Original: T IGHT N ON - ASYMPTOTIC I NFERENCE VIA S UB -G AUSSIAN I NTRINSIC M OMENT N ORM
Revised: T IGHT N ON - ASYMPTOTIC I NFERENCE VIA S UB -G AUSSIAN I NTRINSIC M OMENT N ORM

Original: Anonymous authors
Revised: Anonymous authors

Original: Paper under double-blind review
Revised: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: In non-asymptotic statistical inferences, variance-type parameters of sub-Gaussian distributions play a crucial role.
Revised: In non-asymptotic statistical inferences, variance-type parameters of sub-Gaussian distributions play a crucial role.

Original: However, direct estimation of these parameters based on the empirical moment generating function (MGF) is infeasible.
Revised: However, direct estimation of these parameters based on the empirical moment generating function (MGF) is infeasible.

Original: To this end, we recommend using a sub-Gaussian intrinsic moment norm [Buldygin and Kozachenko (2000), Theorem 1.3] through maximizing a series of normalized moments.
Revised: To this end we recommend using a sub-Gaussian intrinsic moment norm [Buldygin and Kozachenko (2000), Theorem 1.3] through maximizing a series of normalized moments.

Original: Importantly, the recommended norm can not only recover the exponential moment bounds for the corresponding MGFs, but also lead to tighter Hoeffiding’s sub-Gaussian concentration inequalities.
Revised: Importantly, the recommended norm can not only recover the exponential moment bounds for the corresponding MGFs, but also lead to tighter Hoeffiding’s sub-Gaussian concentration inequalities.

Revised: In practice, we propose an intuitive way of checking sub-Gaussian data with a finite sample size by the sub-Gaussian plot.

Original: In practice, intrinsic moment norm can be robustlyand consistently estimated via a simple plug-in approach.
Revised: Intrinsic moment norm can be robustly estimated via a simple plug-in approach.

Original: Our theoretical results are applied to non-asymptotic analysis, including the multi-armed bandit.
Revised: Our theoretical results are applied to non-asymptotic analysis, including the multi-armed bandit

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: With the advancement of machine learning techniques, computer scientists have become more interested in establishing rigorous error bounds for desired learning procedures, especially those with finite sample validity (Wainwright, 2019; Zhang & Chen, 2021;
Revised: With the advancement of machine learning techniques, computer scientists have become more interested in establishing rigorous error bounds for desired learning procedures, especially those with finite sample validity (Wainwright, 2019; Zhang & Chen, 2021;

Original: Yang et al., 2020).
Revised: Yang et al., 2020).

Original: In specific settings, statisticians, econometricians, engineers and physicist have developed non-asymptotic inferences to quantify uncertainty in data; see Romano & Wolf (2000); Chassang (2009); Arlot et al.
Revised: In specific settings, statisticians, econometricians, engineers and physicist have developed non-asymptotic inferences to quantify uncertainty in data; see Romano & Wolf (2000); Chassang (2009); Arlot et al.

Original: ; Yang et al.
Revised: ; Yang et al.

Original: (2020); Horowitz & Lee (2020); Armstrong & Koles ´ ar (2021); Zheng & Cheng (2021); Lucas et al.
Revised: (2020); Horowitz & Lee (2020); Armstrong & Koles ´ ar (2021); Zheng & Cheng (2021); Lucas et al.

Original: ; Owhadi et al. (2013); Wang (2020).
Revised: ; Owhadi et al. (2013); Wang (2020).

Original: Therefore, the concentration-based statistical inference has received a considerable amount of attention, especially for bounded data (Romano & Wolf, 2000; Auer et al., 2002;
Revised: Therefore, the concentration-based statistical inference has received a considerable amount of attention, especially for bounded data (Romano & Wolf, 2000; Auer et al., 2002;

Original: Hao et al., 2019; Wang et al., 2021; Shiu, 2022) and Gaussian data (Arlot et al., 2010; Duy & Takeuchi, 2022; Bettache et al., 2021;
Revised: Hao et al., 2019; Wang et al., 2021; Shiu, 2022) and Gaussian data (Arlot et al., 2010; Duy & Takeuchi, 2022; Bettache et al., 2021;

Original: Feng et al., 2021).
Revised: Feng et al., 2021).

Original: For example,the Hoeffding’s inequality can be applied to construct non-asymptotic confidence intervals based on bounded data 1 .
Revised: For example, Hoeffding’s inequality can be applied to construct nonasymptotic confidence intervals based on bounded data 1 .

Original: However, in reality, it may be hard to know the support of data or its underlying distribution.
Revised: However, in reality, it may be hard to know the support of data or its underlying distribution.

Original: In this case, misusingthe
Revised: In this case, misusing

Original: Hoeffding’s inequality (Hoeffding, 1963) for unbounded data will result in a notably loose confidence interval (CI); see
Revised: Hoeffding’s inequality (Hoeffding, 1963) for unbounded data will result in a notably loose confidence interval (CI); see

Original: Appendix A.1.
Revised: Appendix A.1.

Original: Hence, it is a common practice to assume that data follow sub-Gaussian distribution (Kahane, 1960).
Revised: Hence, it is a common practice to assume that data follow sub-Gaussian distribution (Kahane, 1960).

Original: By the Chernoff inequality 2 , we have P( X ≥ t ) ≤ inf s> 0 (cid:8) exp {− st } E exp { sX } (cid:9) , ∀ t ≥ 0 .
Revised: By the Chernoff inequality 2 , we have P( X ≥ t ) ≤ inf s> 0 (cid:8) exp {− st } E exp { sX } (cid:9) , ∀ t ≥ 0 .

Original: Hence, tightness of a confidence interval relies on how we upper bound the moment generating function (MGF) E exp { sX } for all s > 0 .
Revised: Hence, tightness of a confidence interval relies on how we upper bound the moment generating function (MGF) E exp { sX } for all s > 0 .

Original: This can be further translated into the following optimal variance proxy of sub-Gaussian distribution, i.e., sub-Gaussian parameter σ opt ( X ) .
Revised: This can be further translated into the following optimal variance proxy of sub-Gaussian distribution.

Original: Definition 1.
Revised: Definition 1.

Original: A r.v.
Revised: A r.v.

Original: X is sub-Gaussian (sub-G) with a variance proxy σ 2 [ denoted as X ∼ subG( σ 2 ) ] if its MGF satisfies E exp( tX ) ≤ exp( σ 2 t 2 / 2) for all t ∈ R .
Revised: X is sub-Gaussian (sub-G) with a variance proxy σ 2 [ denoted as X ∼ subG( σ 2 ) ] if its MGF satisfies E exp( tX ) ≤ exp( σ 2 t 2 / 2) for all t ∈ R .

Original: The sub-Gaussian parameter σ opt ( X ) is defined by the optimal variance proxy (Chow, 1966):
Revised: The sub-Gaussian parameter σ opt ( X ) is defined by the optimal variance proxy (Chow, 1966):

Original: [Equation]
Revised: [Equation]

Original: Note that σ 2 opt ( X ) ≥ Var X ; see (16) in Appendix A.2.
Revised: Note that σ 2 opt ( X ) ≥ Var X ; see (14) in Appendix A.2.

Original: When σ 2 opt ( X ) =
Revised: When σ 2 opt ( X ) =

Original: Var X , it is called strict sub-Gaussianity (Arbel et al., 2020).
Revised: Var X , it is called strict sub-Gaussianity (Arbel et al., 2020).

Original: Based on Theorems 1.5 in Buldygin & Kozachenko (2000), we have
Revised: Based on Theorems 1.5 in Buldygin & Kozachenko (2000)

Original: [Equation]
Revised: [Equation]

Revised: we have

Original: [Equation]
Revised: [Equation]

Original: for independent sub-G r.v.s X and { X i } ni =1 .
Revised: for independent sub-G r.v.s X and { X i } ni =1 .

Original: The above inequality (2) provides the tightest upper bound over the form
Revised: The above inequality (2) provides the tightest upper bound over the form

Original: P( X > t ) ≤ exp( − Ct 2 ) (or P( | (cid:80) ni
Revised: P( X > t ) ≤ exp( − Ct 2 ) (or P( | (cid:80) ni

Original: =1 X i | > t ) ≤ exp( − Ct 2 ) ) for some positive constant C via Chernoff inequality.
Revised: =1 X i | > t ) ≤ exp( − Ct 2 ) ) for some positive constant C via Chernoff inequality.

