SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Anytime empirical simple regret The  $\mathrm{EB - TC}_{\varepsilon_0}$  algorithm with fixed  $\beta = 1 / 2$  and  $\varepsilon_0 = 0.1$  is compared to existing algorithms on the instance  $\mu = (0.6,0.6,0.55,0.45,0.3,0.2)$  from [16], which has two best arms. As benchmark, we consider Doubling Successive Reject (DSR) and Doubling Sequential Halving (DSH), which are adaptations of the elimination based algorithms SR [1] and SH [25]. SR eliminates one arm with the worst empirical mean at the end of each phase, and SH eliminated half of them but drops past observations between each phase. These doubling-based algorithms have empirical error decreasing by steps: they change their recommendation only before they restart. In Figure 2(b), we plot the average of the simple regret over 10000 runs and the standard deviation of that average (which is too small to see clearly). We observe that  $\mathrm{EB - TC}_{\varepsilon_0}$  outperforms uniform sampling, as well as DSR and DSH, which both perform worse due to the dropped observations. The favorable performance of  $\mathrm{EB - TC}_{\varepsilon_0}$  is confirmed on other instances from [16], and for "two-groups" instances with varying  $|i^{\star}(\mu)|$  (see Figures 10 and 12).
Supplementary experiments Extensive experiments and implementation details are available in Appendix J. In Appendix J.2.1, we compare the performance of  $\mathrm{EB - TC}_{\varepsilon_0}$  with different slacks  $\varepsilon_0$  for IDS and fixed  $\beta = 1 / 2$ . In Appendix J.2.2, we demonstrate the good empirical performance of  $\mathrm{EB - TC}_{\varepsilon_0}$  compared to state-of-the-art methods in the fixed-confidence  $\varepsilon$ -BAI setting, compared to DSR and DSH for the empirical simple regret, and compared to SR and SH for the probability of 0-error in the fixed-budget setting (Figure 13). We consider a wide range of instances: random ones, benchmarks from the literature [18, 16] and "two-groups" instances with varying  $|i^{\star}(\mu)|$ .
6 Perspectives
We have proposed the  $\mathrm{EB - TC}_{\varepsilon_0}$  algorithm, which is easy to understand and implement.  $\mathrm{EB - TC}_{\varepsilon_0}$  is the first algorithm to be simultaneously asymptotically optimal in the fixed-confidence  $\varepsilon_0$ -BAI setting (Theorem 1), have finite-confidence guarantees (Theorem 2), and have also anytime guarantees on the probability of error at any level  $\varepsilon$  (Theorem 3), hence on the simple regret (Corollary 1). Furthermore, we have demonstrated that the  $\mathrm{EB - TC}_{\varepsilon_0}$  algorithm achieves superior performance compared to other algorithms, in benchmarks where the number of arms is moderate to large. In future work, we will investigate its possible adaptation to the data-poor regime of [41] in which the number of arms is so large that any algorithm sampling each arm once is doomed to failure.
While our results hold for general sub-Gaussian distributions, the  $\mathrm{EB - TC}_{\varepsilon_0}$  algorithm with IDS and slack  $\varepsilon_0 > 0$  only achieves asymptotic optimality for  $\varepsilon_0$ -BAI with Gaussian bandits. Even though IDS has been introduced by [40] for general single-parameter exponential families, it is still an open problem to show asymptotic optimality for distributions other than Gaussians. While our non-asymptotic guarantees on  $\mathbb{E}_{\nu}[\tau_{\varepsilon_0,\delta}]$  and  $\mathbb{E}_{\nu}[\mu_{\star} - \mu_{i_n}]$  were obtained for the  $\mathrm{EB - TC}_{\varepsilon_0}$  algorithm with fixed  $\beta = 1 / 2$ , we observed empirically better performance when using IDS. Deriving similar (or better) non-asymptotic guarantees for IDS is an interesting avenue for future work.
Finally, the  $\mathrm{EB - TC}_{\varepsilon_0}$  algorithm is a promising method to tackle structured bandits. Despite the existence of heuristics for settings such as Top-k identification [40], it is still an open problem to efficiently adapt Top Two approaches to cope for other structures such as  $\varepsilon$ -BAI in linear bandits.
