SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\det \left(\mathbb {1} + \frac {\Phi_ {t} \boldsymbol {\Sigma} _ {0} \Phi_ {t} ^ {\top}}{\sigma^ {2}}\right) = \det \left(\mathbb {1} + \frac {\boldsymbol {\Sigma} _ {0} ^ {1 / 2} \Phi_ {t} ^ {\top} \Phi_ {t} \boldsymbol {\Sigma} _ {0} ^ {1 / 2}}{\sigma^ {2}}\right) = \det (\boldsymbol {\Sigma} _ {0} / \sigma^ {2}) \det \left(\Phi_ {t} ^ {\top} \Phi_ {t} + \sigma^ {2} \boldsymbol {\Sigma} _ {0} ^ {- 1}\right).
$$
Using Lemma C.4 with  $\gamma = \sigma^2$ ,  $\pmb{v} = \pmb{\mu}_t - \pmb{r}_t$  and  $M = \Phi_t\Sigma_0^{1 / 2}$ , we have
$$
\begin{array}{l} \left(\boldsymbol {\mu} _ {t} - \boldsymbol {r} _ {t}\right) ^ {\top} \left(\mathbb {1} + \frac {1}{\sigma^ {2}} \Phi_ {t} \boldsymbol {\Sigma} _ {0} \Phi_ {t} ^ {\top}\right) ^ {- 1} \left(\boldsymbol {\mu} _ {t} - \boldsymbol {r} _ {t}\right) = \left(\boldsymbol {\mu} _ {t} - \boldsymbol {r} _ {t}\right) ^ {\top} \left(\boldsymbol {\mu} _ {t} - \boldsymbol {r} _ {t}\right) \\ - \left(\boldsymbol {\mu} _ {t} - \boldsymbol {r} _ {t}\right) ^ {\top} \Phi_ {t} \Sigma_ {0} ^ {1 / 2} \left(\Sigma_ {0} ^ {1 / 2} \Phi_ {t} ^ {\top} \Phi_ {t} \Sigma_ {0} ^ {1 / 2} + \sigma^ {2} \mathbb {1}\right) ^ {- 1} \Sigma_ {0} ^ {1 / 2} \Phi_ {t} ^ {\top} \left(\boldsymbol {\mu} _ {t} - \boldsymbol {r} _ {t}\right) \\ = \left(\boldsymbol {\mu} _ {t} - \boldsymbol {r} _ {t}\right) ^ {\top} \left(\boldsymbol {\mu} _ {t} - \boldsymbol {r} _ {t}\right) - \left(\boldsymbol {\mu} _ {t} - \boldsymbol {r} _ {t}\right) ^ {\top} \Phi_ {t} \boldsymbol {\Sigma} _ {0} ^ {1 / 2} \left(\boldsymbol {\Sigma} _ {0} ^ {1 / 2} \left(\Phi_ {t} ^ {\top} \Phi_ {t} + \sigma^ {2} \boldsymbol {\Sigma} _ {0} ^ {- 1}\right) \boldsymbol {\Sigma} _ {0} ^ {1 / 2}\right) ^ {- 1} \boldsymbol {\Sigma} _ {0} ^ {1 / 2} \Phi_ {t} ^ {\top} \left(\boldsymbol {\mu} _ {t} - \boldsymbol {r} _ {t}\right) \\ = (\pmb {\mu} _ {t} - \pmb {r} _ {t}) ^ {\top} (\pmb {\mu} _ {t} - \pmb {r} _ {t}) - (\pmb {\mu} _ {t} - \pmb {r} _ {t}) ^ {\top} \Phi_ {t} (\Phi_ {t} ^ {\top} \Phi_ {t} + \sigma^ {2} \pmb {\Sigma} _ {0} ^ {- 1}) ^ {- 1} \Phi_ {t} ^ {\top} (\pmb {\mu} _ {t} - \pmb {r} _ {t}). \\ \end{array}
$$
The resulting expression for  $R_{\mathrm{MM},t}^{2}$  is rather cumbersome, but the upshot is that we now (only) need to compute the inverse and determinant of the  $d \times d$  matrix  $\Phi_t^\top \Phi_t + \sigma^2 \Sigma_0^{-1}$ . Since, we compute  $R_{\mathrm{MM},t}^{2}$  at each round  $t$ , we can update the inverse of  $\Phi_t^\top \Phi_t + \sigma^2 \Sigma_0^{-1}$  incrementally using the Sherman-Morrison formula (Sherman & Morrison, 1950). The determinant of  $\Phi_t^\top \Phi_t + \sigma^2 \Sigma_0^{-1}$  can be updated incrementally using the relation
$$
\det \big (\Phi_ {t} ^ {\top} \Phi_ {t} + \sigma^ {2} \Sigma_ {0} ^ {- 1} \big) = \det \big (\Phi_ {t - 1} ^ {\top} \Phi_ {t - 1} + \sigma^ {2} \Sigma_ {0} ^ {- 1} \big) \big (1 + \phi (a _ {t}) ^ {\top} \big (\Phi_ {t - 1} ^ {\top} \Phi_ {t - 1} + \sigma^ {2} \Sigma_ {0} ^ {- 1} \big) ^ {- 1} \phi (a _ {t}) \big),
$$
which can be found in Eq. (6) in Lemma 11 of (Abbasi-Yadkori et al., 2011). Additionally, recall that (see Eq. (24)) when  $P_{t} = \mathcal{N}\big(\mathbf{0},c\Phi_{t}\Phi_{t}^{\top}\big)$  and  $\alpha = \sigma^2 /c$  for any  $c > 0$ , the radius  $R_{\mathrm{AMM},t}$  of our analytic confidence bounds simplifies to
Note that the inverse  $(\Phi_t^\top \Phi_t + (\sigma^2 /c)\mathbb{1})^{-1}$  still appears in the expression for  $\mathrm{AUCB}_{\Theta_t}(a)$  and in the incremental determinant update rule.