end{array}
$$
where  $\epsilon \in (0,1 / 3]$  and  $(a)$  follows from Eq. 7. Consequently, we find that
$$
\operatorname {t r} \left\{D _ {z} X \left(V ^ {\top} V\right) ^ {- 1} X ^ {\top} D _ {z} \right\} \leq \frac {4 d ^ {3} (1 + 6 \epsilon) ^ {2}}{\Gamma^ {2}}. \tag {8}
$$
This inequality will be useful in the following derivation. Labeling  $\Delta = D_zX(V^\top V)^{-1}X^\top D_z$ , we use Proposition B.7 to bound  $\eta^\top \Delta \eta$  with probability  $1 - \delta / 12$  and explain below.
$$
\begin{array}{l} \eta^ {\top} \Delta \eta <   \sigma_ {\max } ^ {2} \left\{\operatorname {t r} (\Delta) + 2 \| \Delta \| _ {F} \log (1 2 / \delta) ^ {1 / 2} + 2 \| \Delta \| _ {2} \log (1 2 / \delta) \right\} (9) \\ \leq \sigma_ {\max } ^ {2} \left\{\frac {4 d ^ {3} (1 + 6 \epsilon) ^ {2}}{\Gamma^ {2}} + 2 \| \Delta^ {1 / 2} \| _ {F} ^ {2} \log (1 2 / \delta) ^ {1 / 2} + 2 \| \Delta \| _ {2} \log (1 2 / \delta) \right\} (10) \\ \leq \sigma_ {\max } ^ {2} \left\{\frac {4 d ^ {3} (1 + 6 \epsilon) ^ {2}}{\Gamma^ {2}} + 2 \| \Delta^ {1 / 2} \| _ {F} ^ {2} \log (1 2 / \delta) ^ {1 / 2} + 2 \| \Delta \| _ {F} \log (1 2 / \delta) \right\} (11) \\ \leq \sigma_ {\max } ^ {2} \left\{\frac {4 d ^ {3} (1 + 6 \epsilon) ^ {2}}{\Gamma^ {2}} + 2 \| \Delta^ {1 / 2} \| _ {F} ^ {2} \log (1 2 / \delta) ^ {1 / 2} + 2 \| \Delta^ {1 / 2} \| _ {F} ^ {2} \log (1 2 / \delta) \right\} (12) \\ \leq \sigma_ {\max } ^ {2} \left\{\frac {4 d ^ {3} (1 + 6 \epsilon) ^ {2}}{\Gamma^ {2}} + 2 \operatorname {t r} (\Delta) \log (1 2 / \delta) ^ {1 / 2} + 2 \operatorname {t r} (\Delta) \log (1 2 / \delta) \right\} (13) \\ \leq \sigma_ {\max } ^ {2} \left\{\frac {4 d ^ {3} (1 + 6 \epsilon) ^ {2}}{\Gamma^ {2}} + 2 \frac {4 d ^ {3} (1 + 6 \epsilon) ^ {2}}{\Gamma^ {2}} \log (1 2 / \delta) ^ {1 / 2} + 2 \frac {4 d ^ {3} (1 + 6 \epsilon) ^ {2}}{\Gamma^ {2}} \log (1 2 / \delta) \right\} (14) \\ \leq 5 \sigma_ {\max } ^ {2} \frac {4 d ^ {3} (1 + 6 \epsilon) ^ {2}}{\Gamma^ {2}} \log (1 2 / \delta), (15) \\ \end{array}
$$
where line (9) follows from Eq. 8, Proposition B.6 and Proposition B.7. In lines (10) and (11), we use Proposition B.6 again and then the definition of the Frobenius norm in line (12). Consequently, with probability  $1 - \delta / 6$ ,
$$
\begin{array}{l} \mathbf {C} = \left| (\eta \circ z) ^ {\top} X \left(\widehat {\theta} _ {\Gamma / 2} - \theta^ {*}\right) \right| \\ = | (\eta \circ z) ^ {\top} X (V ^ {\top} V) ^ {- 1 / 2} (V ^ {\top} V) ^ {1 / 2} (\widehat {\theta} _ {\Gamma / 2} - \theta^ {*}) | \\ \leq \| (\eta \circ z) ^ {\top} X (V ^ {\top} V) ^ {- 1 / 2} \| _ {2} \| (V ^ {\top} V) ^ {1 / 2} (\widehat {\theta} _ {\Gamma / 2} - \theta^ {*}) \| _ {2} \\ \leq 2 \sqrt {1 0 \sigma_ {\max} ^ {4} \frac {4 d ^ {3} (1 + 6 \epsilon) ^ {2}}{\Gamma^ {2}} \log (1 2 / \delta) \left\{d \log (6) + \log (1 2 / \delta) \right\}}. \\ \end{array}
$$