Original: Given { X i } ni =

Revised: Given { X i } ni =

Original: i .
Revised: i .

Original: i .
Revised: i .

Original: d . ∼ subG( σ 2 opt ( X )) , a straightforward application of (2) gives the following non-asymptotic 100(1 − [Equation]
Revised: d . ∼ subG( σ 2 opt ( X )) , a straightforward application of (2) gives an non-asymptotic 100(1 − α )% CI

Original: [Equation]
Revised: [Equation]

Original: A naive plug-in estimate 3 of σ 2 opt ( X 
Revised: A naive plug-in estimate 3 of σ 2 opt ( X ) :

Original: [Equation] := 2 sup t ∈ R t − 2 log[E exp( tX [Equation] ] (Arbel et al., 2020) is
Revised: = 2 sup t ∈ R t − 2 log[E exp( tX )] (Arbel et al., 2020) is

Original: [Equation]
Revised: [Equation]

Original: However, two weaknesses ofthe naive estimator (4) substantially hinder its application: (i) the optimization result is unstable due to the possible non-convexity of the objective function; (ii) exponentially large n is required to ensure the variance term Var( n − 1 (cid:80) ni
Revised: However, two weaknesses of (4) substantially hinder its application: (i) the optimization result is unstable due to the possible non-convexity of the objective function; (ii) exponentially large n is required to ensure the variance term Var( n − 1 (cid:80) ni

Original: =1 exp( tX i )) not to explode when t is large.
Revised: =1 exp( tX i )) not to explode when t is large.

Original: Figure 3 below present  simulation evidences.
Revised: In Section 3, we present some simulation evidence.

Original: On the other hand, we are aware of other forms of variance-type parameter.
Revised: On the other hand, we are aware of other forms of variance-type parameter.

Original: For instance, van der Vaart & Wellner (1996) introduced the Orlicz norm as ∥ X ∥ w 2 :=
Revised: For instance, van der Vaart & Wellner (1996) introduced the Orlicz norm as ∥ X ∥ w 2 :=

Original: inf { c > 0 : E exp {| X | 2 /c 2 } ≤ 2 } , frequently used in empirical process theory.
Revised: inf { c > 0 : E exp {| X | 2 /c 2 } ≤ 2 } , frequently used in empirical process theory.

Original: Additionally, Vershynin (2010) suggested a norm based on the scale of moments as ∥ X ∥ ψ 2 :=
Revised: Additionally, Vershynin (2010) suggested a norm based on the scale of moments as ∥ X ∥ ψ 2 :=

Original: max k ≥ 2 k − 1 / 2 (E | X | k ) 1 /k in Page 6 of Buldygin & Kozachenko (2000).
Revised: max k ≥ 2 k − 1 / 2 (E | X | k ) 1 /k in Page 6 of Buldygin & Kozachenko (2000).

Original: However, as shown in Table 1 and Appendix A.2.1, both types of norm fail to deliver sharp probability bounds even for strict sub-G distributions, such as the standard Gaussian distribution and symmetric beta distribution
Revised: However, as shown in Table 1 and Appendix A.2.1, both types of norm fail to deliver sharp probability bounds even for strict sub-G distributions, such as the standard Gaussian distribution and symmetric beta distribution

Original: [Table] [Table]
Revised: [Table] [Table]

Original: 1. C ONTRIBUTIONS
Revised: 1. C ONTRIBUTIONS

Original: In light of the above discussions
Revised: In light of the above discussions

Original: [Table]
Revised: [Table]

Original: we advocate the use of the intrinsic moment norm in the Definition 2 in the construction of tight non-asymptotic CIs.
Revised: we advocate the use of the intrinsic moment norm in the Definition 2 in the construction of tight non-asymptotic CIs.

Original: There are two specific reasons: (i) it approximately recovers tight inequalities (2); (ii) it can be estimated friendly (with a closed form) and robustly.
Revised: There are two specific reasons: (i) it approximately recovers tight inequalities (2); (ii) it can be estimated friendly (with a closed form) and robustly.

Original: The following definition 2 is from Page 6 and Theorem 1.3 in Buldygin & Kozachenko (2000).
Revised: The following definition 2 is from Page 6 and Theorem 1.3 in Buldygin & Kozachenko (2000).

Original: Definition 2 (Intrinsic moment norm) .
Revised: Definition 2 (Intrinsic moment norm) .

Original: ∥
Revised: ∥

Original: X ∥ G := max k ≥ 1 [ 2 k k !
Revised: X ∥ G := max k ≥ 1 [ 2 k k !

Original: (2 k )!
Revised: (2 k )!

Original: E X 2 k ] 1 / (2 k ) =
Revised: E X 2 k ] 1 / (2 k ) =

Original: max k ≥ 1 [ 1 (2 k − 1)!!
Revised: max k ≥ 1 [ 1 (2 k − 1)!!

Original: E X 2 k ] 1 / (2 k ) .
Revised: E X 2 k ] 1 / (2 k ) .

Original: From the sub-G characterization (see Theorem 2.6 in Wainwright (2019)) ∥ X ∥ G < ∞ iff σ opt ( X ) < ∞ for any zeromean r.v. X .
Revised: From the sub-G characterization (see Theorem 2.6 in Wainwright (2019)), ∥ X ∥ G < ∞ iff σ opt ( X ) < ∞ for any zeromean r.v. X .

Original: Hence, the finite intrinsic moment norm of a r.v.
Revised: Hence, the finite intrinsic moment norm of a r.v.

Original: X ensures sub
Revised: X ensures sub

Original: -Gaussianity (satisfying Definition 1).
Revised: -Gaussianity (satisfying Definition 1).

Original: Our contributions in this paper can be summarized as follows.
Revised: Our contributions in this paper can be summarized as follows.

Original: Bythe use of ∥ X ∥ G , we achieve the Hoeffding-type inequalities with sharper constants under asymetric distribution; see Theorem 1(b).
Revised: By ∥ X ∥ G , we achieve a sharper Hoeffding-type inequality under asymetric distribution; see Theorem 2(b).

Original: Compared to the normal approximation based on Berry-Esseen (B-E) bounds, our results are more applicable to data of extremely small sample size.
Revised: Compared to the normal approximation based on Berry-Esseen (B-E) bounds, our results are more applicable to data of extremely small sample size.

Original: We illustrate Bernoulli observations with the comparison of two types of CIs based on the B-E-corrected CLT and Hoeffding’s inequality in Figure 1; see Appendix A for details.
Revised: We illustrate Bernoulli observations with the comparison of two types of CIs based on the B-E-corrected CLT and Hoeffding’s inequality in Figure 1; see Appendix A for details.

Original: We propose a novel method called sub-Gaussian plot  for judging whether the  data are sub-Gaussian. 4.
Revised: A novel method called sub-Gaussian plot is proposed for checking whether the unbounded data are subGaussian.

Revised: We introduce plug-in and robust plug-in estimators for ∥ X ∥ G , and establish finite sample theories.

Original: Finally, we employ the intrinsic moment norm  in the non-asymptotic inference for a bandit problem: Bootstrapped UCB-algorithm for multi-armed bandits.
Revised: Finally, we employ the intrinsic moment norm estimation in the non-asymptotic inference for a bandit problem: Bootstrapped UCB-algorithm for multi-armed bandits.

Original: This algorithm is shown to achieve feasible error bounds and competitive cumulative regret on unbounded sub-Gaussian data by the intrinsic moment norm estimation.
Revised: This algorithm is shown to achieve feasible error bounds and competitive cumulative regret on unbounded sub-Gaussian data.

Original: n >=n >= 357 0.
Revised: [Figure]

Original: 500 n
Revised: [Figure]

Original: C on f i den c e Leng t h delta = 0.
Revised: on f i den c e Leng t h delta [Figure]

Original: n >=n >= 163 0.
Revised: n [Figure] n >

Original: 500 n
Revised: [Figure]

Original: C on f i den c e Leng t hdelta = 0.
Revised: [Figure] f [Figure] Leng t h

Original:  n >=n >= 95 0. 500 n
Revised: [Figure] n > [Figure]

Revised: [Figure]

Revised: f i

Revised: [Figure]

Original: C on f i den c e Leng t h delta = 0.
Revised: c [Figure] Leng [Figure]

Revised:  SUB -G AUSSIAN PLOT AND TESTING

Revised: Before estimating ∥ X ∥ G , the first step is to verify X is indeed sub-G given its i.i.d.