Experiments presented in this paper were carried out using the Grid'5000 testbed, supported by a scientific interest group hosted by Inria and including CNRS, RENATER and several Universities as well as other organizations (see https://www.grid5000.fr). This work has been partially supported by the THIA ANR program "AI_PhD@Lille". The authors acknowledge the funding of the French National Research Agency under the project BOLD (ANR-19-CE23-0026-04) and the project FATE (ANR22-CE23-0016-01).
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A Outline
The appendices are organized as follows:
- Notation are summarized in Appendix B.
- In Appendix C, we show how the characteristic times and optimal allocations for  $\varepsilon$ -BAI can be reduced to the ones of a BAI problem with modified instances (Lemma 9).
- The asymptotic upper bound on the expected sample complexity of the  $\mathrm{EB - TC}_{\varepsilon_0}$  algorithm when combined with the stopping rule (4) is proven in Appendix D (Theorem 1).
- The non-asymptotic upper bound on the expected sample complexity of the  $\mathrm{EB - TC}_{\varepsilon_0}$  algorithm when combined with the stopping rule (4) is proven in Appendix E (Theorem 5).
- The upper bound on the probability of error and on the simple regret of the  $\mathrm{EB - TC}_{\varepsilon_0}$  algorithm is proven in Appendix F (Theorem 6), as well as the upper bound on its unverifiable expected sample complexity (Theorem 7).
- Appendix G gathers concentration results used for the calibration of the stopping rule (Lemma 3) and to control the empirical means and gaps.
- Appendix H gathers existing and new technical results which are used for our proofs.
- In Appendix I, we discuss the multiplicative notion of  $\varepsilon$ -goodness, propose the  $\mathrm{EB - TC}_{\varepsilon_0}^{\mathrm{m}}$  algorithm and show asymptotic guarantees on the expected sample complexity when combined with the appropriate GLR stopping rule (Theorem 8).
- Implementation details and additional experiments are presented in Appendix J.
Table 1: Notation for the setting.
Notation	Type	Description
K	N	Number of arms
νi	D	Sub-Gaussian distribution of arm i ∈ [K]
ν	DK	Vector of sub-Gaussian distributions, ν := (νi) i∈[K]
μi	R	Mean of arm i ∈ [K]
μ	RK	Vector of means, μ := (μi) i∈[K]
μ*	R	Largest mean, μ* := maxi∈[K] μi
Δi	R+	Gap of arm i, Δi := μ* - μi
i*(μ)	[K]	Best arms, i*(μ) := arg maxi∈[K] μi
ε	R+	Additive error of ε-good arms
Iε(μ)	[K]	ε-Good arms, Iε(μ) := {i ∈ [K] | Δi ≤ ε}
Tε(μ), Tε,β(μ), Tε,β(μ,i)	R*	Asymptotic (β-)characteristic time for ε-BAI
wε(μ), wε,β(μ), wε,β(μ,i)	ΔK	Asymptotic (β-)optimal allocation for ε-BAI
Cμ	[K]	Number of equivalence classes, Cμ := {μi | i ∈ [K]}
Cμ(i)	[K]	Equivalence class, Cμ(i) := {k ∈ [K] | μ* - μk = Δi}
B Notation
We recall some commonly used notation: the set of integers  $[n] := \{1, \dots, n\}$ , the complement  $X^{\mathbb{C}}$  and interior  $\mathring{X}$  of a set  $X$ , the indicator function  $\mathbb{1}(X)$  of an event, the probability  $\mathbb{P}_{\nu}$  and the expectation  $\mathbb{E}_{\nu}$  under distribution  $\nu$ , Landau's notation  $o$ ,  $\mathcal{O}$ ,  $\Omega$  and  $\Theta$ , the  $(K - 1)$ -dimensional probability simplex  $\triangle_K := \left\{w \in \mathbb{R}_+^K \mid w \geq 0, \sum_{i \in [K]} w_i = 1\right\}$ . The functions  $\mathcal{C}_G$  and  $g_G$  are defined in (23),  $\overline{W}_{-1}$  in Lemma 49,  $h_1$  in Lemma 51,  $\zeta$  is the Riemann  $\zeta$  function. While Table 1 gathers problem-specific notation, Table 2 groups notation for the algorithms.
