Optimal Best Arm Identification under Differential Privacy
Authors: Marc Jourdan
Achraf Azize
Venue: NeurIPS 2025 poster ()
Keywords: Bandit Algorithms
Differential Privacy
Best Arm Identification
Fixed-Confidence
Bernoulli Distribution
Primary area: theory
TL;DR: This paper reduces the gap between lower and upper bounds for fixed-confidence Best Arm Identification under global Differential Privacy to a small constant multiplicative factor by introducing a new lower bound and a near-optimal Top Two algorithm.

Abstract: Best Arm Identification (BAI) algorithms are deployed in data-sensitive applications, such as adaptive clinical trials or user studies. Driven by the privacy concerns of these applications, we study the problem of fixed-confidence BAI under global Differential Privacy (DP) for Bernoulli distributions. While numerous asymptotically optimal BAI algorithms exist in the non-private setting, a significant gap remains between the best lower and upper bounds in the global DP setting. This work reduces this gap to a small multiplicative constant, for any privacy budget $\epsilon$. First, we provide a tighter lower bound on the expected sample complexity of any $\delta$-correct and $\epsilon$-global DP strategy. Our lower bound replaces the Kullback–Leibler (KL) divergence in the transportation cost used by the non-private characteristic time with a new information-theoretic quantity that optimally trades off between the KL divergence and the Total Variation distance scaled by $\epsilon$. Second, we introduce a stopping rule based on these transportation costs and a private estimator of the means computed using an arm-dependent geometric batching. En route to proving the correctness of our stopping rule, we derive concentration results of independent interest for the Laplace distribution and for the sum of Bernoulli and Laplace distributions. Third, we propose a Top Two sampling rule based on these transportation costs. For any budget $\epsilon$, we show an asymptotic upper bound on its expected sample complexity that matches our lower bound to a multiplicative constant smaller than $8$. Our algorithm outperforms existing $\delta$-correct and $\epsilon$-global DP BAI algorithms for different values of $\epsilon$.

Optimal Best Arm Identification under Differential Privacy
Mare Jourdan' EPFL, Lausanne, Switzerland marc. marc. jourdn@ep...
Achraf Azize* FairPlay Joint Team, CREST, ENSAE Paris achraf azize@ensae.fr
Abstract
Best Arm Identification (BAI) algorithms deployed in data-sensitive applica- tions, such as adaptive clinical trials or user studies. Driven by the privacy concerns of these applications, we study the problem of fixed-confidence BAI under global Differential Privacy (DP) for Bernoulli distributions. While numerous asymptot- ically optimal BAI algorithms exist in the non-private setting, significant gap remains between the best lower and upper bounds in the global DP setting. This work reduces this gap a small multiplicative constant, for any privacy budget e. First, we provide a tighter lower bound on the expected sample complexity of any 5-correct and e-global DP strategy. Our lower bound replaces the Kullback- (KL) divergence in the transportation cost used by the non-private characteristic time with a new coormation-thhoredee quantity that optimally trades off between the KL divergence and the Total Variation distance scaled by Second, we intro- duce a stopping rule based on these transportation costs and a private estimator of the means computed using an arm-dependent geometric batching. En route to proving the correctness of our stopping rule, we derive concentration results independent interest for the Laplace distribution and for the sum of Bernoulli and Laplace distributions. Third, we propose a Top Two sampling rule based on these transportation costs. For any budget e, we show an asymptotic upper bound on its expected sample complexity that matches our lower bound a multiplicative constant smaller than 8. Our algorithm outperforms existing 5-correct and e-global DP BAI algorithms for different values
1 Introduction
The stochastic Multi- Arme Bandit (MAB) an interactive sequential decision-making model [Bubeck et al. 2012; Lattimore and Szepesvári 2020], introduced by William R. Thomp- son [Thompson]] 1933]. Thompson's 's motivation for studying MABs is to design clinical trials that adapt treatment allocations on the fly as the medicines appear more or less effective. Specifically, in MABs, learner interacts with K E N unknown probability distributions, referred as arms. In clinical trials, the arms are the candidate medicines, while the observations are patient reactions, 1 if the patient is cured and otherwise. The learner aims to identify the arm with the highest average efficiency, i.e., the medicine that cures most patients in expectation. Given fixed error å E (0, 1), Best Arm Identification (BAI) [Audibert and Bubeck] 2010b Jamieson and Nowak] 20144) algorithms in the fixed confidence setting [Even-arr et al. 2006 Gabillon et al. [2022] Garivier and Kaufmann 2016) suggest candidate answer that coincides with the optimal arm with probability more than I 8. while using as few samples possible.
BAI algorithms have been increasingly deployed in data-sensitive applications, such as adaptive clini- cal trials [Thompson] 1933 Robbins 1952 Aziz al, 2021), pandemic mitigation [Libin et al. 2019
Equal Contribution
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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user studies ct al.
Losada et al., 2022 crowdsourcing [hhou et al. 2014), online advertisement [Chen 2014]. hyperparameter tuning [Li et al. 2017 and communication networks (Lindstahl et al. 2022 to name few. Due to the adaptive nature of these procedures, critical data privacy concerns are raised (Tucker et al. 2016], as exemplified by the adaptive dose finding trial. For each new patient n, a physician chooses a dose level (n E [K] == (1.
K} based on previous observations, and collects a binary observation measuring the effect of the selected dose on the patient. Crucially, the patients' reactions might reveal information regarding their health. Subsequently, these outcomes will guide the physician's is decision for future patients. Eventually, the physician adaptively decides to stop the trial and recommends dose âr, after collecting т samples, referred to sample complexity. Even if those outcomes are kept secret, the experimental findings and protocol are detailed thoroughly to the health authorities. This report contains the sequence of chosen dose levels (e and the recommended dose level ār,, both indirectly leaking information regarding the patients involved in the trial. This example underscores the need for privacy-preservine fixed-confidence BAI algorithms.
We adopt the Differential Privacy (DP) framework [Dwork and Roth 2014), which bounds the influence of any single data point. Given privacy budget e, consider the e-global DP constraint trusted curator (e.g.. the physician running the clinical trial), who
that assumes the existence observes the outcomes and ensures privacy when publishing these findings. While e-global DP is well-studied in regret minimization (Mishra and Thakurta 2015; Azize and Basu, 2022
Azize et al.
2025), its impact on fixed-conte confidenne BAI is less understood [[ajed and Sheffet 2019
Kalogerias
et al.
2021 A significant gap remains between the existing lower and upper bounds [Azize et al.
2024 This paper reduces this gap to small constant for any privacy budget Appendix contains a detailed literature review.
2023
Contributions. Our contributions for fixed-confidence BAI under e-global DP are threefold.
Lower bound under global DP. We derive a novel information--theoreiee lower bound on the expected sample complexity of any 8-correct and e-global DP BAT algorithms (Theorem 2) Our lower bound replaces the Kullback-Leible (KL) divergence in the transportation cost of the non-private characteristic time with an information-theorett quantity de (Eq. (I) that smoothly interpolates between the KL divergence and the Total Variation (TV) distance scaled by e.
2. Private estimator and Generalized Likelihood Ratio (GLR) stopping rule. We introduce a private estimator using arm-dependent geometric batching without forgetting and GLR stopping rule based on the de refined transportation costs. Its correctness (Theorem (6) required novel tails concentration results for Laplace distributions and the sum of Bernoulli and Laplace distributions, which could be of independent interest.
3. Asymptotically optimal algorithm. We propose new Top Two sampling rule (DP-TT Al- gorithm | based on the d. traasspportaii costs suggested by our lower bound. We show that the asymptotic expected sample complexity DT- matches our lower bound for any privacy budget up to a constant smaller than 8 (Theorem 7) DP-TT outperforms all the other 8-correct e-global DP BAI algorithms on all tested instances and all
2 Background: Best Arm Identification under Differential Privacy
In this section, we present the Best Arm Identification (BAT) under fixed confidence problem [Garivier] and Kaufmaan|| 2016), introduce the Differential Privacy (DP) [Dwork et al.) 2006) constraint, and finally extend DP to BAI algorithms.
BAI under Fixed Confidence. F (Ber(p) E (0, 1)} be the set of Berooulii distributions. bandit instance == (Vo)ae[K] E JK is characterized by its means μ == (Ha)ne(K] E (0, 1)x The best (optimal) arm a* is assumed be unique, i.e., a" (v) a" (u) == arg max 14, = {ar) }. Let 8 E (0, 1) be the risk parameter. A fixed confidence BAI algorithm π specifies three rules that rely on previously observed samples and some exogenous randomness. The sampling rule determines the next arm to pull an E [K] for which Xn.a. Ve., is observed. The recommendation rule recommends candidate arm à E [K]. The stopping rule decides when to stop collecting additional samples and output the current candidate arm. The stopping time T,. is the sample complexity. Let Por and Eur denote the probability and expectation taken over the randomness of the observations from vs and the algorithm π (e.g., due to its privacy mechanism). is ò-correct when Pys (Te,6 < to0, à & a" (v)) < 8 for all v E FKK
fixed-coffidence BAI algorithm
2

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Differential Privacy (DP). An algorithm satisfies the Differential Privacy constraint if the algorithm's s outputs are "essentially" equally likely to occur, for any two input datasets that only differ in one individual's ' data. An adversary only observing the mechanism's 's output cannot distinguish whether any individual's s data was included. A privacy budget captures the closeness of the output distributions. Smaller e means stronger privacy.
Definition 1 (e-DP [Dwork et al. 2006)). A mechanism M satisfies e- c-D for a given e 2 O. if. for all neighboring datasets D ~ D', where D ~ D' if and only if d (D,D) D')==E- 1,D, + DJY <
1, i.e. D and D' differ by at most one record, for all sets of output O C Range(M), Pr(M(D) O S et Pr M (D') E O where the probability space is over the coin coinffips of the mechanism M.
To ensure e-DP, the Laplace mechanism [Dwork et al. 2010:
Dwork and Roth 2014] adds calibrated Laplacian noise to the algorithm's 's output. Let (b) be the Laplace distribution with mean/variance
(0, 26²).
Theorem 1 (Laplace mechanism, Theorem 3.6 [Dwork and Roth 2014 D. Let f X + Rd be an algorithm with sensitivity s(f) -
[[((D) -f f(D')ll >|l. where |l-I1, is the l norm.
max
D,D' at (D'=( then the noisy output ((D) (Zi)ketd) satisfies e-DP.
Let (Zi) be i.i.d. from
To be consistent with the literature on private bandits, we use global DP to denote the central DP model with trusted central decision maker. While the notation ō is standard in the d) relaxation of the (e. DDP constraint considered in this paper, we use d for the confidence parameter to be consistent with the literature on pure exploration problems.
DP for BAI, In BAI algorithms,, the private input is the observation dataset and the output is the recommended candidate arm ā and the sequence of sampled actions (an)n<rr.s until stopping Te,5- Let Rt = {ri,.... } be sequence of private observations. Given fixed sequence of observations Rt we denote by Pr(mT (Rt) = (T - 1, ā, (ai,..... .[]) the probability that the BAI algorithm π stops at step 7 + 1, recommending action ā and sampling actions (d1,- (T) when interacting with R. The randomisation in this probability comes only from the BAT algorithm's sampling, recommendation and stopping rules, whereas the observations are fixed. Then, a BAI algorithm is said to be e-global DP if, for every two neighboring sequences of observations R and R', and for every possible stopping time, recommendation and sampled actions (T + 1, (a1,.... Qr) dr)), we have that
Pr[x(R) = (T + 1, a, (ai....,, aT)) < e Pr[=((R) = (T + a. (ot...., aT))]
Main Goal: Design c- c-global DP 8-correct BAI algorithms, with the smallest sample complexity Te,5-
Notation. Let [x]]] := max(0. (0, min x, } } be the clipping operator to [0, 1]. Let 1 (-) be the indicator function. For two probability distributions IP and Q on the measurable space (2, 9), the Total Variation (TV) distance is TV (P Q == sup (A) ((A) and the Kullbac--eibler (KL) divergence
KL Q) = log
aq (w) dlP(w), when P < Q. and +oo otherwise. The KL divergence and
TV distance between two Bernoulli distributions with means (p, 4) E (0, 1)2 are the relative entropy denoted by kl, i.e., KL (Ber(p) || Ber(g)) = kl q) == p pog p/ + (1 - p) log [1 p) 2) and the absolute mean difference, i.e., TV (Ber(p) || Ber(g))
ip- ql. Let 4x == {w E R
2 0, -eE(K] Wa 1} be the probability simplex of dimension K - 1. For all E [K], let
Nn,a : Eme(n-1] 1 (a, = a) be the global pulling count of arm before time n.
3 Lower Bound on the Expected Sample Complexity
In order to be 5-correct, an algorithm has to be able to distinguish v from alternative instances with different best arms, i.c., an instance k E Alt(v) := {K JK ax()) + a a²()). On the other hand, being e-global DP forces an algorithm to have similar behaviour on similar instances. The interplay between the stochasticity of the bandit instance, controlled with the KL divergence, and the stochasticity of the privacy mechanism, controlled with the TV distance, is smoothly captured by
(e == inf (E TV (v || (p) + KL (v || [)) P
recently introduced by Azize et al. 20255 for e-global DP regret minimization. The de divergence measures the shortest two-parts path between the two distributions and k, by finding the best
3

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for all Fk with unique best arm. The inverse of the characteristic time T (v) defined as intermediate distribution & E F. The cost of moving from V to is measured with the TV distance rescaled by e, while is measured with the KL divergence when moving from (p to s.
The tension between the d-correct and e-global DP constraints yields the following non-asymptotic lower bound the expected sample complexity Euw [un[Te, for any algorithm on any instance v/, holding for any values of and 5.
Theorem Let (e, E R2 x (0, 1). For any algorithm π that is ô-correct and e-global DP on FK
Eun[Te,) 2 T. (v) log(11(33))
K
T(w)= = sup inf WEAK REAI(())
Σ
Wade (Vas ka)
Comments. (a) The characteristic time in the lower bound is the value of two-player zero-sum game between a MIN player, who plays instances R close of y is order to confuse the MAX player, allocation A to distinguish between and
who in order plays
(b) The erucial part in characteristic times similar to Eq. 2 finding the "right" measure capturing the "distinguishability" between instances. In the non-private lower bounds, this is captured by the KL divergence for parametric distributions [Garivier and Kaufmann] 2016) and by the Kinf (i.e., inf KL under mean constraint) for non-parametric distributions [Agrawal et al. 2020]. In the DP lower bounds of Azize et al. [2023], it is captured by min (KL, eVVj. Theorem 2 is captured by de (as in Eq. (1) that smoothly interpolates between KL and TV. Azize et al. (2025) recently introduced de for e-global DP regret minimization. Our results show that d, also tightly captures
the hardness of fixed-confidence BAI under e-global DP. Namely, our DP-TTPTTaaa algorrthtm achieves
matching upper bound when + 0 (up to a constant smaller than 8), for all instances with distinct means and all values of c.
(c) Azize et al. [0033] Theorem 2] provides lower bound on the sample com- plexity of any é-global d-correct algorithm, where the inverse characteristic time is Su iefte)) min (En W,, WL, (v, || Ka), 6e Σ W. TV (va ka))- The lower bound of Theorem is strictly tighter than that of Theorem 2 in [Azize al. 2023]. for all instances v and values of The reason is that de (P, Q) S min(KL(P., CTP
for any two distributions.
(d) The lower bound of Theorem 2 suggests the existence two privacy regimes, depending on the value of and the instance v. Specifically, when is big, de reduces to the KL, and we retrieve the classic non-private lower bound. On the other hand, as + O, de reduces to e TV, and the char-
acteristic time reduces to (T == (supwEAx inf infREAA((M) Exe WA TV (va ks)))
E J. where A. = ,== Ma for a e a* and Aa = minagaa A. This improves the high privacy regime lower bound of prior work by factor 6. Also, the value at which the privacy regimes change can be tightly specified, which we quantify for Bernoulli instances in the following.
Proof Sketch and Techniques. The proof uses the standard reduction hypothesis testing Garivier and Kaufmann 2016), using the data-processing inequality. The asymptotic techniques used by Azize al. 20255 for regret cannot be adapted for our non-asymptotic lower bound. Thus, new techniques are needed. The main technical novelty of the proof is a tighter quantification of the "similar" behaviour of DP mechanism when applied to stochastic datasets. Specifically, let M be an c-DP mechanism. Given two data-genee Sata-generating distributions P and Q, letting Mr.M (resp. My.M) M) be the marginal over outputs of the mechanism when the input dataset is generated through IP (resp. Q) then we show that
KL (MI Mc.M) < inf inf (ED,Dw<oh, [driam +.DD')))KKLLII)) Ce.L
where the first infimum is over all distributions LL on the input space, and the second infimum is an optimal transport problem over all couplings between and IL, where the cost the Hamming distance (introduced in Definition D. This bound of general interest could be applied to get tighter lower bounds any DP application using stochastic inputs. For product and bandit distributions, we solve the optimal transport using maximal couplings, where the Total Variation naturally appears,