Combining quantities  $\mathbf{A}$ ,  $\mathbf{B}$  and  $\mathbf{C}$  together and applying a union bound, we have with probability  $1 - \delta / 2$ ,
$$
\begin{array}{l} | \phi_ {x} ^ {\top} (\widehat {\Psi} _ {\Gamma} - \Psi^ {*}) | \leq \max  \left\{\sqrt {8} \sigma_ {\max } \sqrt {\frac {1 6 \log (1 2 / \delta) \sigma_ {\max } ^ {2} d ^ {2} (1 + 6 \epsilon)}{\Gamma}}, \frac {1 6 \log (1 2 / \delta) \sigma_ {\max } ^ {2} d ^ {2} (1 + 6 \epsilon)}{\Gamma} \right\} \\ + \frac {4 d \sigma_ {\max} ^ {2} \log (1 2 | \mathcal {X} | / \delta) (1 + 6 \epsilon)}{\Gamma} + \\ + \quad 2 \sqrt {1 0 \sigma_ {\max } ^ {4} \frac {4 d ^ {3} (1 + 6 \epsilon) ^ {2}}{\Gamma^ {2}} \log (1 2 / \delta) \left\{d \log (6) + \log (1 2 / \delta) \right\}} \\ \leq \underbrace {(3 / 1 0) \max  \left\{\sqrt {8} \sigma_ {\max } \sqrt {\frac {1 6 \log (1 2 / \delta) \sigma_ {\max } ^ {2} d ^ {2} (1 + 6 \epsilon)}{\Gamma}} , \frac {1 6 \log (1 2 / \delta) \sigma_ {\max } ^ {2} d ^ {2} (1 + 6 \epsilon)}{\Gamma} \right\}} _ {\mathrm {W}} \\ + \underbrace {\frac {4 d \sigma_ {\max} ^ {2} \log (1 2 | \mathcal {X} | / \delta) (1 + 6 \epsilon)}{\Gamma}} _ {\mathrm {V}}. \\ \end{array}
$$
We can exploit the dependencies of  $\mathbf{V}$  and  $\mathbf{W}$  to simplify this bound in two ways under different conditions. Note that we only have dependency on  $|\mathcal{X}|$  through term  $\mathbf{V}$ . If  $d - 1 > \log (|\mathcal{X}|) / \log (1 / \delta)$  and  $\Gamma > 16\log (12 / \delta)\sigma_{\mathrm{max}}^2 d^2 (1 + 6\epsilon)$ , then with probability  $1 - \delta / 2$
$$
| \phi_ {x} ^ {\top} (\widehat {\Psi} _ {\Gamma} - \Psi^ {*}) | \leq \sqrt {\frac {2 5 6 \log (1 2 / \delta) \sigma_ {\max} ^ {4} d ^ {2} (1 + 6 \epsilon)}{\Gamma}}.
$$
Alternatively, if just  $\Gamma > 16\log (12|\mathcal{X}| / \delta)\sigma_{\max}^2 d^2 (1 + 6\epsilon)$ , then
$$
\left| \phi_ {x} ^ {\top} \left(\widehat {\Psi} _ {\Gamma} - \Psi^ {*}\right) \right| \leq \sqrt {\frac {2 5 6 \log (1 2 | \mathcal {X} | / \delta) \sigma_ {\max } ^ {4} d ^ {2} (1 + 6 \epsilon)}{\Gamma}}. \tag {16}
$$
Since  $\sigma_{\mathrm{max}}^2\geq \sigma_x^2\geq \sigma_{\mathrm{min}}^2$  for all  $x\in \mathcal{X}$
$$
\left| \phi_ {x} ^ {\top} \left(\widehat {\Psi} _ {\Gamma} - \Psi^ {*}\right) \right| = \left| x ^ {\top} \widehat {\Sigma} _ {\Gamma} x - x ^ {\top} \Sigma^ {*} x \right| \geq \left| \min  \left\{\max  \left\{x ^ {\top} \widehat {\Sigma} _ {\Gamma} x, \sigma_ {\min } ^ {2} \right\}, \sigma_ {\max } ^ {2} \right\} - \sigma_ {x} ^ {2} \right|.
$$
In summary, if  $\Gamma > 16\log (12|\mathcal{X}| / \delta)\sigma_{\mathrm{max}}^2 d^2 (1 + 6\epsilon)$
$$
\mathbb {P} \left(| \hat {\sigma} _ {x} ^ {2} - \sigma_ {x} ^ {2} | \leq \sqrt {\frac {C ^ {\prime} \log (| \mathcal {X} | / \delta) \sigma_ {\max } ^ {4} d ^ {2}}{\Gamma}}\right) \geq 1 - \delta ,
$$
where  $C' = 256(1 + 6\epsilon)$ .