Table 2: Notation for algorithms.
Notation	Type	Description
Bn, BnEB	[K]	(EB) Leader at time n
Cn, CnTCε0	[K]	(TCε0) Challenger at time n
In	[K]	Arm sampled at time n
Xn, In	R	Sample observed at the end of time n, i.e. Xn,In ~νIn
Fn	History before time n, i.e. Fn := σ(I1, X1, I1, ..., In, Xn, In)
i_n	[K]	Arm recommended before time n
τε,δ	N	Sample complexity (stopping time)
c(n,δ)	N × (0,1) → R+	Stopping threshold function
N_n,i	N	Number of pulls of arm i before time n
μ_n,i	I	Empirical mean of arm i before time n
T_n(i,j)	N	Counts of (Bt, Ct) = (i, j) before time n
T_n(i)	N	Counts of i ∈ {Bt, Ct} before time n
Ni_n,j	N	Counts of (Bt, Ct, It) = (i, j, j) before time n
β	(0,1)	Fixed proportion
β_n(i,j)	(0,1)	IDS proportions
β_n(i,j)	(0,1)	Averaged IDS proportions
C Characteristic Times
In Appendix C, we first recall properties satisfied by the  $(\beta -)$  characteristic times and  $(\beta -)$  optimal allocation for the BAI setting with Gaussian bandits. Then, we show how a  $\varepsilon$ -BAI problem can be reduced to a BAI one on a modified (and easier) instance.
BAI setting Let  $\nu \in \mathcal{D}^K$  with mean  $\mu \in \mathbb{R}^K$ . The  $(\beta)$ -characteristic times for the fixed-confidence BAI setting with Gaussian bandits  $\mathcal{N}(\mu, 1)$  are defined as
$$
T ^ {\star} (\mu) = \min  _ {\beta \in (0, 1)} T _ {\beta} ^ {\star} (\mu) \quad \text {w i t h} \quad 2 T _ {\beta} ^ {\star} (\mu) ^ {- 1} = \max  _ {w \in \triangle_ {K}, w _ {i} ^ {\star} = \beta} \min  _ {i ^ {\star} \neq i} \frac {\left(\mu_ {i ^ {\star}} - \mu_ {i}\right) ^ {2}}{1 / \beta + 1 / w _ {i}}, \tag {7}
$$
see [15, 34] for example. Let us denote by  $w_{\beta}^{\star}(\mu)$  and  $w^{\star}(\mu)$  their maximizer, which we will refer to as  $(\beta-)$  optimal allocations. Note that  $(T^{\star}(\mu), T_{\beta}^{\star}(\mu)) = (T_0(\mu), T_{0,\beta}(\mu))$  and  $(w^{\star}(\mu), w_{\beta}^{\star}(\mu)) = (w_0(\mu), w_{0,\beta}(\mu))$ .
We recall in the following some fundamental results on those quantities  $(\beta -)$  characteristic times and  $(\beta -)$  optimal allocations. Lemma 6 proves that the  $(\beta -)$  optimal allocations are unique, have strictly positive elements, and shows a worst-case inequality  $T_{1 / 2}^{\star}(\mu)\leq 2T^{\star}(\mu)$ . Lemma 6 also holds for any single-parameter exponential family of distributions and for the non-parametric family of bounded distributions [21].
Lemma 6 ([15, 34, 21]). If  $i^{\star}(\mu)$  is a singleton and  $\beta \in (0,1)$ , then  $w^{\star}(\mu)$  and  $w_{\beta}^{\star}(\mu)$  are singletons, i.e. the optimal allocations are unique, and  $w^{\star}(\mu)_i > 0$  and  $w_{\beta}^{\star}(\mu)_i > 0$  for all  $i \in [K]$ . Moreover, we have  $T_{1/2}^{\star}(\mu) \leq 2T^{\star}(\mu)$  and with  $\beta^{\star} = w_{i^{\star}}^{\star}(\mu)$ . Additionally, we have  $H(\mu) \leq T^{\star}(\mu) \leq 2H(\mu)$  where  $H(\mu) = 2\sum_{i \in [K]} \Delta_i^{-2}$  where  $\Delta_{i^{\star}} = \Delta_{\min}$ .