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while keeping the first infimum unchanged, giving rise to the de quantity. Finally, plugging the new upper bound on the KL in the hypothesis reduction concludes the sample complexity lower bound proof. A detailed proof and discussion of all these claims is given in Appendix D
Transportation Costs Based Signed Divergences. In non-parametric BAI, the ordering between
the mean parameters is captured by signed Kinf [Jourdan et al. 2022]. We adopt this convention by introducing a signed divergences: d and de defined in Lemma on means rather than probability
distributions, where d. refers to both of them. Given (N, v) E J= with means (A, 14) E (0. 1)². they
satisfy de dx,n, v/ d (A. pd) when μ > X, and de v) = d (A. μi) otherwise (Lemma 22
Lemma For all x E [0, 11, let us define 9. (z) := z(cs-1)+1 and 9²" (x) == 9r (1 2)
ze"
(z). (9))~ For all (>, u)) E R [0, 11, the signed divergences are defined as
(2. = (n (Al)) inf {11(2,H) c(z [xb)]
=c([A]ovv]
0
Tp E [A]]
-log log (1 - 1(1 (1)-
8/ E (gT ([A))), 1]
kl (1,
if E (0, 1) and μ ([A)- 97 ((N))
(0,, = 1 (Mr (Nl))
inf {kl(z, + e (Alb - - =E|p.Ai
When two distributions are close enough compared to the privacy the signed divergences reduce to the KI. divergence: the indistinguishability due to the stochasticity the instance stronger than the indistinguishability due to DP. The function ge (resp- gt)) represents the maximal (resp. minimal) mean for which this property hold for dt (resp. d.).
For (1,, w) E RRK x RK the transportation cost of the pair of arms (a, E [K]² is defined as
W Wen,bu w) == (Iua)d (um)d) inf {wad, (Ma, u) + waat (mb: u)} we(0.1)
The signed divergences d and the transportation costs (W..,t) satisfy all the desired properties required to study BAI algorithms based the empirical version of We,a,b (see Lem- mas 23 24 255 and 26) as well as Lemmas 35| 36 37
and 38). e.g., symmetry, explicit formula,
monotonicity, strict convexity, etc.
Properties of the Characteristic Time and Optimal Allocation. The set w w((v) (v) of optimal alloca- tions is the maximizer of the outer supremum on 4 that defines (z) in Eq. (2). Theorem 4 gathers key properties satisfied by (v) and w (z), for Bernoulli distributions. See lemmas proven in Appendix G i.c., Lemmas 36 and 47
Theorem 4. Let SETVE E JK having means μ E (0, 10. with unique best arm a*. Then, we have
(((=)-1= max min W ((,,,, w) and (: E E.E(K] A-1
(5)
WEAN asta"
where De.. = mina di (Par. (Pa) and Dea = d (PasHa.) allocation is unique, has dense support and ensures the equality of the transportation costs with T (v)) (i.e., information balance equation), namely (V( = {ww), min w >> 0 and for all a²"
Ha.) for all + a". The optimal
W (u, wt) (v)
In Garivier and Kaulmann 2016). the characteristic time and its optimal allocation can be computed with simpler optimisation problem. A simpler optimization problem can also be solved to compute Th (v) and w (v) explicitly (Lemma 47
Allocation Dependent Low Privacy Regime. Let (μ, w, a, b) E (0, 1)x x R x [K]2 such that
[4 14h and min(w,, Way wx) > O. The non-private Bernoulli transportation costs Garivier and
Kaufmann 2016) are defined as
Wafta + wapt
W Wa,b(u, w) == wakl(Ha, (w,)) + wykl(Hbb H",,) with P
Wa + Wb
We provide allocation-dependee low-privacy condition that depends on (e, w) (Lemma 45). ie., We,,, (H4, w) = (1,, w) is implied by
No 1b S (1 - min {(1 + wo/ wo/wb) Mage (1 Ha), (1 + wt/wa) (1 Mo)ge (mo)
(6)
5

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Plugging W. from Theorem 4 in Eq. (6) would give an implicit condition on (c. under which the non-private characteristic time T (v) for Bernoulli distributions is recovered, i... (z)) = T (v). From a privacy-utility tradeoff perspective, the choice of that provides the highest privacy protection while maintaining a low sample complexity is exactly this change-of-regime c, depending on the unknown A weaker (yet explicit) allocation-indepedder sufficient condition for T: (v) = T (v)
is é 2 maXaza Cor,a where Ea.b == log
4
Generalized Likelihood Ratio Stopping Rule
Designing appropriate recommendation and stopping rules for the BAI problem can be framed as a sequential hypothesis testing task with multiple hypotheses {Ma = maX 1s One of the earliest approaches to active hypothesis tertnn-whhre data collection is also optimized-
introduced by Chernoff [159). who advocated for the use of Generalized Likelihood Ratio (GI tests for stopping decisions. This methodology is also popular the context of BAT [Garivier and Kaufmann 2016]. Despite its relevance, fewer works attempted to extend it for private sequential hypothesis testing, see, e.g.. Zhang et al. [0022] under Rényi DP and Azize et al. 2024 under e-local and e-global DP.
Three rules need to be specified define a BAI algorithm: recommendation, sampling. and stopping rules. An important remark in designing BAI algorithms is that the dependence of these rules on the private input observation dataset comes solely through the sequence of mean estimators. Thus, designing a sequence of mean estimators that satisfy DP is crucial when defining a e-global DP BAT algorithm. we rely on two ingredients: adaptive arm-dependent episodes with a geometric update grid and the Laplace mechanism. We call this mechanism estimating the sequence of means the Geometric Private Estimator, i.e.. GPE (e). Most notably, we eliminate "observation forgetting" from GPE,, an important design choice made in all past BAI algorithms Sajed and Sheffet [2019; 11 Azize et al. [2023 2024). Specifically, for some 17 O called the geometrie grid parameter, GPE,, (E) estimates the noisy means in arm-dependent phases: a phase changes when the counts of an arm has increased multiplicatively by 1 77 (Line 5). Then, GPE,, only updates the mean of the arm that changed phases, by accumulating the observations collected from its last phase and adding Laplace noise (Line 7). Due to this accumulation step, we do not forget the observations from past phases. Thus, each estimated noisy mean Pn,a in Line 7 contains Nn,a i.i.d. observations from Va and kn,a A logi+ Nv,. i..d. observations from Lap(1/e). In contrast, using forgetting produces a noisy mean that contains fewer i.idd observations from Vaa (e.g. N samples for forgetting with 17 1), but only one Laplace noise. While removing forgetting allows us to keep more signal, i.e., more i.i.d samples from V/a. we need more noise, i.e., the cumulative sum of Lap(1/c). which is logarithmic in the number of samples from Va. Tighter concentration inequalities allow controlling the cumulative sum of Laplace noise. See below for a detailed discussion about our novel concentration results. As long as the number of samples from the Lap(1 /e) logarithmic in the number of samples from Va. the effect of noise on the sample complexity is similar to having only one additional Laplace noise.
Mean Estimator.
estimate the sequence of means, defined in Lines 5-8 of Algorithm []
Privacy Analysis. By adaptive post-processing, the following lemma is proved naturally for any e. Lemma 5. Any BAI algorithm using only GPE
access observations is e-global DP on [o, 1].
Proof Sketch. The proof combines two steps. First, we show that the sequence of mean estimators produced by GPE,, (c) is c-DP. The crucial observation is that a change in one observation only affects the partial sum collected in just one arm-phase. By the Laplace mechanism, adding one Lap(1/c) to the partial sum is enough to make it eDD. Then, by post-procsssing, the sequence of accumulated partial sums (S)....)) and noisy means (Jm,.a) (Line 7) are also e-D.. The second step shows how to use the sequential nature of the process and adaptive pooc-prcessiig to conclude that BAI algorithms using only GPF., (c) are e-global DP. The detailed proof is in Appendix
Recommendation Rule. The recommendation rule ān is defined as the arm with the highest clipped noisy empirical mean, i.c., on E arg max [mn,a] where ties are broken uniformly at random.
GLR Stopping Rule. The GLR stopping rule runs K sequential GLR tests in parallel, and stops as soon as one of these tests can reject the null hypothesis. When comparing the recommendation ä with alternative arm a, the GLR statistic is defined as the transportation cost W We,.,. evaluated

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(4) Intuitively, W We....a (Pa, Na) represents the amount of empirical evidence to reject the hypothesis that arm a has higher mean than a. One can stop and recommend an when all these statistics exceed given stopping threshold. Given a privacy budget and risk (e,6) E + x (0, 1) and a stopping threshold e: C:No x R (0,1) + R4, we define
empirically at (Pa, N₂) (see Eq.
Te, = inf n Va + un. We, (Rn, N,) > c( N(Nm,... c, + of ((Nn.a,c,0) {
(7)
Given its proximity to the characteristic time Th (v), see Eq. (5) (Theorem 4.. the GLR stopping rule
a good candidate to match the lower bound, i.e., if one could sample arms according to w (v) and
use the stopping threshold (1g() Unfortunately, this threshold is too good to be 6-correct and w (V) should be estimated as it is unknown (Section 5)
Calibration of the Stopping Threshold. Regardless of the sampling rule, the stopping threshold should ensure 5-correctness of the GLR stopping rule for any pair (e, 6), see Theorem 6
Theorem 6. Let (,, E R2 x (0,1) X RT Let 8 > 1, S be the Riemann Ç function and W -z = -W-1(-e-2) for all x 2 1, where W- 1 is the negative branch of the Lambert W fumction, satisfying W -11 ~
+ log (Lemma 52). Given any sampling rule using the GPE,()) using the GLR stopping rule as in Eq. 7 with the GPE, and the stopping threshold c, == cn(n, 8) + c2(n, e) where
(n 8) = W W1 (log (($)/0)) + s slog(kn +3- - 1-33-og22 2) - + log 2,
(8)
(2(n, e) = (n) (log (1 + Zen (n)) +1 with k,(x)) = + log1 3;
yields a 8-correct and e-global DP algorithu for all Bernoulli instances with a unique best arm.
The proof of Theorem 6 builds on novel concentration results of independent interest (Appendix F22
Our explicit instance-ing estaace-inddeendern upper bounds pivotal to derive the stopping threshold in Eq. 8 which avoids the large desence constants used in the regret minimisation literature Azize
et al. 2025
Concentration Results. First, we give tail bounds for the cumulative sum of i.i.d. Laplace observa- tions (Lemma 16). We use Chernoff's method with the convex conjugate of the moment generating function of Lap(1/e), hence improving on Azize et al. 2025 Lemma 18] that approximates it. Second, we derive tail bounds for the sum between independent cumulative sums of i.i.d. Bernoulli and ny i..dd Laplace observations (Lemmas [8 and 19). They involve the modified signed divergences
F that better capture the non-asymptotic tails behaviour, and are equivalent to d= to an additive term O(log[1 + 2rre)/rr) where Tt= := t/n, (Lemma 30p. Whenever Tr + +00, we recover the same noise effect as adding only one Lap(1/e) observation. For x > O, the exponential decrease of the probability of exceeding μ (resp. being lower than μ I) scales tdī (14 + x. 14, r2) (resp. td; (μ (p- x. (1, r.)). The proof builds on fine-graine -reiirr tail bounds of the sum of two independent random variables, i.e., we bound those probabilities by the maximal product between their respective survival functions (Lemma[T)) While Azize et al. (2025) Lemma 19] directly integrates their tail bounds, Lemma [2] can be used with any tail bounds. Third, we obtain time-uniform upper tail bounds for df (Ji,as Nn, a by exploiting the geometric- grid update of (pm, Nu,, k.))
Threshold Scaling. The threshold e₁ in Eq. (8) ensures 8-correctness of the modified GLR stopping rule, defined in Appendix F.1 with the modified transportation costs We,,bb and diver- gences dt (Appendix F.I. Independent of c, it scales log(1/8) + O(log log((1))))) when - 0 and (log log log(n)) when n - +00. The threshold C2 in Eq. (8) is an upper bound on N (dt
d= (Pm.a. Ma: NNn,/kin,,) (Lemma 30) that scales as (cnn) when c - 0 and as O((lgnn)² when n - +oc. Both Ci and C2 scales as C log(1 + r)) when - 0.
Limitation. As the threshold in Eq. (7) is the sum of per-arm thresholds, it scales as 2 1og(1/8) when 5 + 0, hence incurs a suboptimal factor 2 asymptotically. Obtaining a threshold in log (1/5) is left for future work. It requires controlling the re- re-weighted sum of modified divergences d.t Azize
al. 2023. Theorem 4] has suboptimal factor 2 for the same reason and incurs an additive factor meZ log(1 2 due to the separate control of the Laplace and the Bernoulli observations (based on
Sub-Gaussian concentration results). Azize et al. [004] Lemma 18] alleviates this factor their
low privacy regime, yet it also pays nz? log 1
7

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Algorithm Differentially Private Top Two (DP-TT) Algorithm.
1: Input: setting parameters (e, 6) E R2 x (0. 1). algorithmic hyperparameters (1,,3) 3) E IR x (0,
and threshold e, e-g. (1, () = (1, and e as in Eq. (8) (Waaa)) as in Eq.
2: Output: Stopping time Te. recommendation recommendation āz. and pulling history (an)neTe..
3: Initialization: For all E [K], pull arm a, observe Xa.a Va and draw Yí,a Lap(I/e). Set
n = K +1. 1, For a E [K], set Sn,a Xa,a + Vi,as kinaa (a) = n, Na, Nn,a =
An.a = Sn C../Nnn.e Ln,a = 0 and No n,e = 0.
4: for n K +1 do
5:
if there exists a E [K] such that Nn,a 2 (1 + v)) Aaia then & Per-arm geometric update grid
6:
For this arm a, change phase kina E anna + 1, and (Tkn. (a), Nn,a) = (n, N NT aa)
7:
Set Skm...a = Ex -
(a)-1 ((a) Xt, 1 (a. =0) + Ykn.nia + Skna-1,0 with Ykn.a
Lap(1/e), and update the mean Mn, = S Skn.aa/ Nn.ai end if
8:
9:
Set an E arg ma [Mn,
Recommendation rule
10:
if Wean, Nn) >bE Ede(en c(Nn,,, 6, for all a + ån then GLR stopping rule
11:
return (n, ani (at)
end if
13:
Set B, = ān and On E arg R, (ae.B..a(hn N₂) + log Nn
> EB-TCI
Set a, B, N., S BBIn+1,,,, and On On otherwise;
b B-tracckng
Pull an observe and store Xn,, Va.i
16:
Update (NN,-1,,, Lnt1.Bn NBn B,,)) = (Nn.n Ln, A. n.B.
)+ (1,1,1 (B, B, = a.)):
17: end for
5 Top Two Sampling Rule
Equipped with a recommendation and stopping rules, we define a sampling rule using the GPF Within the fixed-confidence BAI literature, we adopt the Top Two approach [Russo] 2016 Qin
al., 2017] Shang et al.) 2020) Jourdan al.] [022] that recently received increased scrutiny due to its good theoretical guarantees (Jourdan and Degenne 2024 You et al.) [023] Jourdan al.
2024: Baandyopadyyay et al. 2024), competitive empirical performance, and low computational cost. The Differentially Private Top Two (DP-TT) algorithm (Algorithm 1 uses the EB-TCI-3 sampling
rule [Jurdaan et al.) 2022]. ). In Appendix|] we introduce the Brack-and-Stop [Garivier and Kaufmann
2016 and LUCB Kalyanakrishhan al. 2012 sampling rules for fixed-confidence BAI under
e-global DP.
After initialization, a Top Two sampling rule specifies four choices [Jourdan 2024
leader arm
B, E [K], challenger arm Cn E [K] [B,), a target allocation B, (Bn, C. E [0. and a mechanism to choose the next arm to sample from, i.e,, E Bn,, Cn) by using 3, (,,, C.). The leader should select good estimator of the best arm a*. We use the empirical best (EB) leader that coincides with our recommendation rule, i.e., B. == an. The challenger should be confusing alternative arm, for which the empirical evidence that the leader has a better mean low. We use the TCI challenger (Jourdan et al. 2022 that penalizes oversampled challenger to foster implicit exploration, i.c., Cn E arg min,y B., WW,BB,.a (1,,N,,, N,) + log Nn.a) where ties are broken uniformly at random. Crucially, we leverage our novel transportation costs (We. o) divergences df that are evaluated empirically at (pn, NN,), see Eq. (3) and (4). The target should be chosen to balance the allocation between the leader and the challenger arms. Let 3 E (0, 1), e.g., 3 = 1/2. We use fixed B-design 3n (Bn, C..) to sample should enforce that this target is reached on average. We use independent 3-tracking procedures (one per leader), i.c., a, = B, if B., B. < 33 and a, = B,, otherwise, where No. = Ete(n-) 11 ((Br ar) (a, a)) and Ln.a Exe(n- 1l (Br = a). Using Degenne et al. 2020b Theorem 6] for each tracking procedure yields - 112 < No 3Ln. < 1 for all a E [K].
fo B featuring the signed
B. The mechanism to choose the next arm
8