We now prove a simple lemma relating the additive bound on  $|\hat{\sigma}_x^2 - \sigma_x^2|$  to a multiplicative bound  $1 - C_{\Gamma, \delta} < \sigma_x^2 / \hat{\sigma}_x^2 < 1 + C_{\Gamma, \delta}$ .
Lemma D.1. Define  $\widehat{\sigma}_x^2 = \min \left\{\max \left\{x^\top \widehat{\Sigma}_\Gamma x,\sigma_{\min}^2\right\} ,\sigma_{\max}^2\right\}$ . Letting  $\sigma_{\mathrm{min}}^2 = \min_{x\in \mathcal{X}}\sigma_x^2$ $\kappa = \sigma_{\mathrm{max}}^{2} / \sigma_{\mathrm{min}}^{2}$ , and
$$
\mathbb {P} \left(\forall x \in \mathcal {X}, 1 - C _ {\Gamma , \delta} <   \frac {\sigma_ {x} ^ {2}}{\widehat {\sigma} _ {x} ^ {2}} <   1 + C _ {\Gamma , \delta}\right) \geq 1 - \delta / 2, \text {w h e r e} C _ {\Gamma , \delta} = \sqrt {\frac {C ^ {\prime} \log (| \mathcal {X} | / \delta) \kappa^ {2} d ^ {2}}{\Gamma}}.
$$
Proof. Using Lemma D.1, we know that with probability greater than  $1 - \delta / 2$  the following event holds.
$$
| \sigma_ {x} ^ {2} - \widehat {\sigma} _ {x} ^ {2} | \leq \sqrt {\frac {C ^ {\prime} \log (| \mathcal {X} | / \delta) \sigma_ {\max} ^ {4} d ^ {2}}{\Gamma}} \Rightarrow
$$
$$
\hat {\sigma} _ {x} ^ {2} - \sqrt {\frac {C ^ {\prime} \log (| \mathcal {X} | / \delta) \sigma_ {\max} ^ {4} d ^ {2}}{\Gamma}} \leq \sigma_ {x} ^ {2} \leq \hat {\sigma} _ {x} ^ {2} + \sqrt {\frac {C ^ {\prime} \log (| \mathcal {X} | / \delta) \sigma_ {\max} ^ {4} d ^ {2}}{\Gamma}} \Rightarrow
$$
$$
\widehat {\sigma} _ {x} ^ {2} - \widehat {\sigma} _ {x} ^ {2} \sqrt {\frac {C ^ {\prime} \log (| \mathcal {X} | / \delta) \sigma_ {\max} ^ {4} d ^ {2}}{\widehat {\sigma} _ {x} ^ {4} \Gamma}} \leq \sigma_ {x} ^ {2} \leq \widehat {\sigma} _ {x} ^ {2} + \widehat {\sigma} _ {x} ^ {2} \sqrt {\frac {C ^ {\prime} \log (| \mathcal {X} | / \delta) \sigma_ {\max} ^ {4} d ^ {2}}{\widehat {\sigma} _ {x} ^ {4} \Gamma}} \Rightarrow
$$
$$
\widehat {\sigma} _ {x} ^ {2} - \widehat {\sigma} _ {x} ^ {2} \sqrt {\frac {C ^ {\prime} \log (| \mathcal {X} | / \delta) \sigma_ {\max} ^ {4} d ^ {2}}{\sigma_ {\min} ^ {4} \Gamma}} \leq \sigma_ {x} ^ {2} \leq \widehat {\sigma} _ {x} ^ {2} + \widehat {\sigma} _ {x} ^ {2} \sqrt {\frac {C ^ {\prime} \log (| \mathcal {X} | / \delta) \sigma_ {\max} ^ {4} d ^ {2}}{\sigma_ {\min} ^ {4} \Gamma}} \Rightarrow
$$
$$
1 - \sqrt {\frac {C ^ {\prime} \log (| \mathcal {X} | / \delta) \kappa^ {2} d ^ {2}}{\Gamma}} \leq \frac {\sigma_ {x} ^ {2}}{\hat {\sigma} _ {x} ^ {2}} \leq 1 + \sqrt {\frac {C ^ {\prime} \log (| \mathcal {X} | / \delta) \kappa^ {2} d ^ {2}}{\Gamma}}.
$$
E Proofs of Best-Arm Identification
We divide the proof of Theorem 4.1 and 4.2 between Appendices E and F for ease of exposition. In Appendix E, we consider the transductive linear best-arm identification problem (BAI) in which we are interested in identifying  $z^{*} = \max_{z\in \mathcal{Z}}z^{\top}\theta^{*}$  with probability  $1 - \delta$  for  $\delta \in (0,1)$ . In Appendix F, we consider the level set (LS) counterpart. Recall that for a set  $\mathcal{V}$ , let  $P_{\mathcal{V}}\coloneqq \{\lambda \in \mathbb{R}^{|\mathcal{V}|}:\sum_{v\in \mathcal{V}}\lambda_v = 1,\lambda_v\geq 0\forall v\}$  denote distributions over  $\mathcal{V}$  and  $\mathcal{V}(\mathcal{V})\coloneqq \{v' - v:v,v'\in \mathcal{V},v\neq v'\}$  be an operator giving differences. We begin with the lower bound.