Revised: copies { X i } n i =1 .

Revised: Corollary 7.2 (b) in Zhang & Chen (2021) shows for r.v.s X i ∼ subG( σ 2 opt ( X )) (without independence assumption)

Revised: [Equation]

Revised: [Equation]

Revised: which implies max 1 ≤ i ≤ j X i

Original:  P ROPERTIES OF I NTRINSIC M OMENT N ORM
Revised: = O P ( √ log j ) . Moreover, we will show the above rate is indeed sharp for a class of unbound sub-G r.v.s characterized by the lower intrinsic moment norm below. Definition 3 (Lower intrinsic moment norm) . The lower intrinsic moment norm for a sub-G X is defined as

Revised: [Equation]

Revised: By the method in Theorem 1 of Zhang & Zhou (2020), we obtain the following the tight rate result with a lower bound.

Revised: Theorem 1.

Revised: (a).

Revised: If ∥ X ∥ ˜ G > 0 for i.i.d.

Revised: symmetric sub-G r.v.s { X i } ni

Revised: =

Revised: ∼ X , then with probability at least 1 − δ ∥

Revised: X ∥ ˜ G / ∥

Revised: X ∥ G

Revised: [Equation]

Revised: ] , where C ( X ) < 1 is constant defined in Lemma 1 below; (b) if X is bounded variable, then ∥ X ∥ ˜ G = 0 .

Revised: The upper bound follows from the proof of (5) similarly.

Revised: The proof of lower bound relies on the sharp reverse Chernoff inequality from Paley–Zygmund inequality (see Paley & Zygmund (1932)).

Revised: Lemma 1 (A reverse Chernoff inequality) .

Revised: Suppose ∥ X ∥ ˜ G > 0 for a symmetric sub-G r.v. X .

Revised: For t > 0 , then

Revised: [Equation]

Revised: where C ( X ) := (cid:16)

Revised: [Equation]

Revised: Theorem 1 of Zhang & Zhou (2020) does not optimize the constant in Paley–Zygmund inequality.

Revised: In contrast, our

Revised: Lemma 1 has an optimal constant; see Appendix C for details.

Revised: Sub-Gaussian plot under unbounded assumption 4 .

Revised: By Theorem 1, we propose a novel sub-Gaussian plot check whether i.i.d data { X i } n i =1 follow a sub-G distribution.

Revised: Suppose that for each j , { X ∗ i } j i =1 are independently sampled from the empirical distribution F n ( x ) = 1 n (cid:80) ni

Revised: =1 1( X i ≤ x ) of { X i } ni =1 .

Original: {+Specifically, we plot the order statistics { max 1 ≤ i ≤ j X ∗ i } nj =1 on the plane coordinate axis, where x axis represents √ log j + 
+}
Revised: {+Specifically, we plot the order statistics { max 1 ≤ i ≤ j X ∗ i } nj =1 on the plane coordinate axis, where x axis represents √ log j + 
+}

Revised: and y axis the value of max 1 ≤ i ≤ j X ∗ i .

Revised: We check whether those points have a linear tendency at the boundary: the more they are close to the tendency of a beeline, the more we can trust the data are sub-Gaussian.

Revised: The Figure 2 shows sub-Gaussian plot of N (0 , 1) and Exp(1) .

Revised: It can be seen that sub-Gaussian plot of N (0 , 1) shows linear tendency at the boundary, while Exp(1) shows quadratic tendency at the boundary.

Revised: For the quadratic tendency, we note that if { X i } ni =1 have heavier tails such as sub-exponentiality, then max 1 ≤ i ≤ j X i =

Revised: O P (log j ) instead of the order O ( √ log j ) ; see Corollary 7.3 in Zhang & Chen (2021).

Revised: [Figure]

Revised:  F INITE S AMPLE P ROPERTIES OF I NTRINSIC M OMENT N ORM

Original: In this section, we characterize two important properties of the intrinsic moment norm that are used in constructing non-asymptotic confidence intervals.
Revised: In this section, we characterize two important properties of the intrinsic moment norm that are used in constructing non-asymptotic confidence intervals.

Original: 2. B ASIC P ROPERTIES
Revised: 3. B ASIC P ROPERTIES

Original: Lemma 1 below establishes that the intrinsic moment norm is estimable.
Revised: Lemma 2 below establishes that the intrinsic moment norm is estimable.

Original: Lemma 1.
Revised: Lemma 2.

Original: For sub-G X , we have arg max m ∈ 2 N (cid:104)
Revised: For sub-G X , we have arg max m ∈ 2 N (cid:104)

Original: E
Revised: E

Original: X m ( m [Equation]
Revised: X m ( m − 1)!!

Original: (cid:105) 1 /m < ∞ , where 2 N := { 2 , 4 , · · · } is an even number set.
Revised: (cid:105) 1 /m < ∞ , where 2 N := { 2 , 4 , · · · } is the even number set.

Original: Lemma 1 ensure that for any sub-Gaussian variable X , its intrinsic moment norm can be computed as
Revised: Lemma 2 ensure that for any sub-Gaussian variable X , its intrinsic moment norm can be computed as

Original: [Equation]
Revised: [Equation]

Original: k ) with some finite k X < ∞ .
Revised: ) with some finite k X < ∞ .

Original: This is an important property that other norms may not have.
Revised: This is an important property that other norms may not have.

Revised: The ∥ X ∥ ψ

Original: For example, the ∥ X ∥ ψ 2 := max k ≥ 2 k − 1 / 2{+
+} (E | X | k ) 1 /k for Gaussian X achieves its optimal point at k = ∞ ; see Example 3 in Appendix A.2.1.
Revised: [Equation] = max k ≥ 2 k − 1 / {+
+} (E | X | k ) 1 /k for Gaussian X achieves its optimal point at k = ∞ ; see Example 3 in Appendix A.2.1.

Original: As for σ 2 opt ( X ) :
Revised: As for σ 2 opt ( X ) :

Original: = 2 sup t ∈ R log[Eexp( tX )]
Revised: = 2 sup t ∈ R log[Eexp( tX )]

Original: t, it is unclear that its value can be achieved at a finite t .
Revised: t, it is unclear that its value can be achieved at a finite t .

Original: Note that if k X = 1 , one has ∥ X ∥ 2 G =
Revised: Note that if k X = 1 , one has ∥ X ∥ 2 G =

Original: Var( X ) .
Revised: Var( X ) .

Original: In the end, we present an example in calculating the values of k X .
Revised: Next, we present an example in calculating the values of k X .

Original: Denote Exp(1)
Revised: Denote Exp(1)

Original: | [0 ,M ] as the truncated standard exponential distribution on [0 , M ] with the density as f ( x ) =
Revised: | [0 ,M ] as the truncated standard exponential distribution on [0 , M ] with the density as f ( x ) =

Original: [Equation]
Revised: [Equation]

Original: Example 1.
Revised: Example 1.

Original: a. X ∼ U [ − a, a ] , k X = 1 for any a ∈ R ; b. X ∼ Exp(1) |
Revised: a. X ∼ U [ − a, a ] , k X = 1 for any a ∈ R ; b. X ∼ Exp(1) |

Original: [0 , 2 . 75] − E Exp(1)
Revised: [0 , 2 . 75] − E Exp(1)

Original: | [0 , 2 . 75] , k X = 2 ; c.
Revised: | [0 , 2 . 75] , k X = 2 ; c.

Original: X ∼ Exp(1) | [0 , 3] − E Exp(1) | [0 , 3] , k X = 3 .
Revised: X ∼ Exp(1) | [0 , 3] − E Exp(1) | [0 , 3] , k X = 3 .

Original: Indeed, for any fixed k 0 ∈ N , we can construct a truncated exponential r.v.
Revised: Indeed, for any fixed k 0 ∈ N , we can construct a truncated exponential r.v.