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sums the observations, and the recommendation and GLR stopping rules are updated when an arm is updated. Using the closed-form formula for We,,,b (Lemma 38 the per-iteration computational and global memory costs of DP-TT are ((K).
Computational and Memory Cost. The GPE
Asymptotic Upper Bound the Expected Sample Complexity. Given fixed target 33, the empirical allocation of at converges towards 3, that differs from w At best, we can estimate the 3-optimal allocation ( i.e., maximizer of the inverse ß-characteristicc time T (v)= defined as in Eq. (5) with the constraint wi- 3. While being only nearly asymptotic optimal, i.c., (v) (0,,) (1)) satisfies T (21/2(2) S 2T (v) (Lemma 44
DT-TT is e-global DP, 8-correct and matches T: (v) to a small constant, for any privacy budget c.
While Theorem 7 is an asymptotic upper bound for 8 - 0, it holds for any
Theorem 7. Let (e, 3) E R2+ (0, 1) R (0, 1) and in Eq. (8. The DP-TT algorithum is e-global DP, ò-correct and satisfies that, for any Bernoulli instance v with distinct means μ (0, 1)K,
ENT [Te.) log(1/8)
lim sup 6-20
S 2(1 n)T)
Proof. The proof (Appendix HD builds on the unified analysis of oourdan et al. [2022] and relies heavily on the derived regularity properties for d (a.b),) (V)) and w²,s F, (v) (Appendix G
For (7. (1, 122) the asymptotic upper bound is we reduced the gap between known lower and upper bounds for fixed-confidence BAI under c-global DP to a constant lower than 8, hence closing the open problem in AAzize et al. (2024). A discussion on how to improve this constant is deferred to Appendix C.2 (Dn-TDDP-TT Pop-oTyenjoys enjoys good empirical performance for moderate value of (Figure [], adapting the non-asymptotic upper bound in Jourdan et al. [2024] to the private setting is an interesting direction for future work.
S 8T7 For any privacy budget
Comparison with Azize al. 2023 2024). AdaP-TT and AdaP-TT" use the DAF(e) estimator, GLR-ine -iR-inspired recommendatton//toooon stoppiig rules and the TTUCB [Jourdan and Degenne [2024] sampling rule (i.c.. UCB-TC-- [Jourdan (024)). all based on arm-dependent doubling, forgetting and unclipped estimators. While AdaP-TT relies on the non-private Gaussian transportation costs, AdaP-TT* accounts for high privacy regime by clipping the mean gap, i.e., (Ma - 1bb)+ min(3e (Haa - pb)+ instead of (M²- ) The AdaP-TT and AdaP-TT* algorithms are c-global DP and 8-correct. The sample complexity of AdaP-TT only matches the high privacy lower bound for instances where the means are of similar order. A improves on AdaP-TT by matching the high privacy lower bound for all instances with distinct means. However, both AdaP-TT and AdaP-TT* fail to match the lower bound beyond the high-privacy regime, due to the use of non-adapted transportation costs. In contrast, DP-TT uses the de -inspired transportation costs, matching the lower bound up to a small constant for all values of e.
Comparison with [Saedd and Sheffet 2019]. DP-SE [Sajed and Sheffet 2019] is an e-global
DP version of the Successive Elimination algorithm introduced for the regret minimisation setting. modified by Azize al. [2023 into a c-global and 8-correct BAI algorithm. Compared to DT-TT DP-SE is less adaptive and not anytime, since it relies on uniform sampling within each phase and the phase length depends explicitly on the risk 8. The high probability upper bound on the sample complexity scales as O'Eaaa- (An min(e, --) with A. Maa Maa- This matches the lower bound when e - 0 (to a constant), but fails to recover the cample-compleitg lower bound beyond this regime.
6 Experiments
The empirical performance of DP-TT with (n. = (1,1/2) is compared to AdaP-TT Azize jet al., 2023), AdaP-TT* (Azize al.) 2024), and DP-SE (Sajed and Sheffel 2019 on different Bernoulli instances for varying privacy budget, ranging from 0.001 to 125 (see Appendix J.I. The first instance has means 14 (0.95, 0.9, 0.9, 0.9, 0.5) and the second instance has means
#2 (0.75, 0.7, 0.7, 0.7, 0.7). As benchmark, we also compare to the non-private EB-TCI-9

---

107
EB-TCI
Privacy Regime
DP
ligh Privacy Regime
DE
AdaPTT
AdaPTT
AdaRTT*
AdaP-TT
10a
Stopping Time.
DP-TT
Stopping Time
104
DP-TT
Low Privacy Regime
10
10*
Low Privacy Regime
104
10a
10-3
10
10-1
10
10
10a
10-1
10
10-1
10
10a
10a
Privacy Budget
Privacy Budget
Figure 1: Empirical stopping time Te,s (mean +2 std) for 10 with respect to the privacy budget
on Bernoulli instances (a) fn and (b) ur2 The vertical line separates the privacy regimes.
algorithm with 3 1/2. For 8 10-2, we run each algorithm 1000 times, and plot the averaged empirical stopping times in Figure Additional experiments are in Appendix !
Figure showss shows that DP-TT outperforms all the other 8-correct and e-global DP BAI algorithms, for different values of c and in all the instances tested. The empirical performance DP demonstrates two regimes. A high-privacy regime, where the stopping time depends the privacy budget c, and a low privacy regime, where the performance of DP-TT is independent of c, and requires twice the number of samples used by the non-private EB-TCI-B. In Figure the vertical lines are both at = 0.45. We compare with our allocation-indeeendent condition, i.e., mäXa fa*, where
Cabb = log (2a(1_Ha) For μ2, we have C₁, = 0.25 for all a * 1 that is close to the empirical [
separation. For μ4,, we have E1,5 = 2.94 and c1.a = 0.75 for all a E {2, 3.44, 4). When an arm has condition, such as Eq. (6). is required to reflect
significantly lower mean, an illocatioo-dependdee
the empirical separation. Intuitively, an arm with significantly lower mean will also be sampled significantly less, hence there is compounding effect.
7 Conclusion
Motivated by the privacy requirements of sensitive applications of BAI, we address the problem of fixed-coffidence BAI under e-global DP. We narrow the gap between the lower and upper bounds on the expected sample complexity to multiplicative constant smaller than 8, for all values. Our novel lower bound incorporates d, an information-thewr and TV distance, scaled by e. We design a private, arm-dependent geometric grid estimator withour forgetting and GLR stopping rule based on the d, -ttrasspoorttiinn costs, whose correctness requires novel concentration results for Laplace and mixed distributions. Finally, we proposed a Top Two sampling rule that achieves asymptotic upper bound matching our lower bound to small constant.
quantity smoothly balancing KL divergence
We detailed research directions to further reduce the constant gap between the lower and upper bounds, by improving both the calibration of the stopping threshold and the analysis of the sampling rule. The most exciting direction for future work is to extend our results to other classes of distributions (e.g., Gaussian or bounded distributions), structured settings (e.g.. linear or unimodal), or other identification problems (c.g., approximate BAI or Good Arm Identification). Another interesting research direction is to extend the proposed technique to other variants of pure DP (c.g.. (c, DD or Rényi DP MMronovv 2017)) or other trust models (c.g., shuffle DP CCeu 2021 Girgis et al. 2021
Acknowledgments and Disclosure of Funding
We thank Junya Honda for the interesting discussions. Mare Jourdan was supported by the Swiss National Science Foundation (grant number 212111) and by an unrestricted gift from Google. Achraf Azize thanks the support of the FairPlay Joint Team.
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A
Outline
The appendices are organized as follows:
Notation are summarized in Appendix B
A detailed related work and an extended discussion is given in Appendix C
The lower bound on the expected sample complexity under e-global DP (Theorem is proven in Appendix D
The proof of Lemma 5 is given in Appendix
The proof of our concentration results are detailed in Appendix In particular, this includes the proof of Theorem [
Appendix G gathers key properties on the (resp. modified) divergence of (resp. d=), the (resp. modified) transportation costs W.,,bb (resp. We.,,)) )) and (resp. 8-)eharacteristic times T (v) (resp. (v)) and their (resp. (8-))ptimal allocation w²(e) (resp. (V))(V)). In
particular, this includes the proof of Theorem 4 based on Lemmas 43 and
and 47
The proof the upper bound on the asymptotic expected sample complexity of DP-TT (Theorem is given in Appendix H
In Appendix ! we propose variants of algorithms to tackle c-global DP BAI We aim at providing several choices for the interested practitioners.
Implementation details and additional experiments are presented in Appendix
Table 1: Notation for the setting.
Notation
Type
Description
N
Number of arms
F
C P(O.1)) 1])
Class of Bernoulli distributions Bernoulli distribution of arm a E [K]
Va
F
Jk
Vector of Bernoulli distributions, V == (Va)aE[K]
Pa
(0, 1)
Mean of arm Vector of means, μ ;= (Pa)aE[K]
[K]
1),
(v)
[
Set of best arms, a" (v) a* (11) == arg max Pa
[K]
Unique best arm, i.e. a (n) {a*}
R1
Privacy budget for e-global DP
(0, 1)
Risk for d-correctness
Alt(v)
S FK
Alternative instances with different best arms
B Notation
We recall some commonly used notation: the set of integers [n] {1.... .n), the comple- ment X and interior X of X, the indicator function 1l (X) of an event, the probabil- ity Pyn and the expectation Eyy taken over the randomness of the observations from w and the algorithm n, Landau's 's notation o, O, Ω and θ, the (K 1)ddiennsional probability sim-
plex AK
ww W R 2 0, Σie[k] W/
The functions [æ]]] max min
(x) = logit F, W - Lemma 52 in Eq.
Eq. 33
C is the Riemann
function. Moreover, recall the definitions: di in Eq. 3 d. in Eq.
d in Eq. 32
W Wea.b
in Eq. 4 W 2 in Eq. 34 (V), (T, w (v), w (v)) in Eg 35 While Table gathers
problem--pecific notation, Table 2 Ogroups notation for the algorithms.
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Table Notation for the algorithm.
Notation
Type
Description
Bn
K
(EB) Leader at time n
Cn
K
(TC) Challenger at time 7
K
Arm sampled at time
Xn.en
{0, }}
end of time n, i.e. Xn..,, ~ von. Noisy perturbation drawn the beginning of phase kn.a for arm a, i.c. Y ~ Lap(1/e)
Sample observed
a
R
Fu
History before time Arm recommended before time n Sample complexity (stopping time) Stopping threshold function Stopping threshold function Approximation threshold function Number of pulls of arm before time Current phase of arm a at time
än
K
Te.8
N
(n,e,0)
Nx R X (0, 1) + R
G (n,
N (0,
Ri
cn(
N X R + R
Nn.
N
kn,
N N R N
(a)
Time n where the arm a changes to phase k
Sk.a
Private sum of observations for arm a at phase k Number of pulls of arm at the beginning of phase kn.a
Nna
Pn,a
TR
Private estimator of the empirical mean of arm a
the beginning of phase kn,0
Ln.
N
Counts of B, before time
No
N
Counts of (Br, ae) = (a, a) before time n
n.a
(0,
Fixed proportion
C Related Work and Extended Discussion
We provide a more detailed literature review in Appendix C.T and an extended discussion in Appendix C.2
C.1 Related Work
Structured Bandits. While we consider unstructured bandits Auer et al. 2002 numerous structural
assumptions have been studied: linear bandits [Soare et al. 20144, generalized linear bandits (Filippi
et al. 2010] such logistic bandits [Jun et al. [021]_ combinatorial bandits [Chen et al.) 2013].
sparse bandits [Jamieson et al. 20155. spectral bandits Kocák and Garivier 2021]. unimodal bandits
[Combes and Proutière 2014 Lipschitz [Magureanu et | 2014), partial monitoring [Audibert
and BBBBBubbee000aaaa etc. Coping for the structural assumption while preserving c-global DP an interesting direction for future works.
While we consider only BAI [Even-Dar et al. 2002], other pure exploration problems have been studied in the literature: e-BAI [annor and Tsitsiklis 2004]. thresholding bandits [Carpentier and Locatelli 2016), Top-k identification [Katz-Samuels and Scott 2019], Pareto set identification Auer et al. 22166), best partition identification [Chen et al. [017]. etc. Extending our e-global DP results to answer these identification problems is an interesting research direction.
Pure Exploration Problems.
Performance Metries. In pure exploration problems, the major theoretical frameworks are the fixed-confidence setting Diven-Dar et al., 2006 Jamieson and Nowak] 2014]] Garivier and Kautmann 2016), which the focus of this paper, and the fuxed-budger setting AAudibert et al.]] 2010; Gabillon et al. 2012]. In the fixed-bur budd-b setting, the objective is to minimize the probability of misidentifying a correct answer with fixed number of samples T. Recent works have also considered the anytime setting, in which the agent aims at achieving low probability of error at any deterministic time (Zhao eet al. 2023, Jourdan al. 2024). Extending our findings to support e-global DP the fixed-budget or the anytime setting is interesting direction for future works, e.g., Chen et al. 2024