E.1 Lower Bound
Proof of Theorem 4.1 for transductive best-arm identification. This proof is similar to the lower bound proofs of [35] and [12]. Let  $\mathcal{C} := \{\widetilde{\theta} \in \mathbb{R}^d : \exists i \text{ s.t. } \widetilde{\theta}^\top (z^* - z_i) \leq 0\}$ , i.e.,  $\widetilde{\theta} \in \mathcal{C}$  if and only if  $z^*$  is not the best arm in the linear bandit instance  $(\mathcal{X}, \mathcal{Z}, \widetilde{\theta})$ . Additionally, label  $K = |\mathcal{X}|$ .
We first invoke Lemma 1 of [27]. Under a  $\delta$ -PAC strategy for finding the best arm for the bandit instance  $(\mathcal{X},\mathcal{Z},\theta^{*})$ , let  $T_{i}$  denote the random variable that is the number of times arm  $i$  is pulled. Defining  $\sigma_x^2 = x^\top \Sigma^* x$  for all  $x\in \mathcal{X}$ , we let  $\nu_{\theta ,i}$  denote the reward distribution of the  $i$ -th arm of  $\mathcal{X}$ , i.e.,  $\nu_{\theta ,i} = \mathcal{N}(x_i^\top \theta ,\sigma_{x_i}^2)$ . For any  $\widetilde{\theta}\in \mathcal{C}$  we know from [27] that
$$
\sum_ {i = 1} ^ {K} \mathbb {E} (T _ {i}) K L (\nu_ {\theta^ {*}, i} | | \nu_ {\widetilde {\theta}, i}) \geq \log (1 / 2. 4 \delta).
$$
Following the steps of [12], we find that
$$
\sum_ {i = 1} ^ {K} \mathbb {E} \left(T _ {i}\right) \geq \log (1 / 2. 4 \delta) \min  _ {\lambda \in P _ {\mathcal {X}}} \max  _ {\tilde {\theta} \in \mathcal {C}} \frac {1}{\sum_ {i = 1} ^ {K} \lambda_ {i} K L \left(\nu_ {\theta^ {*} , i} \mid \mid \nu_ {\tilde {\theta} , i}\right)}. \tag {17}
$$
Label  $\mathcal{Z} = \{z_1, z_2, \ldots, z_n\}$ , where  $z_1 = z^*$  without loss of generality. For  $j \neq 1$ ,  $\lambda \in P_{\mathcal{X}}$  and  $\epsilon > 0$ , define  $A(\lambda) := \sum_{i=1}^{K} \lambda_i \frac{x_i x_i^\top}{\sigma_{x_i}^2}$ ,  $w_j = z^* - z_j$  and
$$
\theta_ {j} (\epsilon , \lambda) = \theta^ {*} - \frac {(w _ {j} ^ {\top} \theta^ {*} + \epsilon) A (\lambda) ^ {- 1} w _ {j}}{w _ {j} ^ {\top} A (\lambda) ^ {- 1} w _ {j}}.
$$
Note that  $w_{j}^{\top}\theta_{j}(\epsilon ,\lambda) = -\epsilon < 0$ , implying that  $\theta_{j}\in \mathcal{C}$ . We find that the KL divergence between  $\nu_{\theta ,i}$  and  $\nu_{\theta_j(\epsilon ,\lambda),i}$  is given by:
$$
\begin{array}{l} K L (\nu_ {\theta , i} | | \nu_ {\theta_ {j} (\epsilon , \lambda), i}) = \frac {\left[ x _ {i} ^ {\top} \{\theta^ {*} - \theta_ {j} (\epsilon , \lambda) \} \right] ^ {2}}{2 \sigma_ {x _ {i}} ^ {2}} \\ = w _ {j} ^ {\top} A (\lambda) ^ {- 1} \frac {(w _ {j} ^ {\top} \theta^ {*} + \epsilon) ^ {2} \frac {x _ {i} x _ {i} ^ {\top}}{2 \sigma_ {x _ {i}} ^ {2}}}{\left\{w _ {j} ^ {\top} A (\lambda) ^ {- 1} w _ {j} \right\} ^ {2}} A (\lambda) ^ {- 1} w _ {j}. \\ \end{array}
$$
Returning to Eq. 17,