Original: X := Exp(1)
Revised: X := Exp(1)

Original: | [0 ,M ] such that k X = k 0 by properly adjusting the truncation level M .
Revised: | [0 ,M ] such that k X = k 0 by properly adjusting the truncation level M .

Original: 2. C ONCENTRATION FOR SUMMATION
Revised: 3. C ONCENTRATION FOR SUMMATION

Original: In what follows, we will show another property of ∥ X ∥ G that it recovers nearly tight MGF bounds in Definition 1.
Revised: In what follows, we will show another property of ∥ X ∥ G that it recovers nearly tight MGF bounds in Definition 1.

Original: More powerfully, it enables us to derive the sub-G Hoeffding’s inequality (2).
Revised: More powerfully, it enables us to derive the sub-G Hoeffding’s inequality (2).

Original: Theorem 1.
Revised: Theorem 2.

Original: Suppose that { X i } ni =1 are independent r.v.s with max i ∈ [ n ]
Revised: Suppose that { X i } ni =1 are independent r.v.s with max i ∈ [ n ]

Original: ∥ X i ∥ G < ∞ .
Revised: ∥ X i ∥ G < ∞ .

Original: We have (a) .
Revised: We have (a) .

Original: If X i is symmetric about zero, then Eexp { tX i } ≤ exp { t 2 ∥
Revised: If X i is symmetric about zero, then Eexp { tX i } ≤ exp { t 2 ∥

Original: X i ∥ 2 G / 2 } for any t ∈ R , and
Revised: X i ∥ 2 G / 2 } for any t ∈ R , and

Original: P( | (cid:80) ni =1 X i | ≥ s ) ≤ 2 exp {
Revised: P

Original: [Equation]
Revised: [Equation]

Original: s ≥

Original: (b) .
Revised: (b) .

Original: If X i is not symmetric, then Eexp { tX i } ≤ exp { (17 / 12) t 

Revised: If X i is not symmetric, then Eexp { tX i } ≤ exp { (17 / 12) t

Original: ∥ X i ∥
Revised: ∥ X i ∥

Original: 2 G / 2 } for any t ∈ R , and
Revised: 2 G / 2 } for any t ∈ R , and

Original: [Equation]
Revised: [Equation]

Original: Theorem 1(a) is an existing result in Theorem 2.6 of Wainwright (2019).
Revised: Theorem 2(a) is an existing result in Theorem 2.6 of Wainwright (2019).

Original: For Theorem 1(b), we obtain (cid:112) 17 / 

Revised: For Theorem 2(b), we obtain (cid:112) 17 /

Original: ≈ 1 . 19 , while Lemma 1.5 in Buldygin & Kozachenko (2000) obtained Eexp { tX i } ≤ exp (cid:8) t 2 2 ( 4 √ 3 . 

Revised: ≈ 1 . 19 , while Lemma 1.5 in Buldygin & Kozachenko (2000) obtained Eexp { tX i } ≤ exp (cid:8) t 2 2 ( 4 √ 3 .

Original: ∥ X i ∥ G ) 2 (cid:9) for t ∈ R with 4 √ 3 .
Revised: ∥ X i ∥ G ) 2 (cid:9) for t ∈ R with 4 √ 3 .

Original: 1 ≈ 1 .
Revised: 1 ≈ 1 .

Original: A technical reason for this improvement is that ∥ · ∥ G does not need Stirling’s approximation for attaining the sharper MGF bounds when expanding the exponential function by Taylor’s formula.

Original: In Appendix C, we gives some comparisons in Figure 6 with σ 2 opt ( X ) , 17 ∥ X ∥ 2 G / 12 , 2 e ∥ X ∥ 2 ψ 2 and Var X in the tail bound constants of

Original: Table 1, to show the tightness of Theorem 1(b), when X is Bernoulli or Beta distribution.

Original: Before estimating ∥ X ∥ G, the first step is to verify X is indeed sub-G given its i.i.d. copies { X i } ni =1 . Corollary 7.2 (b) in Zhang & Chen (2021) shows [Equation]
Revised: Essentially, (cid:112) 17 / 12 > 1 appears for asymmetric variables, since ∥ · ∥ G is defined by comparing a Gaussian variable G that is symmetric.

Original: which implies max 1 ≤ i ≤ j X i

Original: = O P ( √ log j ) . Moreover, we will show the above rate is indeed sharp for  a class of unbound sub-G r.v.s characterized by the lower intrinsic moment norm below. Definition 3 (Lower intrinsic moment norm) . The lower intrinsic moment norm for a sub-G X is defined as
Revised: A technical reason for this improvement is that ∥ · ∥ G does not need Stirling’s approximation for attaining a sharper MGF bound when expanding the exponential function by

Original: [Equation] By the method in Theorem 1 of Zhang & Zhou (2020), we obtain the following the tight rate result with a lower bound. Theorem 2. (a).
Revised: Taylor’s formula.

Original: If ∥ X ∥ ˜ G > 0 for i.i.d.

Original: symmetric sub-G r.v.s { X i } ni

Original: =

Original: ∼ X , then with probability at least 1 − δ

Original: [Equation] [Equation] where C ( X ) < 1 is constant defined in Lemma 2 below; (b) if X is bounded variable, then ∥ X ∥˜ G = 0 . [Figure]
Revised: To show the tightness of Theorem 2(b), in Figure 5 of Appendix C, we gives some comparisons with σ opt ( X ) , (cid:112) 17 / 12 ∥

Original: The upper bound follows from the proof of (5) similarly.

Original: The proof of lower bound relies on the sharp reverse Chernoff inequality from Paley–Zygmund inequality (see Paley & Zygmund (1932)).

Original: Lemma 2 (A reverse Chernoff inequality) . Suppose ∥ X ∥˜ G > 0 for a symmetric sub-G r.v. X . For t > 0 , then [Equation]
Revised: X ∥ G , √ 2 e ∥ X ∥ ψ 2 , ∥

Original: where C ( X ) := (cid:16) [Equation] Theorem 1 of Zhang & Zhou (2020) does not optimize the constant in Paley–Zygmund inequality. In contrast, our
Revised: X ∥ w 2 / √ 2 and √ Var X in terms of confidence length in Table 1, when X is

Original: Lemma 2 has an optimal constant; see Appendix C for details.

Original: By Theorem 2, we propose a novel sub-Gaussian plot check whether i.i.d data { X i } ni =1 follow an unbound sub-G variable.

Original: Specifically, we plot the order statistics { max 1 ≤ i ≤ j X ∗ i } nj =1 on the plane coordinate axis, where x axis represents √ log j and y axis the value of max 1 ≤ i ≤ j X ∗ i .

Original: Here for each j , { X ∗ i } ji =1 are independently sampled from the empirical distribution F n ( x ) = 1 n (cid:80) ni

Original: =1 1( X i ≤ x ) of { X i } ni =1 . We check whether those points have a linear tendency: the more they are close to the tendency of a beeline, the more we can trust the data are sub-Gaussian. The following graphs show sub-Gaussian plot of N (0 , 1) and Exp(1) . It can be seen that sub-Gaussian plot of N (0 , 1) shows linear tendency, while Exp(1) shows quadratic tendency.
Revised: Bernoulli or beta distribution.

Original: For the quadratic tendency, we note that if { X i } ni = have heavier tails such as sub-exponentiality, then max 1 ≤ i ≤ j X i =

Original: O P (log j ) instead of the order O ( √ log j ) ; see

Original: Corollary 7.3 in Zhang & Chen (2021).

Original:  E STIMATION OF THE INTRINSIC MOMENT NORM
Revised:  E STIMATION OF THE INTRINSIC MOMENT NORM

Original: A first thought to estimate ∥ X
Revised: A first thought to estimate ∥ X

Original: ∥ G is by the plug-in approach.
Revised: ∥ G is by the plug-in approach.

Original: Although k X is proven to be finite in Lemma 1, its (possibly large) exact value is still unknown in practice.
Revised: Although k X is proven to be finite in Lemma 2, its (possibly large) exact value is still unknown in practice.

Original: Instead, we use a non-decreasing index sequence { κ n } to replace k X in the estimation.
Revised: Instead, we use a non-decreasing index sequence { κ n } to replace k X in the estimation.