---

DP in Bandits. DP has been studied for multi armed bandits under different bandit settings: finite-
armed stochastic [Mishra and Thakurta [015
Sajed and Sheffet, 2019 Zheng et al. 2020; Hu et al.)
2021; Azize and Basu 2022 Hu and Hegde 2022, Azize and Basu 2024 Wang and Zhu [024]]
Hu et al. 2025], J. adversarial Thaakrra and Smith 2013 Agarwal and Singh| 2017] Tossou and
Dimitrakakis, (017) 77. linear |Hann et al.. [0244]] Li et al. 2022, Azize and Basu [202]] contextual
linear [Shariff and Sheffet 2018 Neel and Roth 2018 Zheng et al. 2020; Azize and Basu 2024]. and
kernel bandits [Pavlovic al. 2025]. J. among others. Most of these works were for regret minimisation,
but the problem has also been explored for best-arm identification, with fixed confidence [Azize et al.
2023 2024] and fixed budget [Chen et al. 2024]. The problem has also been studied under three different DP trust models: (a) global DP where the users trust the centralised decision maker MMishra
and Thakurta 2015 (hhrriff and Sheffet 2018 Sajed and Sheffet 2019; Azzze and Basu 2022 Hu
and Hegde| 2022]. (b) local DP where each user deploys a local perturbation mechanism to send a
"noisy" version of the rewards to the policy [Basu al. 2019 Zheng al. 2020; Han et al. 2021],
and (c) shuffle DP where users still feed their data to a local perturbation, but now they trust an
intermediary apply a uniformly random permutation on all users' data before sending to the central
servers [Tenenbaum et al. 2021 Garcelon et al. 2022 Chowdhury and Zhou| [022).
In the first papers on DP for bandits, the tree-based mechanism [Dwork et al. 2010: Chan et al. 2011] was used to compute the sum of rewards privately. However, this mechanism was proven to be sub-optimal, matching the lower bounds up to logarithmic factors. Then, forgetting was first proposed by [Sajed and Sheffet [2019] to get rid of the tree- based mechanism, then adapted to UCB in Hu eet al. [221 Azize and Basu 2022]. Finally, if the KL is the divergence that controls the complexity
of bandits without privacy LLai and Robbins 1985 Garivier and Kaufmann] (016]. then Azize and
Basu 2022) were the first show that the TV controls the complexity of private bandits, in the high
privacy regime.
In this paper, we focus e-pure DP, under a global trust model, in stochastic ficchstiifiiite-ammee finite-armed bandits, for best arm identification under fixed confidence.
Gap in the Literature. This problem setting is first studied by Azize et al. [2023] who proposed the first deeeeddeet sample complexity lower bound, and introduced AdaP-TT, an e-global DP version of the Top Algorithm. However, the sample complexity upper bound of AdaP-TT only matches the lower bound in the high privacy regime e + O, and for instances where the means have
similar order (see Condition in Azize al. 2023 in the discussion after Theorem 5 in AAzize
ct al. 2023)). ).
Azize et al. [2024] proposes AdaP-TT*, an improved version of AdaP-TT. The improvement is
achieved by using a transport inspired by the sample complexity lower bound from [Azize et al.,
2023 Using the new transport, AdaP-TT' gets rid of Condition 1 needed by AdaP-TT, and achieves the high privacy lower bound for all instances up to a multiplicative factor 48.
However, both AdaP-TT and AdaP-TT* do not match the lower bound, beyond the high privacy regime, i.e. for both the low privacy regime and transitional regimes.
C.2 Extended Discussions
Limitations of Theorem 7
Using adaptive targets (B,, C.,) in DP-TT could replace Ti, by
T (v), which is the optimal scaling asymptotically (Theorem 2h. While we propose two adaptive choices of target based on IDS YYou et al. 2023 or BOLD (Bandyopadhyay al. 2024 (Appendix [
we leave their analysis for future work. Based on finer concentration results, the stopping threshold could scale asymptotically as log(1/6) instead of 2 (1g()))) hence improving by factor 2. The assumption that the means are distinct is used to prove sufficient exploration; it can be removed by using forced exploration or a fine-grained analysis (Jourdan and Degenne, 2024 [Jurdan et al.) 2024). Provided these improvements are proven, the multiplicative factor would be reduced to 1 + 7/. where 1 is the parameter that controls the geometrically increasing phases. While it improves the asymptotic
upper bound, choosing too close to O negatively impacts the performance of DP-TT due to the dependency in (11/ /o//oo((1 + 17)) of the stopping threshold.
Beyond Bernoulli Distributions The non-asymptotic lower bound on the expected sample com- plexity (Theorem 2 holds for any class of distributions F, and thus already true beyond Bernoullis (c.g. exponential families, sub- sub-Gaussians, etc). However, by going beyond Bernoullis, d, might not
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admit closed-form solution, e.g., the TV between Gaussian distributions has no simple formula. We conjecture that most regularity properties used derive Theorem | and needed for the upper bound proofs, also hold for other classes of distributions. From theoretical perspective, this might render the proof of the sufficient regularity properties more challenging. From practical perspective, this increases the computational cost of the stopping rule and the challenger arm, as they require computing an empirical transportation cost based on the de divergence. Finally, since the objective function inside the de is continuous and convex over compact interval, using off-the-shelf convex optimisation solvers to compute de is still straightforward.
Our privacy analysis (Lemma 8 holds for any distribution with support in [0. 1], and thus is already true beyond Bernouliis. (a, b). by multiplying the noise terms by the range b - a. Also, all prior works in bandits with DP are either for Bernoulli or bounded distributions, for the simple reason that this assumption makes estimating the empirical mean privately using the Laplace mechanism straightforward, as the sensitivity controlled. This helps focus on the more interesting tradeoffs between DP, exploration and exploitation, without any additional technical overheads that may be introduced by estimating privately the mean unbounded distributions.
straightforward to extend this to any distribution with a support in
The concentration results (Theorem 6 used to derive a stopping threshold ensuring d-correctness are highly specific to Bernoulli distributions. However, the proof builds on fine ane-grained tail bounds of the sum of two independent random variables, i.e., we bound those probabilities by the maximal product between their respective survival functions (Lemma [0]. Therefore, the technical tools used for this intermediate result can be also used to study other one-parameter exponential families, e.g.. Gaussian observation, with other noise mechanisms, e.g., Gaussian noise.
The proof of Theorem (Appendix H builds on the unified analysis of the Top Two algorithm Jourdan al. [0222] coping with many classes of distributions, e.g., one-parameter exponential families and non-parametric bounded distributions. It relies heavily on the derived regularity properties (Appendix G for the signed divergences, transportation costs, characteristic times, and optimal allocations. Based on the non non-private BAI literature, we conjecture that most regularity properties used to derive Theorem [ also hold for other classes of distributions. Due to the lack of explicit formulac, the proofs are challenging, The TT algorithm becomes computationally costly due to the intertwined optimisation procedure when computing the transportation costs numerically.
Beyond BAI The technical arguments used in the proof of Theorem 2 allow obtaining a similar lower bound for (i) one-parameter exponential families, e.g.. Gaussian, (ii) pure exploration problems with a unique correct answer, e.g., top-m best arm identification, and (iii) structured instances without arm correlation, e-g., unimodal bandits. This is straightforwardly done by adapting the definition of Alt(v), a" (V), and F in Eq. (2). For settings that do not fall into the above three categories, the characteristic time requires more subtle modifications; yet our intermediate technical results can still be used. Based on the non-private fixed-confidence literature, we conjecture the changes of characteristic times described below.
For bounded distribution (non-parametric) or Gaussian with unknown variance (two- parameters exponential family), the KL should be replaced by infimum over KI. Ag.aaaalll et al. 2020]; Jourdan et al. [2023]. The plurality of distributions having the same mean Jourdan
implies the existence of a distribution that is the most confusing given specific constraint on the mean, captured by the Kinf.
For y-Best Arm Identification (multiple correct answers), outer maximization over all the correct answers should be added Degenne and Koolen [2019] The plurality of correct answers implies the existence of correct answer that is the casiest to verify.
For linear bandits (correlated means), the inf in the definition of de per-arm should be replaced by a joint infimum over all the arms and moved outside the sum over arms Degenne lt al. 2020a]
Beyond e-DP Our lower bound technique (Theorem 2 can be seen as a generalisation of the packing argument, and thus depends on the group privacy property. Specifically, Appendix [D] just before Eq. (II. used the group privacy of eDDP to bound Mp:) I edham D').
26

---

For p-zCDP, there is a similar group privacy property that can directly be plugged here, stating that (.M.MM.pp S pdhamm D')2. This means that for p-zCDP, e dham is replaced by p x dhamm in the fundamental optimal transport inequality of Eq. TT We leave solving tightly this new optimal transport for future work.
For (c, 6DDP, the group privacy property is not tight. This is classic issue in (c, d)-DP lower bounds, where other techniques are used (e.g., fingerprinting). Adapting these techniques for bandits is an interesting open problem (even for bandits with regret minimisation).
is straightforward to make DP-TT achieve either p-zCDP or (e, 6)-DP, by replacing the Laplace mechanism with the Gaussian mechanism. To design correct stopping rule with the Gaussian mechanism, it is possible to use the same fine-grained tail bounds of the sum of two independent random variables (Lemma 10 by only replacing the concentration of Laplace random variables with the concentration of Gaussian random variables. In the Top Two family of algorithms, an important design choice the transportation cost, used for stopping and sampling the challenger. [D--TT uses a d, based transport, inspired by the lower bound of Theorem 2 design, it is important to derive tight lower bound that suggests the use of new transportation cost.
Thus, to go beyond e-DP for algorithm
Behavior with Near-optimal Arms The unique best arm assumption is standard in the fixed- confidence BAI setting. Even in the non-private setting, this assumption is necessary obtain d-correct algorithm with finite expected stopping time, as the characteristic time becomes infinite otherwise. In the private setting, this phenomenon persists can be seen by the lower bound in Eq. (5), i... (v) + too when A..a + O. In near-tie scenarios, DP-TT will perform as badly as any other d-correct and e-global DP algorithms due to the fundamental lower bound. Empirically, will be reflected by empirical transportation costs that are too small for the stopping condition to be met. In applications where the near-tie scenario is common, practitioners consider y-Best Arm Identification, i.e., identify an arm that is 7-close to the best arm. For the non-private fixed-confidence y-BAI setting, we refer to Jourdan et al. [2024) for references and guarantees satisfied by Top Two algorithms. To tackle D.....P-TTTssol in both the stopping rule and the definition of the challenger, i.c., W.,,, instead of W W.,.b.
be modified by using the appropriate transportation costs
Implications of Not Forgetting DP-TT has the same memory complexity as the forgetting algo- rithms. The reason is that DP-TT only needs to store one accumulated noisy sum of rewards across phases, while the forgetting ones store the noisy sum of rewards of only the last phase. For privacy considerations, under threat model where the adversary only observes the output of the algorithm (sequence of actions, final recommendation and stopping time), both forgetting and non-forgee non-forgetting algorithms provide the same DP guarantees thanks to post-processing.
One can imagine other threat models where forgetting the rewards provides better privacy guarantees. One possible example of this is the pan-private threat (woork et al. (2010), where the adversary can also intrude into the internal states of the algorithm during its execution. In this threat model, an adversary observes the internal states of the execution of DP-TT at some phase É >> 1. they can see the full sum of rewards and maybe infer the membership of, say, the first user (up to tradeoffs guaranteed by the e-DP constraint, since the sum is noisy). However, for the forgetting algorithms, the adversary observes the internal states of the execution of forgetting algorithms at some phase L >> 1, the adversary infer nothing about the reward of the first patient, since its reward has been deleted completely.
D Lower Bound
Let M : x - O be an e-D mechanism. For D E X an input dataset, we denote by MD the distribution over outputs, when the input is D, and (E) the probability of observing output E when the input is D.
Let IP and Q be two data-generating distributions over X². We denote by M2.. the marginal over outputs of the mechanism M when the input dataset is generated through P. i.e.
Mr.M (=)==
MD (A) dIP (D)
(9)
DEX"
for any event A in the output space. We define similarly Miq. the marginal over outputs of the mechanism M when the input datasct is generated through Q
27

---

The main question is to control the divergence KL (M..M M. MQ. when the mechanism M satisfies DP. In general, for any mechanism M, the data-sroeesiing inqquality provides the following bound
KL (Mip,M MM.. (P
(10)
Now, for e-DP mechanisms,
translate the DP constraint tight bound the divergence
KL (MI.,M || MM..M). To do so, let L be any other distribution on X Let C2.L be coupling of (IP, L), ie., the marginals of C are IP and IL. We can now rewrite our the marginals using the definition of couplings. For Mp.M we have
Mp (A) = S Mo (A) dP
-..er MD dCRL (D,D).
DEX
and for ! get
McM((A)= = SDe Mr (A)
(Q J MD dQ dlL (D'
dD)
J.DEE Mar (A) dl (D')
dQ (D')
dCr, D')
Using the data-processing inequality, we get
KL (Mir.M || Ml..) $ JONDEEN / log
MD (o)
JOEO
MD (o) dodCr, DD
MD (o) dL(D")
dQ(D')
dQ (()) dCe. (D,
dIL
KL (M || MM ) + log
ED,D' [KL (MD || Mb + KL (L Q
Since this is true for any coupling Cz.1 and any distribution IL, we get the final bound
{
KL (MP.. Me.M) - inf
inf LEP(X") (FPLEC(P.L)
(ED.D-woa, [KL (MD || Mb + KL (L (L 10)
where P(x") is the set of all distributions over X and C LP, L) is the set of all couplings between P and L. Using that the M is c-DP, we can use the simple bound KL (Mp MMr)) S edhtam (D. D') which gives
KL (MP. || MQ.M) < inf LEP(X*) {
inf {ED,DrwCen [dHam (D, + KL (L Q) C)LEC(FF))
(11)
D.1
Product Distributions
Suppose that = 8" P. and Q = 8
"__ Q are product distributions. Consider the subset of
product distributions IL. = T.,, and the maximal coupling Coo (IP, L.) = I Coc (Pi, T.)..
Plugging these in Equation 11b, we get
KL (Ma.. Mlq.) < inf Li....ii
Σ Ep, Ep, (1 i=
(D, + D'N + Σ KL (L: Q.)
1=1
inf Li....hh
(I)) Q.))
Σ inf EP())
TV (P, | L.) + KL (L. Q0) = Σ (P..))))
where
dP = inf
(eTV (P LL + KL (L Q)
(12)
LEP(X)
28

---

Algorithm 2 Bandit interaction between a policy and an environment
Input: A policy and an environment - (Pa aa: [K])
2: for = 1,... do
3: 'Te policy samples an action at ~ TT(, a1, (a.0),,.,,,,,1111 (t-1,
4: The policy observes reward ~ Pa.
5: end for
6: if Regret minimisation then
7: The interaction ends after steps
8: else FC-BAI
9: The policy decides to stop the interaction at step Te,á and recomends the final guess
10: end if
D.2 Sequential KL decomposition for bandits under DP
In this section, we adapt the techniques from product distributions to bandit marginals. First, we introduce the bandit canonical model.
The bandit canonical model. stochastic bandit (or environment) a collection of distributions v - (Pa E [K]), where [K] is the set of available K actions. The learner and the environment interact sequentially over T rounds. In each round E ,, T. the learner chooses an action d, E [K], which is fed to the environment. The environment then samples reward r E IR from distribution Pa. and reveals Tt to the learner. The interaction between the learner (or policy) and environment induces a probability measure on the sequence of outcomes Hr _ (a1, 71, 02 r2,-.., .r., In the following, we construct the probability space that carries these random variables.
Let T N be the horizon. Let v = (Pa : E [K]) bandit instance with K E N* finite arms, and each Pa is probability measure on (R, B(R)) with B being the Borel set. For each / E [T]. let
Ω = ([K] x R) C 222 and F, = B(2,). We first formalise the definition
policy.
Definition 2 (The policy). A policy is a sequence (#1)T=1 where is probability kernel from Fi) to ([K), 2(k Since [K] is discrete, we adopt the convention for E [K].
(a 01 Tl....,, 11-1, = (e({a) @1 "i,...,,,-111 Tt-1)
We want to define a probability measure on (RrrFrr Fr.) that respects our understanding of the sequential nature of the interaction between the learner and a stationary stochastic bandit. Specifically, the sequence of outcomes should satisfy the following two assumptions:
(a) The conditional distribution of action a given 01, Ti...., a-1 T(a Hz-1) almost surely.
(b) The conditional distribution of reward rt given 01, Fi,..., ft-1, at Pa. almost surely.
The probability measure on (ST, Fr) depends on both the environment and the policy æ. To
construct this probability, let À be o-finite measure on (R, B(R)) for which Pa is absolutely continuous with respect for all Pa with respect to A. Letting be the counting measure with p B) B), the density Por S2r + IR can now be defined with respect to the product measure (p X)T by
[K]. Let P. d.Pa/ dd be the Radon-Nikool Nikooyyi derivative of
T
Pur (ar F1. @T, PT)
A
(a a1 Ti, at-1, Tz- )a (re)
and Pvn is defined as
P Pun B
Per (ur(w)(p A)" forallB e Fr
Hence (Lr Pvr) probability space over histories induced by the interaction between and v. We define also marginal distribution over the sequence of actions by
myx (a1 ........ (T))
J Pen (a1, T1,-..,... ar, rT) dr ... drr,
29

---

and forallC E )((
Mum (C) -
Σ (aL.,AT)EC
Myz(aj, a2...., 0r))
Finally, ([K]", )((K))), is a probability space over sequence of actions produced when ' interacts with for T time-steps.
The KL upper bound. Now, we adapt the techniques for the bandit marginals. Let = (Pa. E [K]) and v/= (Pa, E [K]] be two bandit instances in FKK We recall that, when the policy ı interacts with the bandit instance v, it induces marginal distribution Mur over the sequence of actions. We define My's similarly.
The goal is upper bound the quantity KL (Myr M,yx). The marginals Myn and Myn in the sequential setting "look like" marginals generated by "product distributions". However, the hardness of the sequential setting lies in the fact that the -ata-generating distributions depend on the stochastic sequential actions chosen. Thus, the results of the previous section cannot be directly applied. To adapt the proof ideas of the previous section to the bandit case, we introduce the idea of coupled bandit instance.
Let v" = (P" coupling between Pa and Pa i.e., Co :- Co Pa). Fix a policy x = =
[K] be any "intermediary" bandit instance from FK Define Ca the maximal
Here, we build coupled environment 7 of v and v". The policy interacts with the coupled environment y up to a given time horizon T to produce an augmented history {(ae, Te, r()) The iterative steps of this interaction process are:
1. The probability of choosing an action a, = a at time is dictated only by the policy n, and a1 T1, 42, T2..... .,, at-1, Tt-1, Le.the policy ignores {r")==]
2. The distribution of rewards (r', r!') is Ca, and is conditionally independent of the previous observed history Ta, r")
This interaction is similar to the interaction process of policy æ with the first bandit instance with the addition of sampling an extra r from the coupling of Pa, and P",-
The distribution of the augmented history induced by the interaction of π and the coupled environment can be defined
T
Prx(ai,71 OT, rT, M4)
II
TZ (at ai 1=1
12)a (re))?")
To simplify the notation, let a = (ai...., aT), r = (rI...., .T), r == (F' (4) and
r" r(). Also, let (r, r²) a1.71 (..7,,....-..777 T1-, We put h = (a, r.). With the new notation
IT Ca. (re, rf') and क r) = - (a
Pyn (a == a (r,
By the definition of the couplings, we have that myr is the marginal of pyn when integrated over (r, (t.r"),ic
mys (a) [ Per (a, r, r"") dr dr".
fr.""
Now, we define new joint distribution Zysr. inspired by the techniques used for product distributions:
4r (a, r. r") = x(a| (r7) 1/ (r())) P/ (1*)
Ca ca(r, r"),
where p'((r') := II Pla, (rr"), and similarly, px = IT- P/ (r!"). First, observe that it is indeed a valid joint distribution, i.e.
r²") dr dr = Σ / a(a 1r (1()
E 9rm r,"
(1") Ca(r, r²t) drdr"
30