For Gaussian bandits, Lemma 7 shows that the  $(\beta -)$  characteristic times and  $(\beta -)$  optimal allocations can be obtained by solving a simpler optimization problem, which is amenable to faster implementation (e.g. Newton's iterates). It also provides upper and lower bound on the optimal allocation of the best arm, and gives hints that a tighter worst-case inequality  $T_{1/2}^{\star}(\mu) \leq r_K T^{\star}(\mu)$  might hold.
Lemma 7 ([3, 20]). Assume that  $i^{\star}(\mu) = \{i^{\star}\}$ . Let  $r(\mu)$  and  $r_{\beta}(\mu)$  be the solution of  $\psi_{\mu}(r) = 0$  and  $\varphi_{\mu,\beta}(r) = 0$  with, for all  $r \in (1 / \min_{i \neq i^{\star}} (\mu_{i^{\star}} - \mu_i)^2, +\infty)$ ,
$$
\psi_ {\mu} (r) = \sum_ {i \neq i ^ {\star}} \frac {1}{(r (\mu_ {i ^ {\star}} - \mu_ {i}) ^ {2} - 1) ^ {2}} - 1 a n d \varphi_ {\mu , \beta} (r) = \sum_ {i \neq i ^ {\star}} \frac {1}{r (\mu_ {i ^ {\star}} - \mu_ {i}) ^ {2} - 1} - \frac {1 - \beta}{\beta},
$$
Then,  $\psi_{\mu}$  and  $\varphi_{\mu,\beta}$  are convex and decreasing. Moreover,
$$
T ^ {\star} (\mu) = \frac {2 r (\mu)}{1 + \sum_ {i \neq i ^ {\star}} \frac {1}{r (\mu) (\mu_ {i ^ {\star}} - \mu_ {i}) ^ {2} - 1}} \quad a n d \quad T _ {\beta} ^ {\star} (\mu) = \frac {2 r _ {\beta} (\mu)}{\beta}.
$$
Additionally, for all  $i \neq i^{\star}$ , we have  $\beta w_{\beta}^{\star}(\mu)_i^{-1} = \beta T_{\beta}^{\star}(\mu)(\mu_{i^{\star}} - \mu_i)^2 / 2 - 1$ . For  $K = 2$ ,  $w^{\star}(\mu) = (0.5, 0.5)$  and  $T^{\star}(\mu) = T_{1/2}^{\star}(\mu) = 8(\mu_1 - \mu_2)^2$ . For  $K \geq 3$ , we have
$$
1 / (\sqrt {K - 1} + 1) \leq w ^ {\star} (\mu) _ {i ^ {\star}} \leq 1 / 2.
$$
Let  $r_K = 2K / (1 + \sqrt{K - 1})^2$ ,  $j^\star \in \arg \min_{j \neq i^\star} \mu_{i^\star} - \mu_j$  and  $x \in \mathbb{R}^{K-2}$  such that  $x_j = \frac{\mu_i \star - \mu_j}{\mu_i \star - \mu_j}$ . Then, we have  $T_{1/2}^\star(\mu) / T^\star(\mu) = \Omega(x)$  (i.e. independent of  $\mu_{i^\star}$  and  $\mu_{i^\star} - \mu_{j^\star}$ ) and
$$
\Omega (1 _ {K - 2}) = r _ {K} \quad a n d \quad \nabla_ {x} \Omega (1 _ {K - 2}) = 0 _ {K - 2}.
$$
Lemma 8 gathers necessary conditions on the  $(\beta -)$  optimal allocations at the equilibrium such as equality of the transportation costs, and a link between the squared allocation which was referred to as overall balance in [40].