Original: Hence, we suggest a plug-in feasible estimator
Revised: Hence, we suggest a plug-in feasible estimator

Original: [Equation]
Revised: [Equation]

Original: Deriving the non-asymptotic property of the (cid:100)
Revised: Deriving the non-asymptotic property of the (cid:100)

Original: ∥
Revised: ∥

Original: X [Equation] is not an easy task: the maximum point ˆ k ( κ n ) :=
Revised: X ∥ G is not an easy task: the maximum point ˆ k ( κ n ) :

Original:  arg max 1 ≤ k ≤ κ n [ 
Revised: = arg max 1 ≤ k ≤ κ n [ 1 (

Original: [Equation]
Revised: [Equation]

Original:  k ) will change with the sample size n even κ n is fixed.
Revised: 1 n (cid:80) ni =1 X 2 k i ] 1 / (2 k ) will change with the sample size n even κ n is fixed.

Original: To resolve this, we first examine the oracle estimator defined asfollows [Equation] It is trivial to show that (cid:103)
Revised: To resolve this, we first examine the oracle estimator defined as (cid:103)

Original: ∥
Revised: ∥

Original: X ∥ G a.s. −−→
Revised: X ∥ G = (cid:104)

Original: ∥ X∥ G .
Revised: [Equation] X .

Original: Hence, based on Orlicz norm ∥
Revised: Here, based on Orlicz norm ∥

Original: Y ∥ ψ θ := inf { t > 0 : E exp {| Y | θ /t θ } ≤ 2 } of sub-Weibull r.v.
Revised: Y ∥ ψ θ := inf { t > 0 : E exp {| Y | θ /t θ } ≤ 2 } of sub-Weibull r.v.

Original: Y with θ > 0 (Hao et al., 2019; Zhang & Wei, 2022),  the following result gives the nonasymptotic concentration of (cid:103) ∥ 
Revised: Y with θ > 0 (Hao et al., 2019; Zhang & Wei, 2022), we present the non-asymptotic concentration of (cid:103) ∥ X ∥ G around it ture value ∥

Original: X ∥ G .
Revised: X ∥ G .

Original: Proposition 1.
Revised: Proposition 1.

Original: Suppose { X i } ni =

Revised: Suppose { X i } ni =

Original: i .
Revised: i .

Original: i .
Revised: i .

Original: d . ∼ X and X satisfies ∥
Revised: d . ∼ X and X satisfies ∥

Original: X ∥
Revised: X ∥

Original: ψ 1 /kX < ∞ , then for any t > 0 ,
Revised: ψ 1 /kX < ∞ , then for any t > 0 ,

Revised: P (cid:18)(cid:12) (cid:12)(cid:12) (cid:103)

Original: [Equation]
Revised: [Equation]

Original: where the constant γ ≈ 1 . 78 , and the constant functions C ( · ) and A ( · ) are defined in Appendix C.
Revised: where the constant γ ≈ 1 . 78 , and the constant functions C ( · ) and A ( · ) are defined in Appendix C.

Original: The exponential-moment condition ∥
Revised: The exponential-moment condition ∥

Original: X ∥ ψ 1 /kX < ∞ is too strong for the error bound of (cid:103) ∥
Revised: X ∥ ψ 1 /kX < ∞ is too strong for the error bound of (cid:103) ∥

Original: X ∥ 2 k X G − ∥
Revised: X ∥ 2 k X G − ∥

Original: X ∥ 2 k X G in
Revised: X ∥ 2 k X G in

Original: Proposition 1, although it has exponential decay probability 1 − 2exp( − t ) .
Revised: Proposition 1, although it has exponential decay probability 1 − 2exp( − t ) .

Original: Except for the direct plug-in estimator,for estimators that are robust to outliers, here we resort to the median-of-means (MOM, Page244 in Nemirovskij & Yudin (1983))  plug-in estimator of intrinsic moment norm.
Revised: Except for the direct plug-in estimator, here we resort to the median-of-means (MOM, Page244 in Nemirovskij & Yudin (1983)) as the robust plug-in estimator of intrinsic moment norm.

Original: Let m
Revised: Let m

Original: and b be a positive integer such that n = mb and let B 1 , . . .
Revised: and b be a positive integer such that n = mb and let B 1 , . . .

Original: , B b be a partition of [ n ] into blocks of equal cardinality m .
Revised: , B b be a partition of [ n ] into blocks of equal cardinality m .

Original: For any s ∈ [ b ] , let P B s m X =
Revised: For any s ∈ [ b ] , let P B s m X =

Original: m
Revised: m

Original: − 1 (cid:80) i ∈
Revised: − 1 (cid:80) i ∈

Original: B s X
Revised: B s X

Original: i for independent data { X i } ni =1 .
Revised: i for independent data { X i } ni =1 .

Original: The MOM version intrinsic moment norm estimator is defined as
Revised: The MOM version intrinsic moment norm estimator is defined as

Original: [Equation]
Revised: [Equation]

Original: As stated in Proposition 1, the naive plug-in estimator (cid:100)
Revised: As stated in Proposition 1, the naive plug-in estimator (cid:100)

Original: ∥
Revised: ∥

Original: X ∥ G = (cid:100) ∥
Revised: X ∥ G = (cid:100) ∥

Original: X ∥ 1 ,G is not robust.
Revised: X ∥ 1 ,G is not robust.

Original: MOM estimators (8) with b ≫ 1 have two merits: (a) it only needs finite moment conditions, but the exponential concentration bounds are still achieved; (b) it permits some outliers in the data.
Revised: MOM estimators (7) with b ≫ 1 have two merits: (a) it only needs finite moment conditions, but the exponential concentration bounds are still achieved; (b) it permits some outliers in the data.

Original: Non-asymptotic inferences require to bound for ∥ X ∥ G exactly by a feasible estimator (cid:100)
Revised: Non-asymptotic inferences require to bound for ∥ X ∥ G exactly by a feasible estimator (cid:100)

Original: ∥
Revised: ∥

Original: X ∥ b,G up to a sharp constants.
Revised: X ∥ b,G up to a sharp constants.

Original: Next, we establish a high-probability upper bound for the estimated norm, if the data has finite moment assumptions as follows.
Revised: Next, we establish a high-probability upper bound for the estimated norm, if the data has O ∪ I outlier assumptions as follows.

Original: [Figure]
Revised: • (M.1) Suppose that the data { X i } ni =1 contains n − n o inliers { X i }

Original: s t
Revised: i ∈

Original: i m a t ed
Revised: I drawn i.i

Original: [Figure]
Revised: d. according to a target distribution, and there are no distributional assumptions on n o outliers { X i } i ∈

Original: • (M.1) Suppose √ Var X 2 k ≤ σ kk , 1 ≤ k ≤ κ n for a sequence { σ k } k ≥ 1 ; • (M.2) 
Revised: O ; • (M.2) b = b O + b S , where b O is the number of blocks containing at least one outliers and b S is the number of sane blocks containing no outliers .

Original: Define g k,m ( σ k ) and ¯ g k,m ( σ k ) as the sequences tending to zero as m → ∞ [Equation] [Equation]
Revised: Let ε := n o /n be the fraction of the outliers and n o b < 12 . Assume here exists a fraction function η ( ε ) for sane block such that b S ≥ η ( ε ) b for a function

Original: E
Revised: η

Original: [Equation] [Equation]
Revised: ( ε ) ∈ (0 , 1] .

Original: k
Revised: To serve for error bounds in the presence of outliers, (M.2) considers the specific fraction function of the polluted inputs; see Laforgue et al.

Original: [Equation] 1 + g  k [Equation] m ( σ k )  for any m ∈ N and 1 ≤ k ≤ κ n respectively .
Revised: Define g k,m ( σ k ) and ¯ g k,m ( σ k ) as the sequences for any m ∈ N and 1 ≤ k ≤ κ n : g k,m

Original: [Equation]
Revised: [Equation]

Original: 1) is due to Chebyshev’s inequality applying to the empirical mean of sub-sample.
Revised: [Equation]

Original: (M.2) is technical moment conditions in the proof, and the remainder terms in (9), (10) are from Taylor expansions.
Revised: (8) and g k,m ( σ k ) : = [2[ m/η ( ε )] − 1 / 2 σ kk / (E X 2 k ) + 1] 1 / (2 k ) − 1 .