---

Σ // a(a rr'p'((r)) dr = 1/a (2((")) =1,
and that mylr is the marginal of 2yr when integrated over (r, r²") i.e.,
[ F.P"
x(a 1d) 1/ (r")
(r")") Ca r²)) drdr"
Jr,""
π ('") dr my'x(a)
Using the proc-ssoes inequality, we get
KL (Mer || Mlv')) S KL (pyn ll (rr)
(13)
Now, we compute
T(a (r, r")
KT. (pyz |l (7
log
T(a r²))) 1((r")
(a(r,r"))
(6)
1a (r")
En: (a.r.r")) ~Ppp ed (r, r1) + log
(1()))
T
Enrpyr
ell fre r + log
Va, (r"))
T
P'a, (r²') P'a, (r(") 1 acl
(d)
Enr-pyr
Enppyy [el fre * #, + log
Ehrepra
(CTV (Pas || |(/h)) KL (p". P'a)]
Σ TTV (Par || P(a)) + KT. (p', pla.)
E
where:
(a) by the definition of the KL
(b) the group privacy property, applied to the e-global DP policy, we have
edhum(t."" aTal r")
(c) by the definition of dham
(d) by the towering property of conditional expectations
(e) given a,, we have r, ~ Pac. re ~ p'a, and r" p'a,
(f) by linearity of the expectation, and the fact that the expression inside the expectation only depends on the actions 0z
Since this is true for any "intermediary" bandit instance v" E FK we take 1/" to be the environment where the infinimum of the de P!) is attained for each arm E [K]. Specifically, let v/ (p" a [K]) where
pd arg min {e TV (pa || [] KL (L p/.)3
LEF
Plugging V/ gives
KL (Myr || M.xx) S Eyn Σ (Pa.
(14)
31

---

Let Ni, = E 1 fas a} be the counts of arm before step t. Then, we can rewrite the bound
K
KL (Myr My'n)
B (d
(15)
Stopping time version of the KL decomposition for BAI under DP. Let π be an c-DP BAI strategy. Let and À be two bandit instances. Denote by M,T the marginal distribution of the output of the BAI strategy when и interacts with v. By using Wald's lemma in the proof technique seen before for the canonical bandit setting under FC-BAI, we get that
KL (M
|| M)x) S RBy
Σ delVair Mail)
Σ a=1
Eyn (N (N (16)
where 7 is the stopping time.
D.3 Sample Complexity Lower Bound Proof
Theorem 2 (Sample complexity lower bound for BAI under c-DP). Let (c, R7 x (0,1). For any algorithm that is 8-correct and e-global DP JKK.
E 2
for all E FK with unique best arm. The inverse of the characteristic time (v) is defined as
(²(1)~ sup inf wEAK NEAlt(w)
Σ wade (ia
(17)
(a)s = inf
(KL (yoa ka) +e- TV (Va pa)}
(18)
F.EF
Proof. Let т be an e-global DP 8-correct BAI strategy. Let v be bandit instance and E Alt(v).
Let Men denote the probability distribution of the output when the BAI strategy и interacts with v. For any alternative instance À Alt(v), the pata-procsssing inequality gives that
KL (M MM,,) > kl (MIr (ā a"(v)) My, (ā a"())) kl(1 8,8) (19)
where the second inequality is because т is 8-correct, i.e., My (ā = a (v)) > 1 - and Ml,, (ā = (v)) < 6, and the monotonicity of the kl. Now, using the stopping time version of the KL decomposition for FC-BAI, we get that
K
kI( 5,6) < KL. (Miu,= MM S
Buz (N (o
Since this is true for all X E Alt(v), we get
K Σ
kl(1 (18)- < inf
Evn tlo Aa)
AEAlt(v)
a=1
K R
[Nr
E E
inf xEAlt(w)
de (Var da)
E[Te,o]
(b)
K wade (udr
<
B[Tc,a]
sup inf WEAK xEAle(v)
Σ a=1
(a) is due to the fact that E[Tc,8] does not depend on A. (b) is obtained by noting that the vector
Be [N,,NN [t
(Wo) A
belongs the simplex Ag. The theorem follows by noting
Lv TT.6l
ac [K]
that for d E kll kl(1 8,8) > log (1/36).
32

---

E Privacy Analysis
In this section, we prove Lemma First, we justify using a geometric grid for updating the means (Lemma 8). Second, we obtain Lemma as a combination of Lemma g and the post-processing
property DP (Proposition
E.1 Releasing partial sums privately
First, the following lemma justifies the use of geometric grids, and provides that the price of getting rid of forgetting is summing the Laplace noise from previous phases.
Lemma 8 (Privacy of our grid-based mean estimator). Let T E {l.... }. l < T and ti te+1
be [1, such that = t1 < <... < <t t&+1 11 1==1-1111
Let M be the following mechanism:
31
(I) + ... + tt 1z2 +()) (11t... I1z-1 + (Y1 + 122
T2
FT
(x) + IT) (V + Y2 + ---1)//
where (Yi,..., Then, for any (f1...., E [0, 1J" is
......,)))) Ye) Hid Lap(1/e).
Proof. First, consider the following mechanism, that only computes the partial sums:
3 + It2-1
x2
It2 + + Its-1
PT
Ftz + ST
Because v, [0, 1J. the sensitivity of each partial sum is Since each partial is computed over non-overlapping sequences, combining the Laplace mechanism (Theorem T with the parallel composition property of DP (Lemma gives that the following mechanism:
I1
I1 +
+ xt, Y1
J2
P
Itz
+ + Ftz + 122
ET
Tte 1 + + FT + Ye-1
is c-DP, where (Yi...., YY-1) Gi Lap(1/c).
Consider the post-processing function f (#1... xz-1) - (I1, + ........ 21 D2 + xe Then, we have that that M = o P. So, by the post-processing property of DP. M is cDDP.
Remark 1. Mechanism P, defined the proof of Lemma is the fundamental mechanism used by all previous bandit algoritimss (Sajed and Sheffet 2019: Azize and Basu 2022 Hu and Hegde] 2022 Azize et al. 2224]] to justify the use of forgetting. Our mechanism M is just summing over the partial sums computed each phase, and thus the price of having sums of x; that start from the beginning (i.e. do not forget) that we have to sum now the noise from all previous phases too.
E.2 Proof of Lemma
We are now ready to prove Lemma 5 i.e. that any BAI algorithm based solely on using GPE access observations c-global DP on 1).
(c) to
Proof. Let be a BAI algorithm using only GPE (c) to access observations. Let R = (xI,...)
and R' {.f....} } be two neighbouring sequences private observations, i.e. there exists t" E {1....} such that I z for all t + t², i.c. that R. and R' only differ at t".
33

---

Fix a stopping time, recommendation and sampled actions (T + 1, (e.....,rr)), we want to show that
(r[ - (T' +,, (aI, (T).......))) Pr( (T' + 1, (o),)..;))))..
Step 1: Probability decompositions: First, let us denote by т, Å and A...... A, the random variables of stopping, recommendation and sampled actions, when interacts with R. Similarly, let r', A' and .i..... A' the random variables of stopping, recommendation and sampled actions, when æ interacts with R'.
We have
Pr(a(R)= (T 1, aa..........................
(a.....,, aT)) p T ā, a...... ......,AA ...,AAT--- or]
Pr(m = (T 1,ū, (a1,.... ar))]]]] = = 1, = = A/ = 01 N/ = -rr
Since for all 1s, Ft = t/c the policy samples the same actions, up step 1s, i.e.
Pr[A1 an ... Ar- a.-) Pr[A) .1.--.;- Ai. at.
And thus
Pr(x(R) (T + 1, i. (01, U7)
Pr[m (R') (T 1, (0) ...... 0T
Pr[r T + 1 ā, Ar
Ar ar A1 01 A.. at+)
Pr(r' T +
A/.41 41.
A/ or
A 01
A/. 0+.
Let us denote by 41,..., le the time step corresponding to the beginning of the phases when æ interacts with R. and "/....,'y the the time step corresponding to the beginning of the phases interacts with r'.
Also, let tx. be the beginning of the phase for which 5* belongs in phases. Similarly, let tgg be the beginning of the phase for which t" belongs in R' phases.
Since the actions a1....,d0 ar are fixed, and = r for t < t", t falls in the same phase under both R and R'. Thus, 4t. = t/y and K.
Step 2: Using the structure of GPF,
Let Sg. = Es=i Is Ys. be the noisy partial sum of observations collected at phase k* for 11-1 x/ + Yk, be the noisy partial sum of
where YE. Lap(1/e). Similarly, let S:" -e=te observations collected phase ks for r", where Yg. Lap(1/c). Using the structure of GPE) we have that:
(a) If the value of the noisy partial sums phase k* is exactly the same between the neighbouring R and R', then BAI algorithm π will sample the same sequence of actions from step /* onward, recommend the same final guess and stop at the same time, with the same probability under R and R'. Thus, for any 8 E R:
A = 01 Ar, arr, Sp. = =s Pr(r T + 1, ā, Ar = +1 Ar =o
Pr(7' T A' ā, A/.41 a. Al. ar A 01 A/. at., SyP
(20)
This is due to the fact that, in GPF (c), the reward at step t* only affects the statistic Sg_, and nothing else.
(b) Since rewards are [0, 1], using the Laplace mechanism, we have that
Pr[Se- = A . Ar- ar. e PP = A ai.... A,. =a.)). (21)
Step 3: Combining Eq. 20 and Eq. 21 aka procsscoe:::ing
34

---

We have
= + a, Ar - A or A = = Ap. = ac-l
-/0 Pr[t = T Ar+1 :1... Ar or Ai a....... ,.....,,. Ar.
0r-, -0-, S.
JER
Pr|S R. = A = , = ap-
e² P = â, A/+11 = 0p,41...., = or
SER
A = a1..... A/e axr, 5
Pr(S 8 A1 = 01,..... : A/c ac-)
= Pr(T = T + 1, A' = ã, A/.+1
Oe +1, ................................ A/- = or A al ,
Ai. = =a,.
This concludes the proof:
(r( (T 1, (01 (r1....... S 6e Pr(m(R') = (7 11, (01 (1:...., (T))
E,3 Recalling the post-processing and composition properties of DP
Proposition (Post-processing Dwwork and Roth 2014)). ). Let M be mechanism and f be an arbitrary randomised functiooo function defined on M's output. M e-DP, then f o M c-DP.
The post-processing property ensures that any quantity constructed only from private output is still private, with the same privacy budget. This is consequence of the data processing inequality.
Proposition (Simple Composition). Let M , M be k mechanisms. We define the mechanism
G:
M
as the i composition of the mechanisms M
Mr.
If each M' is e:-D,, then G is 1
Proposition 3 (Parallel Composition). Let M .t....,, M be k mechunisms, such that k < where n the size of the input dataset. Let l,,... .. Lx, 1x+1 be indexes in [1, n) such that 4 th ty+1
Let's 's define the following mechanism
g (
xn
M
1
g is the mechanism that we get by applying each M to the i-th partition of the input dataset (x1....,III) according the indexes t tx tx-1.
If each M is e-DP, then G is e-DP.
In parallel composition, the k mechanisms applied to different "non-overlapping" parts of the input dataset. If each mechanism is DP, then the parallel composition the & mechanisms is DP, with the same privacy budget. This property will be the basis for designing private bandit algorithms.
F Concentration Results
In Appendix F
we detail the proof of all our concentration results. In Appendix F.I we start by
introducing variant of GLR-based stopping rule using the modified transportation costs W
35

---

(see Appendix G.2.1 for details) which are defined based on the modified divergences Appendix G.I.I for details). The proof of Theorem i given in Appendix F22 Appendix F3 we
d= (see
show tail bounds for a sum between independent Bernuulii and Laplace observations that feature the product of the tail bounds of each process. We prove time-uniform and fixed-time tails concentration for Laplace distribution in Appendix F.4, and recall existing results for Bernoulli in Appendix F.5 In Appendix F.6 we provide tail bounds for sum between independent Bernoulli and Laplace observations that feature the modified divergence d, defined in Eq. (32). In Appendix F.7
we give
geometric grid time uniform tails concentration for the reweighted modified divergence.
F.1 Modified GLR Stopping Rule
The modified GLR stopping rule defined as
MGLR Fe.
inf Va a Un, Wejeni (Pn Nn)
(kk with ān arg max(īn.alu aE[K]
DE (Rn.a)
(22)
where (Ma, Nn) are the outputs of GPE (c). The modified transportation costs (We, (a,.a,)). are defined in Eq. (34), i.e., for all (n, w) IK E and all (a, [K] such that b.
W We,a,ll w) Il (wual (mold) inf wE(0.1)
{wade (Ma, u, r.((w)) + wadt (Mu, u, r(wb))
where r(x) = 1+logi+4 is defined in Eq. 33 for all x 2 1. The modified divergence oे are defined in Eq. 32 i.e., for all (A, (4, E R x (0, x TR4
kl(2, + 1 - (areAA=2)] 2)] -2)1
= (AT inf =E(M.AA))
[I(2,,)) + 1 -n(re(2- {))}
(A,,,)) =1 (リ> (7))
inf =E((A)or))
1 where h(z) = - log 2 (vI+**-1)) is defined in Eq. 31 for all x > 0.
Lemma 9 gives stopping threshold under which the modified GLR stopping rule is d-correct.
Lemma 9. Let d E 1) and e > 0. Let vj O. Let 1 and & be the Riemann function.
Let W (x)) - W -W-((e--) for all x 2 1, where W-1 is the negative branch of the Lambert
W function. It satisfies W (1)) = + log see Lemma 52 Given any sampling rule using the
GPE, combining GPE (c) with the modified GLR stopping rule as in Eq. (22) with the stopping
Threshold
log <((s)) 8
ō(k,8)= W-1
+ slog(k) +3 3 log 3 log
(23)
yields 8-correct and e-global DP algorithm for Bernoulli instances with unique best arm.
Proof. Lemma yields the e-global DP. Let Es,a+, U Naaaa Es,a,- with
Es =f Vn N, Nn.- Tt (um.ar Mar, Nn.a "in.a//nndd /kn,a'))) S Alkn,a .)} Va + Fu, Ea,a-- = Vn VVn
N, N, No (Dm,a, Nn.o/ Nn.//n,a) 7 ((knaa;))) 6)
where (Jin, Nn. kn) are given by GPE (c), T as in Eq. 23 and di as in Eq.
Using Lemmas 20 and we have P Per (Ef. < 5/K for all a + a", and Pur (Ef + $ 5./K.
By union bound over E K], we obtain un (EF) S
Let 2.,
MGLR as in Eq. (22 and ãn E arg max [K] [in, Then, we directly have that
PPT
MGLR Te,6
+00, à MaR # a²) S Pur (8]) + Pur (Es n (T MGLR < +00, ā M,M + a²))
36

---

S 6 Pu (Egn (TAOLR +00, d,yaa # aa))
Under &s n TO < +oo, a.yaa + at) }. by definition of the stopping rule as in Eq. 7 and the stopping threshold in Eq. (8) we obtain that there exists a" and n E N such that Pn.ald > [Nn.a= and
Σ E(kin,, 6) < W We,a, (Mn, Na)
be (a,a"
{ (Nnade (Anas u, r( r(Nna)) + New (in u, r rrrn((no ((Nnaa))) wE(0,1)
inf
(uarus (ua.Ma-)E(01111- uosuar (Nmade (Un,ar Mas (rrrrNNn (Nn,a)) + an (Unarr Uay r(Nn,a))) ))3
inf
I Na (Fm,a Ma: r (Nma)) + Nn di (Dmaa' Sonnat,ea Par r r(Nu,aa))
I Naadt (Pm,as Pas Nna/kaa ain a + Nn dt (Fm,at Hary Nn ainna)))) S
Σ bEla,a*
a(kr,,; 0),
where we used the definition of W ae in Eq. (34) and Lemma in in the two equalities and Ho- > Po. in the following inequality. The second to last inequality uses that r r(Nn,a) I Nn./kin, for all [K] by definition of (kn, N,), i.e., Rig < 1 1++ 10+1o No, S knna 1, and that di (X, u, r) is non-decreasing, see Lemmas 31 and 32 The last inequality is obtained by the concentration event Eg. Since this yields contradiction, we obtain Es n (TMLLR +00, a,waR + a²} 0. This
concludes the proof, i.e., PuT
MGLR +00, 0-MaR F a² )at)s S T,,
F.2 Proof Theorem
Lemma 5 yields the e-global DP. The proof of 8-correctness is the same as the of Lemma detailed above. In particular, we use the same concentration event Eg Es n Nazza E5,a,- that satisfies Pv)(E S 6.
Under Eg n [Te,6 < to0, âz.s + }²}, by definition of the GLR stopping rule as Eq. and the stopping threshold in Eq. (8), we obtain that there exists + a" and N such that [Nn,a]ū > a l..
Σ (
(c(NNn,b))) (2(Nn.1.))=
c(hin,,,,,)) < W We,n (Pns Nn) bE(a,a))
Σ
Then, obtain
W We (Kn, Nn)
inf { (Naade (Un,a u) + Nn (un.ar, wE(0.1]
(an.u, JE00,11, uaSu, (Nnod (Ki, ua) + Nno dft (Rn,ar, Uay) }}
inf
I Nn.ade (Nn,: Ma) + Nn. (Bmmar
where we used the definition of W ee..a- in Eq. (4) and Lemma 35 in the two equalities, and (uar, Ma) (Ho*, E 1] that satisfies ue. >> in the following inequality.
Using Lemma 39 and initialization yields mii(r((,,,))) r ((N,a)) > 0 by. When jiān, > and pa- > i Lemma 30 yields
Nn.o- (dt (in,ar )a-) - dt (jm,ar, Mar, r (N ))) S kin (Nn.e- (log(1 + 2c Nn.a (=,(aa,a)))
+ ++1)
Nn.(ee (d. (ji,a Ha) - d. (Jn.or Bay ((Nnaa)) S kin (Nnjo) log 1+
An.a
+11
Knp (Nn.))
37