Lemma 8 ([3, 40]). Assume that  $i^{\star}(\mu) = \{i^{\star}\}$  and  $\beta \in (0,1)$ . Then, for all  $i \neq i^{\star}$ ,
$$
2 T ^ {\star} (\mu) ^ {- 1} = \frac {(\mu_ {i ^ {\star}} - \mu_ {i}) ^ {2}}{1 / w ^ {\star} (\mu) _ {i ^ {\star}} + 1 / w ^ {\star} (\mu) _ {i}} \quad a n d \quad 2 T _ {\beta} ^ {\star} (\mu) ^ {- 1} = \frac {(\mu_ {i ^ {\star}} - \mu_ {i}) ^ {2}}{1 / \beta + 1 / w _ {\beta} ^ {\star} (\mu) _ {i}}.
$$
$$
w ^ {\star} (\mu) _ {i ^ {\star}} ^ {2} = \sum_ {i \neq i ^ {\star}} w ^ {\star} (\mu) _ {i} ^ {2}.
$$
Reduction of an  $\varepsilon$ -BAI problem to a BAI problem Lemma 9 gives a reduction of a  $\varepsilon$ -BAI problem to a BAI one on a modified instance, which is easier. Thanks to Lemma 9, it is possible to leverage existing results on  $T^{\star}(\mu), T_{\beta}^{\star}(\mu), w^{\star}(\mu), w_{\beta}^{\star}(\mu)$  (such as Lemmas 6, 7 and 8) in order to study  $T_{\varepsilon}(\mu), T_{\varepsilon,\beta}(\mu), T_{\varepsilon,\beta}(\mu,i), w_{\varepsilon}(\mu), w_{\varepsilon,\beta}(\mu)$  and  $w_{\varepsilon,\beta}(\mu,i)$ .
Lemma 9. Let  $\mu \in \mathbb{R}^K$ ,  $\varepsilon \geq 0$  and  $\tilde{\varepsilon} \in [0, \varepsilon]$  and  $\beta \in (0, 1)$ . For all  $i \in \mathcal{I}_{\tilde{\varepsilon}}(\mu)$ , we define  $\mu_{\varepsilon}(i)$  as  $\mu_{\varepsilon}(i)_j = \mu_j - \varepsilon$  for all  $j \neq i$  and  $\mu_{\varepsilon}(i)_i = \mu_i$ . Then, for all  $i \in \mathcal{I}_{\tilde{\varepsilon}}(\mu)$ ,  $T_{\varepsilon, \beta}(\mu, i) = T_{\beta}^{\star}(\mu_{\varepsilon}(i))$  and  $w_{\varepsilon, \beta}(\mu, i) = w_{\beta}^{\star}(\mu_{\varepsilon}(i))$ . Moreover, for all  $i^{\star} \in i^{\star}(\mu)$ ,
$$
T _ {\varepsilon} (\mu) = T ^ {\star} \left(\mu_ {\varepsilon} \left(i ^ {\star}\right)\right) \quad a n d \quad T _ {\varepsilon , \beta} (\mu) = T _ {\beta} ^ {\star} \left(\mu_ {\varepsilon} \left(i ^ {\star}\right)\right).