Original: Below, the result concerns for estimated norm bound for finite sample size.
Revised: We obtain a robust and non-asymptotic CI for ∥ X ∥ G .

Original: Theorem 3 (Guaranteed coverage) .
Revised: Theorem 3 (Finite sample guaranteed coverage) .

Original: Under (M.1), (M.2) with (9) , we have for  κ n ≥ κ X
Revised: Suppose √ Var X 2 k ≤ σ kk for a sequence { σ k } κ n k =1 , we have

Original: [Equation]
Revised: [Equation]

Original: and P {∥ X ∥
Revised: and P {∥ X ∥

Original: G ≥ [1 + max
Revised: G ≥ [1 + max

Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: g
Revised: e

Original: [Equation]
Revised: [Equation]

Original: [Equation]

Original: [Equation]
Revised: [Equation]

Original: (cid:100) ∥ X ∥ b,G } > 1 − κ n · exp( − b/ 8) by (M.1), (M.2) with (10) .
Revised: 2 for κ n ≥ κ X under (M.1-M.2).

Original: Theorem 3 ensures the consistency of the estimator (cid:100)
Revised: Theorem 3 ensures the concentration of the estimator (cid:100)

Original: ∥
Revised: ∥

Original: X ∥ b,G when κ n ≥ k X under enough sample.
Revised: X ∥ b,G when κ n ≥ k X under enough sample.

Revised: If η ( ε ) = 1 with ε = 0 , then the data are i.i.d., which have no outlier, and outlier assumptions in M.1-M.2 can be dropped in Theorem 3.

Original: When the data is i.i.d.
Revised: When the data is i.i.d.

Original: Gaussian vector, Proposition 4.1 in Auer et al.
Revised: Gaussian vector, Proposition 4.1 in Auer et al.

Original: (2002) also gave a high-probability estimated upper bound for ℓ p -norm of the vector of Gaussian standard deviations, our result is for intrinsic moment norm.
Revised: (2002) also gave a high-probability estimated upper bound for ℓ p -norm of the vector of Gaussian standard deviations, our result is for intrinsic moment norm.

Original: In practice, the block number b can be taken by the adaptation method based on the Lepski method (Depersin & Lecu´e, 2022).
Revised: In practice, the block number b can be taken by the adaptation method based on the Lepski method (Depersin & Lecu´e, 2022).

Original: To guarantee high probability events in Theorem 3, it is required that the index sequence κ n should not be very large for fixed b .
Revised: To guarantee high probability events in Theorem 3, it is required that the index sequence κ n should not be very large for fixed b .

Original: The larger κ n needs larger b in blocks B 1 , . .
Revised: The larger κ n needs larger b in blocks B 1 , . .

Original: . , B b .
Revised: . , B b .

Original: In the simulation, we will see that an increasing index sequence κ n with slow rate will lead a good performance.
Revised: In the simulation, we will see that an increasing index sequence κ n with slow rate will lead a good performance.

Original: Here, we compare our two estimators (6) and (8), as well as the estimator (4).
Revised: Finally, we compare our two estimators (6) and (7), as well as the estimator (4) in Figure 3.

Original: We consider the standard Gaussian and Rademacher variable distributed X , in the two case we have ∥ X ∥ 2 G = σ 2 opt ( X ) =
Revised: We consider the standard Gaussian and Rademacher variable distributed X , in the two case we have ∥ X ∥ 2 G = σ 2 opt ( X ) =

Original: Var( X ) = 1 .
Revised: Var( X ) = 1 .

Original: The following figure shows the performance of three estimators under sample n = 10 to 1000 with κ n just chosen as ⌈ log n ⌉ .
Revised: The following figure shows the performance of three estimators under sample n = 10 to 1000 with κ n just chosen as ⌈ log n ⌉ .

Original: For the MOM method, we use five blocks in this simple setting.
Revised: For the MOM method, we use five blocks in this simple setting.

Original: For a more complex case, one can use Lepski’s method to choose b (see Page & Gr¨unew¨alder (2021)), but some considerable computation cost may be introduced.
Revised: For a more complex case, one can use Lepski’s method to choose b (see Page & Gr ¨ unew ¨ alder (2021)), but some considerable computation cost may be introduced.

Original: From Figure 3, we know that the performance of the MOM estimator is best, while the naive estimator (4) is worst.
Revised: From Figure 3, we know that the performance of the MOM estimator is best, while the naive estimator (4) is worst.

Original: For the highquality data of extremely small sample size, we can apply the leave-one-out Hodges-Lehmann method (Rousseeuw & Verboven, 2002)in the estimation for further numerical improvement, and the details can be seen in Appendix B.
Revised: For the high-quality data of extremely small sample size, we can apply the leave-one-out Hodges-Lehmann method (Rousseeuw & Verboven, 2002) for further numerical improvement; see Appendix B for details.

Original:  A PPLICATION IN M ULTI - ARMED B ANDIT P ROBLEM
Revised:  A PPLICATION IN M ULTI - ARMED B ANDIT P ROBLEM

Original: In the multi-armed bandit problem (MAB), a player chooses between K different slot machines (an K -armed bandit), each with a different unknown random reward r.v.s { Y k }
Revised: In the multi-armed bandit problem (MAB), a player chooses between K different slot machines (an K -armed bandit), each with a different unknown random reward r.v.s { Y k }

Original: Kk =1 ⊆ R , while each realization of a fixed arm k is independent and shares the same distribution, i.e., the rewards from each arm are i.i.d. sample.
Revised: Kk =1 ⊆ R , while each realization of a fixed arm k is independent and shares the same distribution.

Original: We assume the rewards are sub-Gaussian, i.e.
Revised: Further, we assume the rewards are sub-Gaussian, i.e.

Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: Our goal is to find the best arm with the largest expected reward, say Y t ∗ , by pulling arms.
Revised: Our goal is to find the best arm with the largest expected reward, say Y t ∗ , by pulling arms.

Original: In each round t ∈ [ T ] , the player pulls an arm (an action)
Revised: In each round t ∈ [ T ] , the player pulls an arm (an action)

Original: A t ∈ [ K ] .
Revised: A t ∈ [ K ] .

Original: Conditioning on { A t = k } , we define the observed reward { Y k,t } t ∈
Revised: Conditioning on { A t = k } , we define the observed reward { Y k,t } t ∈

Original: [ T ] i .
Revised: [ T ] i .

Original: i .
Revised: i .

Original: d .
Revised: d .

Original: ∼
Revised: ∼

Original: P k .
Revised: P k .

Original: The goal of the exploration in MAB is to minimize the cumulative regret after T steps: 
Revised: The goal of the exploration in MAB is to minimize the cumulative regret after T steps: [Figure] t i

Original:  Reg T ( Y, A ) :
Revised: [Figure] t [Figure] Reg T ( Y, A ) :

Original: = (cid:80) Tt =1 ( µ t ⋆ − µ A t ) , i.e. the performance of any exploration strategy { A t } t ∈ [ T ] .
Revised: = (cid:80) Tt =1 ( µ t ⋆ − µ A t ) , i.e. the performance of any exploration strategy { A t } t ∈ [ T ] .

Original: The exploration performance is better, if we have smaller Reg T ( Y, A ) .
Revised: The exploration performance is better, if we have smaller Reg T ( Y, A ) .

Original: Without loss of generality, we assume t ⋆ = 1 .
Revised: Without loss of generality, we assume t ⋆ = 1 .

Original: Since Reg T ( Y, A ) is a random quantity, we seek to evaluate the expected bounds from the decomposition (see Lemma 4.5 in Lattimore & Szepesv ´ ari (2020)),
Revised: We seek to evaluate the expected bounds from the decomposition (see Lemma 4.5 in Lattimore & Szepesv ´ ari (2020)),

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: where E is taken on the randomness of the player’s actions { A t } t ∈ [ T ] , and ∆ k = µ 1 − µ k is the sub-optimality gap for arm k ∈
Revised: where E is taken on the randomness of the player’s actions { A t } t ∈ [ T ] , and ∆ k = µ 1 − µ k is the sub-optimality gap for arm k ∈

Original: [ K ] / { 1 } .
Revised: [ K ] / { 1 } .