---

where we used that MH)) E (0, and 2/r(2) = 1+logi+y, 2 = (z) When fnnalo S Ha. we have di (p Pa) = 0 = di (Pn.as Ha: r(Nn,a)). When Na S aM los we have d (Anma*, (aa) O di (in,a" i Mar, r(Nn,e=) )). In either case, the above inequalities are still valid since the left hand side is null and the right hand side is positive. Therefore, we have
do (Pn,a-aa + Nn,dd di (an,),Ha>)))
S Nmadt (Pn.a Pas r (N) + Nna-d (Pmar, Par,
r r(No,,
+
Σ
C2 c)
BE be(a,))
Σ aknn, 6) +
Σ
(2
Σ
(cr c2 (2(N
bE(a,0*
bEfa,a"}
DE(u,a"}
where the second inequality uses the proof of Lemma 9 and third leverages that
((($)
T(kin, 8) S W
log
+ (N )) + 3 log - -3 + log2,
8
by using that W-1 1 is increasing (Lemma 52 and kn,a I 1 + logi4., Nn,a (N,,a,,) for all E [K], as well as rxx) x z/kin(z) (z)) Combining all the above inequalities, we have shown that
Σ (AI(Nn,,,)) 6) + c2 c)) < We, (Mas S bE(a,-)
Σ
(a (cI(Nn,b.)) + o2 (2 c))
bEla,a"}
This yields a contradiction, hence we have &g n (Te.8 < to0, āze, # a²} = 0. This concludes the proof, i.c., Pew (Te,6 < +00, are,s # a²) S 6.
F33 Fixed Time Tails Bounds for a Convolution of Probability Distributions
We derive general upper and lower bounds on the upper and lower tails of the convolution (i.e., sum) between two independent random variables (Lemma IO}. We provide upper (Lemma [2] and lower (Lemma] 12) tail bounds for sum (i.e., convolution) between independent Bernoulli and Laplace Li.d. observations for a fixed time. The bounds are expressed as a function of the infimum over a bounded interval of a - log(.) transform of the product between the (upper or lower) tail bounds of each process. Therefore, in Lemmas [2 and [2] we can plug any bounds on the (upper or lower) tail concentration of each process. While those bounds are standard for Bernoulli distribution (Lemma 17 in Appendix F.5). we propose new bounds for Laplace distribution (Lemma 16 in Appendix F.4.
The main difficulty when studying the sum of two random variables lies in the fact that it involves the integral of the convolution of their probability measures. In all generality, it is difficult to upper bound such quantity. The main idea behind our proof technique to split the event of interest into partition of carefully chosen events. Then, on each of those smaller events, we derive a "tight" upper bound on the integral of the convolution of their probability measures. It is reasonable to wonder how one could choose those events such that the upper bound is easier to obtain. When the event is defined as the intersection of two independent events, then we obtain a straightforward upper bound by the product of their respective probablities. When the event truly mixes the distributions, we need to function. First, we upper bound a sub-component of this function by a maximum of the product of their respective probablities (on small interval that is defined by the smaller event). Second, after this upper bound, the integrated function coincides with the hazard function, whose integral is the cumulative hazard function. To conclude the proof, it only remains to merge together the different upper bounds.
Sketch of Proof of Lemma 10
smarter approach
control the integrated
To the best of our knownledge, the proof technique closest to ours is the one used to prove Lemma 64 Jourdan et al. 2022). They control the probability that two random variables have an unexpected empirical ranking While tackling a distinct problem, they adopt the same proof structure. They decompose the event into carefully chosen events on which they can upper bound the integral of the convolution of their probability distributions. The upper bounds are obtained similarly as ours, with fewer events to consider.
function of the boundary crossing probabilities of each random variable.
Lemma 10 gives upper and lower bounds on the upper and lower tails of the sum of two independent
random variables. This result is of independent interest.
38

---

Lemma 10. Let I and X be two independent real random variables such that (i) & has bounded support included in (a, 31 and mean μ E {a, 33 and (ii) X has zero mean. Let
Vu [0. 1], Vu (0, p(u,u) == a(1 - -log(u) + log(v))
Then, for all x > 0, we have
P(0 + X 2 + x S P( 2 P0 E (a, nl) + P(A - 0)P(0 E (3, x),33)
(T(0 E [z, P/ 2 + x - 2)} P(0 E 3))P() 2 21)
+ +
sup
(B,
P(0 + A 2 x 2
sup s((uminn(3pptat
(P(0 E PA 2 / + - 2)}
P(0 + 2 μ - x) - P( S PP E jn, 8) + P(A 2 0)P(0 E (a, max z))))
(P(O E (a, 2))((A S p - --2) 2]}. P(0 E (1) - 0)
sup (((ma((a.l_0)..)
sup P(O+ASM-20 (=(max(aw-))),)
(P(0 E (a, 2)) S p 1 x x
Proof. I. Upper Bound Upper Tail. We start by studying P(O+ 2 x where > 0. We can
suppose that there exists y/ E (max ( + B, 0), x) such that IP(0 2 x + lr y/ (P 2 y1) > 0.
Otherwise, the probability of (0 X 2 1 2r } is O, and both bounds are 0 as well. Let y1 be such value, and
1/3 E [ + 14) and y/2 (minfx + μ 3,03.0
First, we note that log 2 x - 91) and - log P(X 2 y1) are finite, since P(0 2 + μ y1 ()) 2 v1) > implics that min 2 x + μ y1), IPA y1}}}> > 0. Second, we note that y/2 only exists when / < 8, i.e., (min(x + μ --,,-,,,,0000 0, * V. In order to study the cases + /z < and x + - 2 / simultaneously. we adopt the convention that the maximum of a positive quantity on an empty set is defined as zero Note that the situation x + μ < B has more subcases.
31)
We partition of the event (0 + A 2 l + 2 into eight sets, namely
(0 X u x. 0 E (a, 3. A R)
{> (max + P 3,0), y1), 0 E [a + 1 11
e μ u- A 2 v1
U10 E (1, vi), >2 x + / 0}
U [x x 0}
( E [10
UfA 0E X, x + / - }}}}
Uft (2
U e (x+ 0 [ 1/2 31}
First, it is direct to see that
{) E [yz, 0]. 0 S f200 [min(8.p+2),,{}}
u (= 0 [ + μ 122 $1
(0 E fx+ 93, U UAE [va, x +), 0 E (0
μr}
v)) C ( x, 0 E [a.
By union bound, the probability of the union of those five events upper bounded by the sum of the probability of those two events, i.e., PPA 2 x, 0 E n4) + P(A < 0, E [min[ p + x), :8).
A. Separate Conditions. Those two events and one of the three remaining do not require to control
(0, A) simultaneously,, as they separate the conditions on (0, A). Thanks to the independence of (0, A), the probability of those events can be simply upper bounded by the product of the respective probability of those conditions. Therefore, we obtain
P(A 2 x, 0 E [a, MI) + P(A S 0, 0 E [min( μ + +x),8)) + P(0 E [x 1 p- - vI, 3). A A2 v1)
= P(A 2 x) P(O [4) + IP(A < 0) e [min{
39

---

P(O E - U1, V1, (I) 2 31).
B. Mixed Conditions. The two remaining events truly require to control (0, A) simultaneously. i.e., consider their convolution. The proof idea is the following: (1) we integrate one integral to obtain one survival function, (2) we make appear the other survival function artificially, (3) we upper bound the product of their survival functions on the whole set and (4) we integrate the remaining hazard function, whose integral is the cumulative hazard function. Let dG and of be the probability measures of and on IR.
For all s E (max fir + μ p 3,0),9)) we have F(A 2 8) 2 P(AZ 2 91) > O. Then, we obtain IP (A E (max(s + μ 3,0), v1), 0 E I + - 31)
1 (6((mxx(0t4-0000) P(O [ - (s)
P(( 2 8) s) E [x 1 (3)
(F(s)
P(X 2 8)
$ sup
{P(A 2 8) s) E [2 + -リ- - s. sE(max(=+ 8.0)..) P(A 2 s)
dF(s)
*E(max -E(max(=+xx-B00 (=tn--B0)).n)
sup
{LP(A 2 8) P ( s, 91)} (- Log 2 91)) + (P 2 0)))
<E(max(=+x-B.0)- )
where we used that P(A 2 max x + μ 3.0)) < 2 0).
For all E (u. x + م 1- v1), we have P(O [z, B) 2 P(0 E |a + μ y1, B1) > 0. Then, we obtain
F( E (H, + -- y1), X ~ 8)
PPA #PA2
- ]ze(n.ttun)) v1)
1
P E [z, B1) P(O E [2, dG(z)
1
5 sup 2 =)) (2, (=E(Hatt--n) vn
dG(z)
P(O E [2, 3)
=E(H,0tH-1) v1)
< sup (P(A 2 (( E (=+)=) #E(v1.2)
(ollogPP( [a + p yr, 8)) + log(P(( e [u, $1)))
C. Combining Results. Putting everything together, we have, for all yn (maxf + μ -
P (0+ p+ S ( P0 (a, + P(A < ()P E [min + 19)
P(8 E 1 yr PP 2 y1)
sup
(F(A s)P(O -- $1) (- log(P((> 2 vi) + log 2 0)))
sup
{P(A> 5) (( E [a + μ -
-, 3)) (- log((P( E fx+ μ -1 + log(P(0 E [u. B)))
*E(v).))
< TP(A 2 x) E |a, (4) P(A < O)P(@ [min
P(0 E [x + / - yr PP 2 y))
sup
(I s)P(0 [z s,
(olog(P(A2 2 y1 P( E | yn 3)) + log( e [u, (1,()) 2 0)))
where the second incquality is obtained by extending the two suprema to (maxfæ + - 3, 2).
which is possible since multiplied by positive value, and factorizing them together. Taking
v E
arg max ([1max(*+==3300
(P(A 2 s)P [r μ ,,)))}>
and the change of variable 2 = x + p
S, i.i.e
sup
[P s) E f (1)
40

---

sup (in
[P(O E [2, (A 2 x 2)}
concludes the proof of the upper bound on the upper tail.
II. Lower Bound on Upper Tail. Let & E (1,, min [3, + x)). Then, we have directly that
{0 E [z, B], 2 - - 2} _ 10 + x
Using independence, we obtain
PPO E [z, 3))P(A 2
x +
P(O p= P -
Taking the supremum over (4, [3, μ + xz) on the left hand side concludes the proof of the lower bound on the upper tail.
III. Upper/Lower Bound on Lower Tail. The third and forth inequalities are direct consequence of the first and second inequalities applied the two independent real random variables 00 and since (1) -0 has bounded support included in [-8, -x] and mean -M (a),-a), and (2) A has zero mean. Namely,
P(0 + - μr -
X -
IP(A < -x) PI) E (4, = P000)
IP(A 2 () E [a, (a) P(- (0 E jmin{ a μt), -a)).
PP(O E ())PA S 0) = -- 0),
sup
(P(( ( -
=E =maaxaa.
sup =E(-N,miu( -aa
(P(-0 E [2, aal) + -
where we used the change of variable =
Properties on the Rate Function and
Lemma 11 gathers properties on the rate function / in Lemmas 12
Lemma 11. Let us define
Vx 1 O, ; = 3 exp(-x).
(24)
On R2, the function f twice continuously differentiable, positive, decreasing and strictly convex. satisfies f0) > 1, lim, f(x) = 0 and
f(x) <
x 2 W Wi (1/0) + log 2) log
where W is defined in Lemma 52
Proof. Direct manipulation yields f(0) 3 - log 2 > 1, 1..im [(x) 00
Vx 2 0, f((x) = -xx + - log exp(=x) 0 and f"(x) = (x - log 2) exp(-z) > 0.
Using that f(x) = 3 23-log2exp(-h1
h1 + 3 log 2)) where h(x)
-log(:),, Lemma 52 yields
f(x) S - h(2 + log > log (e 3-10 2/6)
W-1 (1/5) + - log 2) - + loo S x.
Fixed Time Upper and Lower Tails Concentration Lemma 12 gives an upper and lower tails bound for sum between independent Bernoulli and Laplace i.i.d. observations for a fixed time. Lemma 12. Let μ E 10,1) and E > O. Let Z₁ = E.ejy X, where X, ~ Ber(μ) are i.i.d. observations. Let S2 = E I Y, where Y, Lap(1/e) are i.i.d. observations where (Tu)EN be piece-wise constant increasing function from N to N. Let f as Eq. 24 Then, for all I N and all & >>0 0,
PPZz + S, 1xx (p) S (1.
1
inf (tttu.mimmI.arttt)) "E(max(0,aa 20))
log (P((2, (2)P(S, 2 1(x + 1 -1
IP(2 + S, < 14p - x)) <
inf
log (P(Z, s t2)p(S: S &1( - x -2)1)
41

---

Proof. Let E N and x > O. Then, Ze and St are two independent real random variables such that (1) Z, has bounded support included in [0, t| and mean tu E (0, and (ii) S, has zero mean. By symmetry of (ap(1/e) around 0, the cumulative sum of observations (i.e., St) is also symmetric around 0. However, Ze follows Bin (i) which can be skewed. Therefore, we have
[ax E [1, P(z E (4])) 5 Vz E [0, 1]. P E [t, (]) P( 2 1z) and (P E [0, (2]) = P(Z I st2).
P(S: 2 0) 1/2 P(S, - 0) and
Using that + IP, 2 tz) is decreasing on (1, mim [1, + 24)) and + P(S. 2 t(1 2) is increasing on (1, min [1, x + +,)), we obtain
max 2 2 min (1, + p)) - 0). () > tz) S tt))
<
sup
[P 2 t2) 2 t(x + -
=E SE(p.min(1.mtt/)
Let us define g(x) == z(3 log - log(z)). Using Lemma 10 for (Z, and considering f.r 0
and & E (H, min μ + x)} (i.e., tz E (tj,, m p + x}), we obtain
P(Z: + S; 1(r nk) S
(P 2 12 Ps 2 txx 1 =))} selg.min(L.
sup
Let in Eq. 24 of Lemma 11 Then, have ((:) g(exp( (-))). This concludes the proof of
the upper bound on the upper tail. The second result is obtained similarly based on Lemma and the above results.
E,4 Tails Concentration of Cumulative Laplace Distributions
We derive time-uniform (Lemmaa 15 and fixed-time (Lemma [6] tails concentration for the cumulative sum of i..d. Laplace observations. Our proof technique is based on the Chernoff method and Ville's inequality as in Eq. (26). Therefore, we need to derive the convex conjuguate of the moment generating function of a Laplace distribution (Lemma [3). While the time-uniform result requires using the peeling method, the proof of the fixed-time concentration is simpler. To use the peeling method, need control the deviation of the process on slices of time (Lemma 14].
Convex Conjuguate of the Moment Generating Function of Laplace Distribution Let > 0.
The moment generating function of the Laplace distribution Lap(1/c) is defined
AY (0. d'Lap.e (A) log E (ta(p(1//) [exp(XX)] log(1 - x2/2))
(25)
Lemma 13 explicits the convex conjuguate of ahap. and its associated maximizer.
Lemma 13. Let L'aap.e as in Eq. 25). Let us define
Vx > 0, Visp.e()) = max {hr ( and M(x) = arg max Vrrape (A)) XE(0,c)
xE(00))
Then, for all x > O, we have
X(x) 1
(xc)2 (0, and (T) = h((x)((xx)))))))))()))) > 0.
x
where h is defined in Eq. (31
Proof. Let ʃ(A) = Ax Lop. (A) for all X E (0, c). Direct manipulation yields that
2A
+2 X2
AA (0, f(()= = 22 - X2
and f"(A) -2 (62 (62- -2 2232 < 0.
Moreover, for all À E c0,c), we have
x= 1 ( -1)
=
We used that the second solution to the second order polynomial equation is negative, hence not in (0, e). Moreover, it is direct to see that 1 (VI+ (xe)² 1) E (0, e) since v1 + x² - x, as
42