$$
Moreover, we have  $w_{\varepsilon}(\mu) = \bigcup_{i^{\star}\in i^{\star}(\mu)}w^{\star}(\mu_{\varepsilon}(i^{\star}))$  and  $w_{\varepsilon ,\beta}(\mu) = \bigcup_{i^{\star}\in i^{\star}(\mu)}w_{\beta}^{\star}(\mu_{\varepsilon}(i^{\star}))$
Proof. For  $\varepsilon = 0$ , the result is direct by definition. Let  $\varepsilon > 0$  and  $\tilde{\varepsilon} \in [0, \varepsilon]$ . The first part is obtained by definition of  $T_{\varepsilon, \beta}^{\star}(\mu, i)$ ,  $w_{\varepsilon, \beta}^{\star}(\mu, i)$ . Let  $i^{\star} \in i^{\star}(\mu)$  and  $\mu_{\varepsilon}(i^{\star})$  defined as above, hence  $i^{\star}(\mu_{\varepsilon}(i^{\star})) = \{i^{\star}\}$ . For all  $i \in \mathcal{I}_{\varepsilon}(\mu) \setminus i^{\star}(\mu)$ , let us denote by  $\lambda_{\varepsilon}^{(i^{\star}, i)}$  the instance such that  $\lambda_{\varepsilon, i}^{(i^{\star}, i)} = \mu_i$  and  $\lambda_{\varepsilon, j}^{(i^{\star}, i)} = \mu_j - \varepsilon$  for all  $j \neq i$ . If  $\mu_{i^{\star}} - \mu_i = \varepsilon$  then  $i^{\star}(\lambda_{\varepsilon}^{(i^{\star}, i)}) = \{i\} \cup i^{\star}(\mu) \setminus \{i^{\star}\}$ , otherwise  $i^{\star}(\lambda_{\varepsilon}^{(i^{\star}, i)}) = \{i\}$ . We consider the permutation  $\sigma$  that swaps arm  $i$  with arm  $i^{\star}$ . By symmetry, we have  $T^{\star}(\lambda_{\varepsilon}^{(i^{\star}, i)}) = T^{\star}(\sigma(\lambda_{\varepsilon}^{(i^{\star}, i)}))$ . Moreover, we have that the gaps of  $\sigma(\lambda_{\varepsilon}^{(i^{\star}, i)})$  are all strictly smaller than the gaps of  $\mu_{\varepsilon}$  since  $\varepsilon \geq \Delta_i > 0$ . Therefore, Lemma 11 of Barrier et al (2022) yields that  $T^{\star}(\sigma(\lambda_{\varepsilon}^{(i^{\star}, i)})) > T^{\star}(\mu_{\varepsilon})$ . We have proved that
$$
\forall i ^ {\star} \in i ^ {\star} (\mu), \quad T ^ {\star} \left(\mu_ {\varepsilon} \left(i ^ {\star}\right)\right) <   \min  _ {i \in \mathcal {I} _ {\varepsilon} (\mu) \backslash i ^ {\star} (\mu)} T ^ {\star} \left(\lambda_ {\varepsilon} ^ {\left(i ^ {\star}, i\right)}\right).
$$
By symmetry  $T^{\star}(\mu_{\varepsilon}(i^{\star}))$  is constant for all  $i^{\star}\in i^{\star}(\mu)$ , hence we have shown that
$$
\forall i ^ {\star} \in i ^ {\star} (\mu), \quad T _ {\varepsilon} (\mu) = T ^ {\star} \left(\mu_ {\varepsilon} \left(i ^ {\star}\right)\right).
$$
It also shows that  $w_{\varepsilon}(\mu) = \bigcup_{i^{\star}\in i^{\star}(\mu)}w^{\star}(\mu_{\varepsilon}(i^{\star}))$ . The same reasoning yields the result for  $T_{\varepsilon ,\beta}(\mu)$  and  $w_{\varepsilon ,\beta}(\mu)$ .
$\varepsilon$ -BAI problems cannot be arbitrarily difficult Lemma 10 shows that, for all  $i \in \mathcal{I}_{\varepsilon/2}(\mu)$ ,  $T_{\varepsilon,1/2}(\mu, i)$  cannot be arbitrarily large and  $\min_{j \neq i} w_{\varepsilon_0,1/2}(\mu, i)_j$  cannot be arbitrarily small. The proof relies on Lemma 9 and existing results for BAI.
Lemma 10. Let  $\varepsilon > 0$  and  $\mu \in \mathbb{R}^K$ . Then, for all  $i \in \mathcal{I}_{\varepsilon/2}(\mu)$ , we have  $T_{\varepsilon,1/2}(\mu, i) \leq 32K / \varepsilon^2$  and  $\min_{j \neq i} w_{\varepsilon,1/2}(\mu, i)_j \geq (16(K - 2) + 2)^{-1}$ .