Original: The upper bound of Reg T is called problem-independent if the regret bound depends on the distribution of the data and does not rely on the gap ∆ k .
Revised: The upper bound of Reg T is called problem-independent if the regret bound depends on the distribution of the data and does not rely on the gap ∆ k .

Original: For each iteration t , let T k ( t ) :
Revised: For each iteration t , let T k ( t ) :

Original: = card { 1 ≤ τ ≤ t : A τ = k } be the number of pull for arm k until time t during the bandit process.
Revised: = card { 1 ≤ τ ≤ t : A τ = k } be the number of pull for arm k until time t during the bandit process.

Original: Then if we define Y T k ( t ) :
Revised: Then if we define Y T k ( t ) :

Original: = 1 T k ( t ) (cid:80) τ ≤ t,A τ = k Y k,τ as the running average of the rewards of arm k at time t .
Revised: = 1 T k ( t ) (cid:80) τ ≤ t,A τ = k Y k,τ as the running average of the rewards of arm k at time t .

Original: Suppose we obtain a 100(1 − δ )% CI (cid:2)
Revised: Suppose we obtain a 100(1 − δ )% CI (cid:2)

Original: Y T k ( t ) − c k ( t ) , Y T k ( t )
Revised: Y T k ( t ) − c k ( t ) , Y T k ( t )

Original: + c k ( t ) (cid:3) for µ k from a tight concentration inequality, where c k ( t ) decreases with time t .
Revised: + c k ( t ) (cid:3) for µ k from a tight concentration inequality.

Original: Therefore, we confidently reckon that the reward of arm k is Y T k ( t )
Revised: Therefore, we confidently reckon that the reward of arm k is Y T k ( t )

Original: + c k ( t ) , and play the arm
Revised: + c k ( t ) , and play the arm

Original: A t =
Revised: A t =

Original: k , hoping to maximize the reward with a high probability for finite t .
Revised: k , hoping to maximize the reward with a high probability for finite t .

Original: This is upper confidence bound (UCB, Auer et al.
Revised: This is upper confidence bound (UCB, Auer et al.

Original: algorithms.
Revised: algorithms.

Revised: And many works based on this methods appears recently, for example, Hao et al.

Revised: (2019) use bootstrap method with the second order correction to give a algorithm with the explicit regret bounds for subGaussian rewards.

Revised: However, many existent algorithms contain unknown norms for the random rewards, they are actually infeasible.

Revised: And Theorem 4 is one example with explicit regret bound.

Revised: For instance, the algorithm Hao et al.

Original: In details, suppose that Y k − µ k is symmetric around zero, by one-side version of Theorem 1, the (11) implies that for all k and all t , P( Y T k ( t ) > µ k + ∥
Revised: (2019) needs to use the unknown Orlicz-norm of Y k − µ k in the algorithm. Thus, it is actually infeasible in practice. Fortunately, our estimator can solve this problem. Suppose that Y k − µ k is symmetric around zero, by one-side version of Theorem 2, the (9) implies that for all k and all t , P( Y T k ( t ) > µ k + ∥

Original: Y k − µ k ∥ G (cid:113) 2 T k ( t ) log 1 δ ) ≤ δ .
Revised: Y k − µ k ∥ G (cid:113) 2 T k ( t ) log 1 δ ) ≤ δ .

Original: Let sub-sample size m k and block size b k be positive integer such that T k ( t )
Revised: Let subsample size m k and block size b k be positive integer such that T k ( t )

Original: = m
Revised: = m

Original: k b k for MOM estimators (cid:92) ∥
Revised: k b k for MOM estimators (cid:92) ∥

Original: Y k − µ k ∥ b k ,G in Section 3.
Revised: Y k − µ k ∥ b k ,G in Section 3.

Original: Theorem 3 (a) guarantee that true norms can be replaced by MOM-estimated norms such that 
Revised: Theorem 3 (a) guarantee that true norms can be replaced by MOM-estimated norms such that P( Y T k ( t ) ≤ µ k +

Original: [Equation]
Revised: [Equation]

Original: If the UCB algorithm is correctly applied, for a finite T k ( t ) , with high probability, we will pull the best arm.
Revised: If the UCB algorithm is correctly applied, for a finite T k ( t ) , with high probability, we will pull the best arm.

Original: In practice, we nearly do not know any knowledge about the data.
Revised: In practice, we nearly do not know any knowledge about the data.

Original: As a flexible way of uncertainty qualification, the multiplier bootstrap (Arlot et al., 2010) enables mimicking the non-asymptotic properties of the target statistic by reweighing its summands of the centralized empirical mean.
Revised: As a flexible way of uncertainty qualification, the multiplier bootstrap (Arlot et al., 2010) enables mimicking the non-asymptotic properties of the target statistic by reweighing its summands of the centralized empirical mean.

Original: The multiplier bootstrapped quantile for the i.i.d.
Revised: The multiplier bootstrapped quantile for the i.i.d.

Original: observation Y n := { Y i } ni
Revised: observation Y n := { Y i } ni

Original: =1 is the (1 − α ) -quantile of the distribution of n − 1 (cid:80) ni
Revised: =1 is the (1 − α ) -quantile of the distribution of n − 1 (cid:80) ni

Original: =

Revised: =

Original: w i
Revised: w i

Original: ( Y i − Y n ) , which is defined as
Revised: ( Y i − Y n ) , which is defined as

Original: [Equation]
Revised: [Equation]

Original:  where w := { w i } ni =1 are bootstrap random weights independent of Y n .
Revised: , where w := { w i } ni =1 are bootstrap random weights independent of Y n .

Original: We denote the statistics (cid:98) φ G ( Y n ) as something satisfying [Equation] Algorithm 1:
Revised: We denote the statistics (cid:98) φ G ( Y n ) as something satisfying P Y n ( | Y n − E Y 1 | ≥ (cid:98) φ G ( Y n )) ≤ α .

Original: Input: (cid:98) φ G ( Y T k ( t ) ) is given by (13).
Revised: Input: (cid:98) φ G ( Y T k ( t ) ) is given by (11).

Original: for t = 1 , . . .
Revised: for t = 1 , . . .

Original: K do
Revised: K do

Original: Pull each arm once to initialize the algorithm.
Revised: Pull each arm once to initialize the algorithm.

Original: endfor t = K + 1 , . .
Revised: endfor t = K + 1 , . .

Original: , T do
Revised: , T do

Original: Set a confidence level α ∈ (0 , 1) .
Revised: Set a confidence level α ∈ (0 , 1) .

Original: Calculate the boostrapped quantile q α/ 2 ( Y T k ( t ) − Y T k ( t ) , w ) with the Rademacher bootstrapped weights w independent with any Y .
Revised: Calculate the boostrapped quantile q α/ 2 ( Y T k ( t ) − Y T k ( t ) , w ) with the Rademacher bootstrapped weights w independent with any Y .

Original: Pull the arm
Revised: Pull the arm

Original: [Equation]
Revised: [Equation]

Original: Receive reward Y A t .
Revised: Receive reward Y A t .

Original: end
Revised: end

Original: Motivated by Hao et al.
Revised: Motivated by Hao et al.

Original: (2019), we design Algorithm 1 based on some estimators of the UCB.
Revised: (2019), we design Algorithm 1 based on some estimators of the UCB.

Original: It guarantees a relatively small regret by bootstrapped threshold q α/ 2 ( Y T k ( t ) − Y T k ( t ) , w ) adding a concentration based second-order correction (cid:98) φ G ( Y T k ( t ) ) that is specified in Theorem 3.
Revised: It guarantees a relatively small regret by bootstrapped threshold q α/ 2 ( Y T k ( t ) − Y T k ( t ) , w ) adding a concentration based second-order correction (cid:98) φ G ( Y T k ( t ) ) that is specified in Theorem 3.

Original: In the following regret bounds, we assume the mean reward from the k -th arm µ k is known.
Revised: In the following regret bounds, we assume the mean reward from the k -th arm µ k is known.

Original: In practice, it can be replaced by a robustestimator such as the MOM estimator, and we obtain the following results.
Revised: In practice, it can be replaced by a robust estimator, and we obtain the results of MOM estimator.

Original: Theorem 4.
Revised: Theorem 4.