---

is equivalent to 1 + 2 +x+1) + 1)2 which is true when æ > O. Since f is strictly concave, the above computation gives its unique maximizer on (0, e), namely we have A(x) - (VI+ 1) Moreover, the convex conjuguate of V/Lap,e is
(z) = f(A(x)) = v (xe)² 1 log 1 (ze) (VI+ (20)) - ))² (xee2 -
= V 1 +1+ (xxe)² - 1 log
(xD)2 (xee(2
This concludes the proof.
Test Martingale for Cumulative Laplace Observations Let and St EvE(e] Y, where
Y, ~ Lap(1/e) are i.i.d. observations. Let us define
AX E (0, e), Mc(A) = exp(AS; (A))
direct to see that Mo( = 0 almost surely and
(A) Fi-1] M, (AlTx=AA))Xxxxxxt//ooeooll
Lap (A))] M
Therefore, (A) is a test martingale. Using Ville's inequality
Ville 1939 yields that
A5 E (0, 1), AA E (0, P e N, AS, tt (A) 2 log(1/6)) 88.
(26)
Time Unitorm Tails Concentration Lemma 14 controls the deviation of the process slices of time.
Lemma 14. Let and S. = EsEja Y, where Y, ~ Lap(11/) are ii.d observations. Let N For all x > 0, there exists M(x) such that for all 1 2 N,
{S: 2 tx] S (M(x)S, - ( 2 NN(er))
where Mxx as in Lemmal 13 and h as Eq.
Proof. Using Lemma N Va (I) for t 2 N. Then, direct computations yield
obtain ((x) (0, e) and V = h(Ex) 0, hence trh (x) 2
S Lx
S(z)S: - but (A(x)) 2 (x)(xx) Vrap. (A(x))) = Whi (2)
AS Lr (A) 2 NiViap (x) = NN(ex).
This concludes the proof.
Lemma (5 giige time-uniform tails concentration for the cumulative sum of i.i.d. Laplace observations. It is obtained by applying Lemma 14 on slices of time with geometric growth rate.
Lemma 15. Let 8 E (0. 1). Let 0, > 1 and Ç be the Riemann Ç function. Lei h be the inverse of defined as Eg. (3I) which well-defined by Lemma 28 Lete >> O and St Exele) where Y, ~ Lap(1/e) are i..dd observations. Then,
P EEN, St 2 h1
(1 1 1+7
log
(C(s) 8
+ s log (1 logito t) 1) 28,
[(s))
+ s log (1 + log t) 1))) s 8.
F 3 St S - hh-
h-
log
t
8
Proof. Let define the geometric grid N, (1 + 7)- hence we have N ULEn [N,, No+1). For all i E N, let z0))) > 0 to be defined later, and (0)) as in Lemma 14 For all E N, let g(t, 8) to be defined later such that gt, 8) 2 x; (5) for E [N, N241). Using Lemma 14 with x; (8) > 0 and g(t, 8) 2 (5) for t [N;, N441), a union bound yields that
( E N, S. 2 tg(t, 8))
43

---

( N441) S tx,()))
iEN
P (3t [N, N411 (ax(O))) (X(xp.((((:((0))) ((h(ez,()))
iNN
N,
KEN
where the last inequality uses Ville's inequality as in Eq. (26) for all E N. Let us define
1
+
((8)
+ log (1 1081++(0)))
g(t,
h
log
c
1
(²((s)
[6) h-
log
N
Using Lemma 28
we obtain that (6) > 0 and that h-1 is increasing on RT. Using t N441)
and + 1o N₁. we obtain
((s)
oll. 8) 2 - h1
log
s log (1 logglog (1))
N2
(r(((s) ))
hー log
20)
N₂
Therefore, we have
POLEN,, S. 2 (g((,))) S Nch(ez.(8))
ô ((8) 1
Σ VEN
5
sNN
This concludes the proof of the first result.
By symmetry the Lap(1/c) around zero, the cumulative sum of i.i.d. observations symmetric around zero. Combining the first result with the symmetry around zero yields the second result. G
Fixed Time Tails Concentration When the time is fixed and not random, there is no need to consider slices time and we can directly control the deviation of the process.
Lemma 16. Let e and S. = Eee(g Yo where Y, Lap(1/e) are ii.d. observations. Let as in Eq. 31.. Then,
VE N, Vr 0, P(S: N tx) S exp(-th(ex)) e N, 0, P(S. S tx) S exp(-th(ez)).
Proof. The first result can be obtained with the same manipulation in the proof of Lemma 15 i... combining Ville's 's inequality in Eq. 266 with Lemma [4] N = f.
By symmetry of the Lap(1/e) around zero, the cumulative sum of i.i.d. observations is symmetric around zero. Combining the first result with the symmetry around zero yields the second result. C
F.5 Fixed Time Tails Concentration of Cumulative Bernoulli Distributions
The fixed time upper and lower tail concentration of cumulative Bernoulli distributions are well- studied. Using the Chernoff method yields Lemma 17 whose proof is omitted since a classic result.
Lemma 17 (Chernoff Tail Bound for Bernoulli Distributions (Boucheron et al. 2013 ). Let / E (0, and 7, Eee(t) X, where X, Ber(μ) are i...d. observations. Then,
WE N, Vx (u, 1), P(Z, 2 tæ) S N. (), P(Z, I tz)
u))
44

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F.6 Fixed Time Tails Concentration for Convolution between Bernoulli and Laplace Distributions
We provide upper (Lemma 18) and lower (Lemma 19 tail concentrations for sum (i.e., convolution) between independent Bernoulli and Laplace i.i.d. observations for a fixed time.
Fixed Time Upper Tail Concentration Lemma 18 shows an upper tail concentration on the sum (i.e., convolution) between independent Bernoulli and Laplace i..dd observations.
Lemma 18. Let μ E (0, and Z: = EsE' Xs where X, Ber(u) are ii.d. observations. Let (NE)EE be piece-wise constant increasing function from N N. Let > 0 and St Eve(ns] Y, where Y, Lap(1/0 are ii.d. observations. Then,
VIEN, Vr > 0, P(Z. + St 2 t(1 + x)) S (ide (u (,,.tn/e)n)))
where f is defined in Eq. 24 and dे is defined in Eq.
32
Proof. Let t E N and > O. Combining Lemmas 16 and 17 E (i, ut),
we obtain, for all > and all
-et c(x + 2))
log (tz) (t(x + - 2))) 2 kl p) + h
n
where we used that μ 2 0. Taking the infimum on (u, min(1, + ++} on both sides and using that [r plj = min 1, x + / > Jr, we obtain
inf (1.,min(1,,stttt))
log (G.(tz)FF (t(x + - 2))) 2 o. (μ +u+x, x.
where d= is defined in Eq. 32 Since is decreasing on R (Lemma 11
using Lemma 12 yields
P(Z Se 11x + x)) S (Itz 2,, 2,,1//a)))
which concludes the proof.
Fixed Time Lower Tail Concentration Lemma 19 shows lower tail concentration the sum (i.e., convolution) between independent Bernoulli and Laplace i.i.d. observations.
Lemma 19. Let μ E (0,1) and Z, Exe'r X. where X, Ber(p) are i.i.d. observations. Let (ne)EeR be a piece-wise constant increasing function from N to N. Let > 0 and S, = Exe(ne) Ys where Yo ~ Lap(1/e) are i..dd observations. Then,
V, VI P(Z. + S. Stu 5f (u (u 2.
where f is defined in Eq. 24 and ď= is defined in Eqg. 32
Proof. Let t E N and > 0. Combining Lemmas 16 and 17 E (mnax(0,) -0,,, 1),
obtain, for all > and all
1 log (t.( (t(p
=))) kl ne c(2
+ c(2
-1)
n4
where we used that x z < 0. Taking the infimum on E (maxf0, 4- - x), on both sides and using that (u xr] max (0, μ 2) nr, obtain
1
inf (0,
log (tz), (t( x - -=)))] 2 at x, (,,,t)))
where dt is defined in Eq.
32 Since is decreasing R4 (Lemma 11). using Lemma 12 yields
P(Z, + S. 2 t( - x)) S f (t (μ - x, n, t,,/nn))
which concludes the proof.
45

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F.7 Geometric Grid Time Uniform Tails Concentration for a Convolution between Bernoulli and Laplace Distributions
We provide upper (Lemma 20) and lower (Lemma 21 tail concentrations for sum (i.e., convolution) between independent Bernoulli and Laplace i.dd. observations that holds time uniformly on geometric grid.
Geometric Grid Time Uniform Upper Tail Concentration Lemma 20 gives threshold ensuring that a geometric grid time uniform upper tail concentration holds with probability least 1 8.
Lemma 20. Let ò E (0. Let (Jin Nn. kn) are given by GPE, (E). Let s > and Ç be the Riemann function. Ler W -1) = - -W-1( (-----=) for all x 2 1, where W is the negative branch of the
Lambert W fumction. It satisfies W -11(2)) (2) = x + log see Lemma 52 Let us define
c(k, 0) = W- (log (1/6) log(k) + log(((s)) + 22 3 2
(27)
For all a [K], let us define
Es,o. [VVV N, Nn.to (Pma; Pa: /n.a (in,a) S c(kmmas0)}
(28)
where dz is defined in Eq.
32 Then, we have P (E 8 for all [K].
Proof.
Let us define the geometric grid N; = (1 + r))--1, hence we have = UseN N4+1). Let [K]. If Nn. E [N,, N1+1), then we have Nn.a [N:] and kn.a i. By union bound, we
obtain
P. (eC (C,,,,)) = = (3n N, Na,ado (Rn,a Mas N Nn.a/knna 2 c(Kinaa0))
Pur (31 NN, (Nn,y kn,a) ([Ns), A Nn.ate (Ftnaa Maa Nn, /na//Knn an,a) 2 c(ki,asd))
iEN
= =E iEN
(INi) (WV]d((ZNMTT) + S.) Ma, (N///) (1011)22(01 c(i.))
where Z[N.1 is the cumulative sum of [N&] i.i.d. observations from Ber(ua) and S, is the cumulative sum of i.i.d. observations from Lap(1 [/e).
For all i N, let x; > 0 be the unique solution of [Ne]d (xc pas Mas [N:1/t c(i, 6), which exists by Lemma 33
Then, we obtain
(INITT ((Z + S.)/N)) Pa, (N//1) 2 c(i,
= -p + S.) [N:1/1 di (zt + Has Mai (N.1/1))
<P(Z +S, [N: (a)) < f (IN (x: + Mas Ma. (N/)1)) f (c(i,
where f(x) = (x + log 2) exp for all x 2 0. The first and the last equalities are otained by definition of r;, i.e., 1d, Lemma and the second inequality obtained by using Lemma [8 Using Lemma II yields
d (x
+ Ma: Ha. 1/t) c(i, 5). The first inequality is obtained by using
f(x) 8
+
WLo (1/5) - - 3 +lg 2 < x.
Taking
cli,8) W -log ())(()/))) +3 log - log
can conclude the proof since Per (Ef,._ Eien f (c(i,))) S EiEN ((s) 5 6.
Geometric Grid Time Uniform Lower Tail Concentration Lemma 21 gives threshold ensuring that geometric grid time uniform lower tail concentration holds with probability at least 1 8.
46

---

Lemma 21. Let 8 E (0, 1). Let (Pins Nu. kn) are given by GPE, (t). Let c as in Eq.
27 For all
E [K], let us define
Es,a, = -V N, Naadt (Pm,a Pa, Nnaa /na/hnaa) & ((Kk,,as))
(29)
where d is defined in Eq.
0,0,+) 32) Then, we have Pux (E (E8,0,t) & 8 for all a E [K].
Proof.
Let us define the geometric grid N₂ = (1 + r)/=1 hence we have N = UVEN [N, N4+1). Let [K]. If Nn. E [N,, N1+1), then we have Nn,a [Nv]] and kn,a By union bound,
obtain
Pun((E,,,4)) P (IME E N, Na.a (Bn,as Hay Nn. 2 ckknaa:)) 80)
Σ PUR (3i E N. (Nn, kn,a) ([NN),a) A Nn. (fm Har Nna/ (o 2 c(ki,a;00)
= =Er [NNi]ct ((Z + S.)/ Ma, [NT/0) c(((,)) iEN
where Z|N,1 is the cumulative sum of [N&] i..d. observations from Ber(Ha) and S, is the cumulative sum of i.i.d. observations from Lap(1/e).
For all i E N. let x; > 0 be the unique solution of [Naldt (Ma Ma. (N:1/t) c(i, 6), which exists by Lemma 33 Then, we obtain
P (NNil (2 + (.)/[NN1..4. [N710) 2 c(i,
= PP ( + S:) [N:1/i) 2 2t (Ma IIn Mas [Nol/i)
$ P(Z + S, - NN - x,)) - [IN.]at (Na- I:s Mor (N.1/t)) f v ((s) -
where f(x) = (x + - log 2) cxp(-2) for all x 2 0. The first and the last equalities are otained by definition of I;, i.e., [N. (N (/aa xz Pa. 1/i) c(i, 8). The first inequality is obtained by using Lemma 34 and the second inequality is obtained by using Lemma 19 The last inequality uses the same derivations based Lemma I as in the proof of Lemmal 20 by taking
c(i, W - (log (i" ((s)/8) + log 2) + log2.
This concludes the proof since P, (E6 -) S ΣiEn (()) < 8.
5
G Divergence, Transportation Cost and Characteristic Time
Appendix G is organized follow. First, we derive regularity properties for the signed (modified) divergences d= (Appendix G.1 and di (Appendix G.1.1 p. Second, we derive regularity properties the (modified) transportation costs We.a.b (Appendix G.2 and We.,s (Appendix G.2. for pair of arms (a, b). Third, we study the characteristic time for e-global DP BAI (Appendix G.3
G.1 Signed Divergence
Recall [z]] == max (0, min(1,A]} and
kI(A, = Alog ( 11)
V(A, n.) (0, 1)3,
+ +1 1)
where kl is infinity when {u, }} n {0, + 0. The signed divergences di are defined in Eq. 3 i.c.
v(A.)) [0, dī (A. p)) = (μr (A1b)
inf (kI(2, <(>)] =))
= =E(m|A(
(A, = (4+ > (ld) inf (Kl(2, + c(2 -A1d)
Lemma 22
reeates d, and d==
47

---

Lemma 22. Let de and de as in Eq. 33) and ① Let (K,V) E F² with means (A, E (0. Then,
0
if p if < X if X
(2, dt (2,
Proof. When = 1a, we have de (15 = by taking yp - v and using the non-negativity of de.
Let E F with mean 2 E (0, 1). When < X, we have
(x,,,, v) = minf inf (kl(2, 11) + e(A =)} inf (kl(2, + =((A - = =E(0,1)
=E((AN)
inf (ki(2, po) + cc(=- -)}} =E(A.1)
inf (kl(2,)) + c(> - 2)} = de (A, 11)
=EAAAA]
where we partitioned (0, 1) and used that (1) 2 kl(z, ( μم) + E(2 - A) is increasing on (A, 1), hence the infimum on this interval is achieved at X, and (2) + kl,,, (1) + e((2 - z), is decreasing on (0, pt).
hence the infimum on this interval is achieved at μu.
When μ > X, we have
=E(0,A) de(x,,) v) = mink inf (kl(z, + +e(A- - =)}; inf {kl(2, () + + {}}} =E(A.n]
inf (kl(2, + c -}}}}
=E(a,1)
inf (kl(2, + c(z - 2)} = d= (>, 14) =E[A.]]
where we partitioned (0, 1) and used that (1) = + kl(2, + e(2 - A) is increasing on (H, 1), hence the infimum on this interval is achieved at μ., and (2) + kl(z, u)) + E(( 2), is decreasing (0, A).
hence the infimum on this interval is achieved at A.
Lemma 23 shows strong link between d:tt. This symmetry property can be used to carry regularity properties from dt to dr. and vice versa. Lemma 23. Let d= as in Eq. 3 For all μ E [0, 1] and all X E R,
(= X, - n) (2, and d. (1 X, 1 (A) 14)
Proof. By definitions and change of variable ă = 1 2 and k1(1 2, - p)) kl(,, 1)). we obtain
dt(11 X, - (1) = (1x < (*bb) inf
([[1-AAi (kl(z, 14) + <mmax (0, min (1, }} - (1 - =)))
=1 (14 < (blb) inf [kl(1 - 2, 11) + <mmax (0, min (ainn(1,)))) -1 -})}
1 (μ < (Nb)
inf
(kd(2, ()) + c(max(0, min (1, 2)} =d[(A, (A, p2)
The second equality is a consequence of the first.
Lemma 24 gathers regularity properties on the functions 9.t that appear in the explicit solutions of dtt as shown below. Intuitively, those functionals govern locally the separation between the low privacy regime where d is equals to the kl and the high privacy regime where the divergence has to be modified to account for the privacy budget e.
Lemma 24. Let E > O. Let g² defined as
DE
Vx E [0, 1], g²(x)= x+1-e+++ e and 9e (z) x(e - 1)
(30)
On J0. 1), the function g.t twice continuously differentiable, increasing and strictly convex. It satisfies g.t (0) O, 9," (1) = and ge (x) < x on (0, 1). On 10, 11. the function g. is twice continuously differentiable, increasing and strictly concave. satisfies g (0) 0, g. (1) = and
9. (x) > on (0, 1). For all x [0, we have of (ge (x)) = x and 9. (1 - x) + 9. (x) = 1.
For all x E [0, 1), we have lim,-o g (x) = lim 9 (x) x; it satisfies lim 9e (x) = 1 if
* 0 and tom to gf (x) if *
48