Proof. Let  $i \in \mathcal{I}_{\varepsilon/2}(\mu)$ , hence  $\mu_i \geq \mu_\star - \varepsilon/2$ . Let  $\mu_{\varepsilon}(i)$  as in Lemma 9 which satisfies that  $i^\star(\mu_\varepsilon(i)) = \{i\}$ . Let  $\Delta_{i,j} = \mu_i - \mu_j$  and  $\Delta_{i,i} = \mu_i - \max_{j \neq i} \mu_j$ . Then, we have
$$
T _ {\varepsilon , 1 / 2} (\mu , i) = T _ {1 / 2} ^ {\star} \left(\mu_ {\varepsilon} (i)\right) \leq 2 T ^ {\star} \left(\mu_ {\varepsilon} (i)\right) \leq 8 \sum_ {j \in [ K ]} \left(\Delta_ {i, j} + \varepsilon\right) ^ {- 2} \leq 8 \sum_ {j \in [ K ]} \left(\Delta_ {j} + \varepsilon / 2\right) ^ {- 2},
$$
where we used Lemma 6 for the two first inequality and  $i \in \mathcal{I}_{\varepsilon/2}(\mu)$  for the last one. Since  $\Delta_j \geq 0$  we conclude that  $T_{\varepsilon, \beta}(\mu, i) \leq 32K/\varepsilon^2$ . Moreover, using Lemma 6, we have
$$
T _ {\varepsilon , 1 / 2} (\mu , i) = T _ {1 / 2} ^ {\star} \left(\mu_ {\varepsilon} (i)\right) \geq T ^ {\star} \left(\mu_ {\varepsilon} (i)\right) \geq 2 \sum_ {j \in [ K ]} \left(\Delta_ {i, j} + \varepsilon\right) ^ {- 2}.
$$
Likewise, using Lemma 9 and Lemma 7, we obtain that, for all  $j \neq i$
$$
\begin{array}{l} w _ {\varepsilon , 1 / 2} (\mu) _ {j} ^ {- 1} / 2 = w _ {1 / 2} ^ {\star} \left(\mu_ {\varepsilon} (i)\right) _ {j} ^ {- 1} / 2 = T _ {1 / 2} ^ {\star} \left(\mu_ {\varepsilon} (i)\right) \left(\mu_ {i} - \mu_ {j} + \varepsilon\right) ^ {2} / 4 - 1 \\ \leq 2 \sum_ {k \notin \{i, j \}} \left(\frac {\mu_ {i} - \mu_ {j} + \varepsilon}{\mu_ {i} - \mu_ {k} + \varepsilon}\right) ^ {2} + 1 \\ \end{array}
$$
where the last inequality uses what we proved above. When  $\mu_k \leq \mu_j$ , the ratio is smaller than one. When  $\mu_k > \mu_j$ , we have  $\mu_\star \geq \mu_k > \mu_j \geq \mu_\star - \varepsilon/2$ , hence  $\mu_k - \mu_j \leq \varepsilon/2$  and
$$
\frac {\mu_ {i} - \mu_ {j} + \varepsilon}{\mu_ {i} - \mu_ {k} + \varepsilon} \leq 1 + \frac {\varepsilon / 2}{\mu_ {\star} - \mu_ {k} + \varepsilon / 2} \leq 2.
$$
Therefore, we obtain  $w_{\varepsilon,1/2}(\mu)_j^{-1}/2 \leq 8(K - 2) + 1$ , which concludes the result.
Lemma 11 links the characteristic times for  $\varepsilon$ -BAI where  $\varepsilon \in \{\varepsilon_0, \varepsilon_1\}$ .
Lemma 11. Let  $\mu \in \mathbb{R}^K$  such that  $|i^{\star}(\mu)| = 1$ . Let  $\varepsilon_0 > \varepsilon_1$ . Then, for all  $\beta \in (0,1)$ , we have
$$
T _ {\varepsilon_ {0}} (\mu) \left(\Delta_ {\min } + \varepsilon_ {0}\right) ^ {2} \geq T _ {\varepsilon_ {1}} (\mu) \left(\Delta_ {\min } + \varepsilon_ {1}\right) ^ {2} \quad a n d \quad T _ {\varepsilon_ {0}, \beta} (\mu) \left(\Delta_ {\min } + \varepsilon_ {0}\right) ^ {2} \geq T _ {\varepsilon_ {1}, \beta} (\mu) \left(\Delta_ {\min } + \varepsilon_ {1}\right) ^ {2}.
$$
Let  $\varepsilon_0 < \varepsilon_1$ . Then,