Original: Consider a K -armed sub-G bandit under (11) and suppose that Y k − µ k is symmetric around zero.
Revised: Consider a K -armed sub-G bandit under (9) and suppose that Y k − µ k is symmetric around zero.

Original: For any round T , according to moment conditions in Theorem 3, choosing (cid:98) φ G ( Y T k ( t ) ) as
Revised: For any round T , according to moment conditions in Theorem 3, choosing (cid:98) φ G ( Y T k ( t ) ) as

Original: [Equation]
Revised: [Equation]

Original:  as a re-scaled version of MOM estimator (cid:92) ∥
Revised: (11) as a re-scaled version of MOM estimator (cid:92) ∥

Original: Y k − µ k ∥ b k ,G with block number b
Revised: Y k − µ k ∥ b k ,G with block number b

Original: k satisfying the moment assumptions
Revised: k satisfying the moment assumptions

Original: C[UCB1] and C[UCB2] in Appendix C. Fix a confidence level α = 4 /T 2 , if the player pull an arm A t ∈ [ K ] according to Algorithm 1, then we have the problem-dependent regret of Algorithm 1 is bounded by
Revised: C[UCB1] and C[UCB2] in Appendix C. Fix a confidence level α = 4 /T 2 , if the player pull an arm A t ∈ [ K ] according to Algorithm 1, then we have the problem-dependent regret of Algorithm 1 is bounded by

Original: [Equation]
Revised: [Equation]

Original: where ∆ k is the sub-optimality gap.
Revised: where ∆ k is the sub-optimality gap.

Original: Moreover, let µ ∗ 1 :=
Revised: Moreover, let µ ∗ 1 :=

Original: max k 1 ∈ [ K ] µ k 1 − min k 2 ∈
Revised: max k 1 ∈ [ K ] µ k 1 − min k 2 ∈

Original: [ K ] µ k 2 be the range over the rewards, the problem-independent regret
Revised: [ K ] µ k 2 be the range over the rewards, the problem-independent regret

Original: [Equation]
Revised: [Equation]

Original: From Theorem 4, we know that the regret of our method achieve minimax rate log T for a problem-dependent problem and √ KT for a problem-independent case (see Tao et al.
Revised: From Theorem 4, we know that the regret of our method achieve minimax rate log T for a problem-dependent problem and √ KT for a problem-independent case (see Tao et al.

Original: , so Algorithm 1 can be seen as an optimal algorithm.
Revised: , so Algorithm 1 can be seen as an optimal algorithm.

Revised: Compared with the traditional vanilla UCB, we do improve the constant.

Revised: When Y k ∼ N ( µ k , 1) , the constant factor in regret bound in Auer et al. (2002) is 256 , which is larger than 16(2 + √ 2) 2 in our theorem.

Revised: When the UCB has unknown sub-G parameters, Theorem 4 first studies a feasible UCB algorithm with sub-G parameter plugging estimation.

Revised: Many previous UCB algorithms based on non-asymptotic inference in the literature assume that the sub-G parameter is a preset constant, see the algorithm in Hao et al.

Original:  Next, we give an simulation for Theorem 4 in two sub-G cases to verify the performance of estimated norms.
Revised: for instance. Next, we give an simulation for Theorem 4 in two sub-G cases to verify the performance of estimated norms.

Original: Similar to Hao et al.
Revised: Similar to Hao et al.

Original: ; Wang et al.
Revised: ; Wang et al.

Original: (2020), we design the three methods as follows:
Revised: (2020), we design the three methods as follows:

Original: Use our method (cid:98) φ ( Y T k ( t ) ) with Estimated Norm in Theorem 4; 
Revised: Use our method (cid:98) φ ( Y T k ( t ) ) with Estimated Norm in Theorem 4; 2.

Original: Use Asymptotic Naive varphi (cid:101) φ ( Y T k ( t ) ) satisfying P (cid:0)
Revised: Use Asymptotic Naive varphi (cid:101) φ ( Y T k ( t ) ) satisfying P (cid:0)

Original: | Y T k ( t ) − µ k | ≤ (cid:101) φ ( Y T k ( t ) ) (cid:1) → α by CLT, i
Revised: | Y T k ( t ) − µ k | ≤ (cid:101) φ ( Y T k ( t ) ) (cid:1) → α by CLT, i

Original: e. (cid:101) φ ( Y T k ( t ) )
Revised: e. (cid:101) φ ( Y T k ( t ) )

Original: = (cid:98)
Revised: = (cid:98)

Original: σ k Φ − 1 (1 − α/ 2) / (cid:112) T k ( t ) with (cid:98) σ k = (cid:113) 1 T k ( t ) (cid:80) τ ≤ t,
Revised: σ k Φ − 1 (1 − α/ 2) / (cid:112) T k ( t ) with (cid:98) σ k = (cid:113) 1 T k ( t ) (cid:80) τ ≤ t,

Original: A τ = k ( Y k,τ − Y T k ( t ) ) 2 as the estimated standard deviation; 3.
Revised: A τ = k ( Y k,τ − Y T k ( t ) ) 2 as the estimated standard deviation; 3.

Original: Regard all the unbounded rewards as bounded r.v. and use Hoeffding’s inequality ( wrongly use Hoeffding’s inequality ) to construct φ , i.e. ˇ φ ( Y T k ( t ) )
Revised: Regard all the unbounded rewards as bounded r.v. and use Hoeffding’s inequality ( wrongly use Hoeffding’s inequality ) to construct φ , i.e. ˇ φ ( Y T k ( t ) )

Original: = (cid:2) max { Y T k ( t ) }
Revised: = (cid:2) max { Y T k ( t ) }

Original: − min { Y T k ( t ) } (cid:3)(cid:113) log(2 /α ) 2 T k ( t ) .
Revised: − min { Y T k ( t ) } (cid:3)(cid:113) log(2 /α ) 2 T k ( t ) .

Original: And for our detailed MAB simulation, we consider as follows, in each case, the number of arms is assigned as K = 5 , and ( µ 1 , . . .
Revised: And for our detailed MAB simulation, we consider as follows, in each case, the number of arms is assigned as K = 5 , and ( µ 1 , . . .

Original: , µ 5 ) =
Revised: , µ 5 ) =

Original: 02 , 0 . 01 , 0 . 01) ⊤ ,
Revised: 02 , 0 . 01 , 0 . 01) ⊤ ,

Original: [Equation]
Revised: EG1 .

Original:  µ 
Revised: Y k ∼ N ( µ k , µ 2 k ); EG2 . Y k ∼ 0 . 9 N ( µ k , µ k ) + 0 .

Original: [Equation] where µ 6 := µ 5 .
Revised: 1 N ( µ k +1 , µ k +1 ) where µ 6 := µ 5 .

Original: [Figure]
Revised: [Figure]

Original: As we can see EG1 and EG2 are both sub-Gaussian rewards.
Revised: As we can see, EG1 and EG2 are both sub-Gaussian rewards.

Original: In the simulation, µ k is assigned sightly small for bounded max k ∆ k , which is a standard-setting in the MAB problem (see the condition of Corollary 1 in Wang et al.
Revised: In the simulation, µ k is assigned sightly small for bounded max k ∆ k , which is a standard-setting in the MAB problem (see the condition of Corollary 1 in Wang et al.

Original: (2020) for instance).
Revised: (2020) for instance).

Original: The simulation results are shown in Figure  which illustrates that our method outperforms the other two methods under the unbounded sub-Gaussian rewards and small sample T ∈ [1 , 150] .
Revised: The simulation results are shown in Figure 4, which illustrates that our method outperforms the other two methods under the unbounded sub-Gaussian rewards and small sample T ∈ [1 , 150] .

Original: In the Gaussian case,
Revised: In the Gaussian case,

Original: Algorithm 1 can also give better results compared with the CLT-based UCB when the round is relatively small.
Revised: Algorithm 1 can also give better results compared with the CLT-based UCB when the round is relatively small.

Original: For the Gaussian mixture case, the algorithm based on the intrinsic moment norm has smaller regret than the other two methods under T ∈ [1 , 800] .
Revised: For the Gaussian mixture case, the algorithm based on the intrinsic moment norm has smaller regret than the other two methods under T ∈ [1 , 800] .