[@li2017provably] provided in the supplementary material.

With probability $1-\delta$ it holds that $$\|\theta^*-\theta_0\|_2 \leq \frac{2}{\kappa}\sqrt{\frac{d+\log(1/\delta)}{\lambda_{\min}(V)}}
    \;\;\;\;\text{where}\;\;\kappa = \frac{1}{2e(1+\rho)}\;\;\text{and}\;\; V = \sum_{t=1}^{T_0}v_{t,i_t}v_{t,i_t}^\top.$$ \[lem:pilot\]

The following corollary immediately follows Lemma \[lem:pilot\], by lower bounding $\lambda_{\min}(V)$ using standard matrix concentration inequalities. Its proof is again deferred to the supplementary material.

There exists a universal constant $C_0>0$ such that for arbitrary $\tau\in(0,1/2]$, if $T_0 \geq C_0\max\{\nu^2d\log T/\lambda_0^2, \rho^2 (d+\log T)/(\tau^2\lambda_0)\}$ then with probability $1-O(T^{-1})$, $\|\theta^*-\theta_0\|_2\leq \tau$. \[cor:tau\]

The purpose of Corollary \[cor:tau\] is to establish a connection between the number of pure exploration iterations $T_0$ and the critical radius $\tau$ used in the local MLE formulation. It shows a lower bound on $T_0$ in order for the estimation error $\|\theta^*-\theta_0\|_2$ to be upper bounded by $\tau$ with high probability, which certifies that the true model $\theta_0$ is also a *feasible* local estimator in our MLE-UCB policy. This is an important property for later analysis of local MLE solutions ${\widehat}\theta_{t-1}$.

### Analysis of the local MLE {#sec:ana_local}

The following lemma upper bounds a Mahalanobis distance between ${\widehat}\theta_t$ and $\theta_0$. For convenience, we adopt the notation that $r_{t0}=0$ and $v_{t0}=0$ for all $t$ throughout this section. We also define $$\begin{aligned}
I_t(\theta) & := &\sum_{t'=1}^t M_{t'}(\theta), \label{eq：pop_fisher} 