---

Proof. Using that e" > 1, direct computations yield that, for all x E [0, 11,
c
e - E
(xt)(x)= = (x(1 - Ed) +
> 0 and (x))²(x) = - (x(1 (x(1 ec) ee) > 0
e5)s
C¹
c'(c'- -1)
(x)) = (a(e- +)3 1)2 0 and (ge (z) = -- (x(e--1)) <0.
Therefore, ot is twice continuously differentiable, increasing and strictly convex on [0. 1] and 9. twice continuously differentiable, increasing and strietly concave on [0. 1). is direct to see that at (0) = 9. (0) 0 and g²) (1) 9. (1) 1. Since they are strictly convex and strictly concave, we obtain g." (x) < x and ge (x) > x for all x E (0. 1). is direct to see that, for
all [0, 1]. we have 9. (9. (x)) = and
(x) 9. (1 r). It is direct to see that,
lim 9- (x) = lim 9. (I) = x for all x E [0,1]. and lim ge (x) = 0 if x + and
lim 9 (x) = 1 if 0.
Lemma 25 gathers regularity properties of d," In particular, it gives closed-fore solution, which is key property used our implementation to reduce the computational cost.
Lemma 25. Let d in Eq. (3) and gttt as in Eq. (30) For all μ [0. and À E R, we have
if E [o, [A]b]
at (A.
- log (1 u(1-uu(1-- -)) kl ())
if E (9r ([A)),
if (0, A p ([>l)- 9. ((A))
The function (1) + dt (A, n.) is jointly continuous on x [0, 1]. For all μ E [o, 1]. the function X + dī (X, p1) is constant on (-o0, 0] and on [1, +00). Then,
is E [0, E (A,
AA (0,1), 0, 1), (A, = kl (A, pd)
log (1 -11 if E (g (A).
For all μ E [0,1], 1), the function X + dt (x. is continuously differentiable, positive, decreasing and convex on (0, p): it is affine with negative slope -E on (0, 9.. (μ)) and twice continuously differentiable and strictly convex on (ge (p),
For all X E (0,1) 1), the function + dt (>, 1)) is positive, three times differentiable with continuous first derivative, increasing and strictly conver (X, 1]; its second derivative is discontinuous at d² 9e (A) with gap Oxz (2, (A)) lim --m (A)- 0n (2, 0. Moreover, we have
0²d
dd du
if E (gT (A), if E (2, (A)
4u (A,1), 1].
(A, 1))
I-M(
#(1-H)
The function dt is jointly convex on (0, 1) x 1].
Proof. Recall that dt (A, = 1 (n > (A]) inf 2ellA|b f ((A] 2)
where ft(,,,,))
kl(z, n) e(z A). Direct computations yield that, for all ([A],,
05. (>, = log
(x(1-1) - 11) (1 2)μ
+e and 0ft
p, =0 = 2 == (p)
0z
Oz
0
(), 2, z(1 z) > 0.
0z2
Therefore, = - f:t( (A. 2) is twice continuously differentiable, positive and strictly convex on ([ib, p). Moreover, = + f:((A, μ. decreasing on ([>] max(g" (μ),A) and increasing on (ga ()), Using Lemma
24 we obtain
1 (A,11)) (A,
kl(g. (p), -aa (u) ge (1 1D) log (M(1 - e¹) + 6 Eg. (1 1))
49

---

log (1 μ01 - -ege =g]((H)
(A, 14, 22 (1)) - log (1 101 E -) -ex.
By definition of the indicator function, we have dt (A, pi) if μs [0, [A]]] When A S 0, for all
E (0, we have
Vp 1). (A. f f. ([A) 14 gt (n)) = -log (1 - pal -) - -c(Al
by using the properties of & - f (>, 2) on (0.1)= (g. ((N))). 1) by Lemma 24 This function can be extended by continuity 1
De ((i))) with value d. (2, 0. When A (0, 1) and
E (A), 1), we have
Vy (), (A,,)) = fe fT u. ge (n)) = log (1 pbl - -
- -((A]
by using the properties of = + (d, (. on (g. (A), = (9_ ((>bb), 1) by Lemma 24 This function can be extended by continuity to (X, (1) (0, limx-s+ (A, 9. ((Al))) with value
d (0.0) = 0. In both cases, this function can be extended by continuity to μ = with value di (A,1) 1) e(1 (Al)).
When X (0, 1). i... [N]b X, and μ E (A, ge (A)) _ (0, by Lemma 24 we have
(A, = = (A, = (1,
This function can be extended by continuity to μ = with value d (A, A) = 0 since kl (A, (x, A) = 0.
Using Lemma 24 this function can be extended by continuity μ 9e (A) (i.e., x 9: (1)) with value
(A, (A)) = kl (2,95 = kl (gg (4), = - log (1 p(l - - eg. (p)
Therefore, we have
AY E (0, Vp E gT
g. (A)], (). k2,
Using that limx- [), (A)] = (0). this function can be extended by continuity to 0 with value O. Using that 1imx-11
(X. g. (A) (1). this function can be extended by continuity to x =
with value lim - log (1 10 - e 5) - -((l]-
Putting all the continuity arguments together, we have shown that (X. p) + dt (d. p)) is jointly continuous on xx 1]. Moreover, is direct to see that, for all p E [0, 1], the function À + o (A.
constant on (-o0, 0] and on (o,+ Then,
if E [0,
[0, 1]. ()
kl (A, 1d) log (1 - p(1-e-9)-- c
ME (N, (A)]
if E (gr (A),
Let μ E [] and À E (). Using that μ (ot (A), if and only if X E (0, (n)). For all J E 1), the function - dt (2, is positive and affine with negative slope -e on 00.9. (n)). Let X E (gt (p), Direct computation yields that
adt 0x (N,
Okl
(A11- 14) (1
(A, 1)) log
< <0,
Ox
1)p1
dd
dd
lim
(2, p,) =
lim
Ox (>, n1)
A (a)+
0x
X (a)
Odt
Kl
1
ax 14)
Ox2 (), 14) X(1 > 0.
For all μ e [0, 1]. the function x dt (A. μ)) is continuously differentiable, positive, decreasing and convex on (). For all μ E [0, 1]. the function X + di (A, μ)) is twice continuously differentiable. positive and strictly convex on (g. (p), Combining the above results concludes the part of X d (A, on (0,
Let X E (0, 1). Let > and & E N. The k-th derivative of u(x) a(1 ax) -1 on [0, 1] is (+)) (z) (k- 1)la =+1 (1 az)-((+))) (k+1), Then,
a Fp
(A, u) = (1 (1- 11)!
Ay E (g (A), 11, V
(1 (1- - > 0,
50

---

ddt
1 X >0,
by (A.g. ge (A)),
(X. 11)
0p
He1 - 11)
d (2,
(14 1)2 X(1 X)
> 00
112 - -1)2
d
(A, (1,
0p3
Direct computation yields
1 X
lim
= (11 -)),
u- (A) pL1 - 11)
--
n- (A) 1 - 111 ce-9) = e A(e -1))
lim
(u- ))2 + x(1
eI- (1- (1--)
A(1 A) ge (A)3(1 ge (2))2 0.
lim
p- (A)
1²(1- 1²)
For all X (0,1), 1). the function - d (A, pi) is positive, three times differentiable with continuous
first derivative and increasing on (A, 1]. For all X E (0, 1). the function μ - d (A, μ)) is strictly
convex (A, De (A)) and (g. (A), 1]. The second derivative is discontinuous 9. (A) with gap
o'd, "o (A, S5 (A)) lim u-25 (A) Zu (A, > 0. Thanks to the continuity of the first derivative and
a²d
the sign of the second derivative, the function μ - d (A. p)) is strict convexity on (X, 1].
Let (Hi, u2) E [0, 1J2 and (A₁, A2) E (0, On the convex set Fo = {(A, p)) E (0, x 00 e [0, A)}. the function di is null hence jointly convex. Let ((11, A1) (122, A2)) E ((0, (0, Fo) Let 22) E [AI,/i p1] x [^2, be the minimizers realizing d (41 and (A2, 12). Since it is
convex set, we have (a\1 + (1 (1-))22 ap1 (1 p2))) ((0, 1) [0. 10) ( Fo for all E [0, 1].
Morcover, we have az1 + -11 a)22 E aa + (1- a2 au1 + (1--++11aaappla a)422 for all a E [0, Using the definition of dt as an infimum, we obtain
d (ad1 + (1 - a)22 au1 + a2)
< kl(az) + (1 - a)zz, a/1 + (1 - (1-a)a2) + e(az) + (1 a)22 (ad1 + (1 aa)))) s (kl(z), u1) + 6(21 A1) (1 - a) (kl( p2) + E A2))
adt (4) + (1 - aad (22)
where the second inequality comes from the joint convexity of the Kullback-Leible divergence.
Combining both results, we have shown that the function d is jointly convex on (0, x [0. 1]. O
Lemma 26 gather regularity properties of di- In particular, it gives closed-form solution, which is key property used in our implementation to reduce the computational cost.
Lemma 26. Let de as in Eq. 33, and gtt as in Eq. 30 For all μ [0, 1] and all R, we have
0
if E [A]], 1]
(A,
log log (1 + ple" 1)) (1(),
if [0, gt ([A]))
if E (0, and μ E los (IAI))((11)
The function (), + di (A. ()) is jointly continuous on R x [0,1]. 1]. For all μ E [0, 1], the function A+ di (7, 14) is constant on (-00,0) 0] and on (0)) Then,
if E [A, 1]
vA (0,1), [0, (),
kl (A, log + p(e 1)) +e if E (0. (2))
if E 1g (A), 2)
For all E 10, 1}, the function X (2, is continuously differentiable, positive, increasing and convex on (pt, 1); it is affine with positive slope on (g_ (11), 1) and twice continuously differentiable and strictly convex on (p. g (4)).
51

---

For all X E (0.1), the function + d, (A, 1o) is positive, three times differentiable with continuous
first derivative, decreasing and strictly convex on [0, A); its second derivative discontinuous at add,
ddd,
9e (A) with gap lim 4-mt (A) 0n (A, (1)
0x (A, ge (A)) < O. Moreover, we have
dd,
e-1 11)
if E (0, (2)) if E (A),
E (),
(A, p))
0p
A-1
m(1-x)
The function dr is jointly convex on (0, 1) x [0,
Proof. Using Lemmas 23 and 24 we have
di (A, = d.t (1 - X, 1 - (4) and Od=
g: (A) 1- ge (1 (1-18 X),
(2, 11) = Odg (1 - X, 1 - 11) and Oμ
dd
dd
0n2 (1, 14) = Op² (1 - X, -1).
Moreover, we have kl(A, 11) = kl - X, 1 14) and
llog (1 + p(e=- 1)) + c[Alb log (1 (1-(1-1 11) (1) N7o
Combining the above with properties of d in Lemma 25 concludes the proof.
G.1.1 Modified Divergence
Let us define
(2 x
(V1+
Vx > O, h(x) == 21+22---- 1 log
(31)
For all (A, ER 10 x R1, we define
(A, = (1x < (a]d) inf =E(M.AAA)
kl(2, + (((r((A
(Apr) 1 (1))
inf sE(Alb
kl(z,))) + - A11
(32)
Lemma 27 shows strong link between d... This symmetry property can be used carry regularity properties from dt to di. and vice versa.
Lemma 27. Let di in Eq. 32 For all (A, E x [0, we have
111 (2, and J. (1 X, 1111-,))--- dt. ().)))).
Proof. Using the definitions, the change of variable = = 1 and kl(1 - - -1) = kl(2, 11), we obtain
1 11(2, 1 14) (re(d (1 (-11))
(1 Hr)) = (p= (N]b)
inf =E(1- (A]] -w]
kl(1 1011-211-pp 11) 5,)-M) + - (re() -2)}
= 1 (μx (}]b]
inf
kl(2, + 1 (re(s -21) }]} = = = (2,1115))
= 1 (p < (A]b)
inf =E(M[(A[
The second equality is consequence of the first
Lemma 28 gathers regularity properties of the function h defined in Eq. 31
52

---

Lemma 28.
Let h as in Eq. G1. Then.
1
Yx O, (1(x) Vx² ++2+1+11 and h²(x) + + x²
0.
On RRi, the function h is twice continuously differentiable, increasing and strictly convex. Moreover satisfies
h(z) ==+0 / 04+ and h(x) --tto (log(2))
Proof. For all x > 0, (z) + log(z))) h2(x) = v1+ - 1 and h3(x)) v1 + x²
Then
(x)= 1
[12x)) = + and n₂(x)= = x
x
+1+
Then, have
Vx > 0,
h(z) = (1)) - 2 log(2) + 2.
Therefore, we have
(xx)) hb(x)) (h2(z)) (hxx))(()((z)))
2
x
1+
1
2
x
v1 + x
v
+ x²
1
1
11
h/3
VI + x
x2
Note that
1
x2
V
+ 1 +1
Moreover, we have W
x² (
- [(x)=-- 1/(1/))
1
(1/z)2 1)
1/x
1 +++* x2 + v1 + x²
By taking the limit, we have lim,--o+ h(z) = O. Moreover, we see that lim h' (z)z) 0 and lim, h=xx) = 1/2. Therefore, one can conclude that h(x) = / + (x*t) by Taylor expansion. The second result is obtained directly by limit.
Lemma 29 provides upper and lower bound on the function - h(rz)/r involved in the definition of d
Lemma 29. Let h as in Eq. 31. Let k(r, = h((x)/r- for all r > O and all x E RR. Then, we have
dR
(r, x) = rzh' rxh'(rz) - h(rx) = log (ra)+ + > 0.
Vr > 0,
Ör
x2
On IRi,, the function r + s(rr x)) is increasing. Moreover, we have
Vr 0, VI I.,,
5. rk((r))) + +log(1 + 2xr) + 1
Proof. Using Lemma 28 and the definition in Eq. B1. we obtain that
2
(V++ log ( + > o0,
Vx > 0,
ah'(x) (x) h(x))= = log
2
where we used that v1 + x- 1 > 2 for the last inequality. Let define
Vx E R4, (1(x)= = 2(1 + 2x)
v1 x² +1
Then, we obtain 91 (0) = 1, lim, (x) and
2 2/ +
2+
0.
V1+ v1 + +
v1+ x² (V1 + x² + 1)²
53

---

Since is strictly increasing on R2, we obtain log (x) 2 log 91 (0) and log (1 S log 4 for all +.
By definition, we obtain
re(r, x) + log(1 + 2xr) 1 h(rz) - rx 1 log(1 + -xxr)
2(1 2(1+ + 2xr) 41+ +1
- (ræ) rz log
Using that 0 S vI+ x S 1 on R,,, we obtain
2(1 + 2xr) = log(91(122)) 0,
rx(r, + log(1 + 2xr) + 2 log
v1 + r +
re(r,0) x) + log(1 + 2rr) + 1 < 1 + log(g
< + log 4
This concludes the proof.
Lemma 30 provides lower and upper bounds on the gap between d' and d==
Lemma 30. Let dt and dt in Eq. and (32 For all (X, p,r) IR (0, x TR, such that [N] < μ. Then,
dt (A, p.) S át (2, log(1 + 2er) +1
For all E (0, 1) x 2 such that [}] > n.. Then,
di (A p)) $ के (2, log(1 2cr)
For all (A, E [0 x (0, 1) x R2 such that X < μ. Then,
dt (A, 1)) 2 dz (A, log 4
For all (j, A,r) E [0. x R44 such that x > μ.. Then,
d te (,, log 4
Proof. Since μ (0, 1). we have [i]] max/0, A}. Therefore, we have
A [X]] and
[A]d E (4 - (Ald) c (0, 1) for all E (D1)., 11. Using Lemmas 28 and 29 and 0, we obtain, for all > and all E ([>))- 14),
log(1 + 2e(2 (Nbb)r +
E(= - (A1b) < ((rr((z (xl)) +
r
< h(re(2 A)) log(1 2er) +1
r
Therefore, for all 2 E ([A]-= p). we obtain that
log(1 2er) +
kI(2, + c(2 - (A]b) s kl(2,
14) +
T n((e(( - 2)) +
r
Taking the infimum over ([A],, μ) on both sides of both inequalities and using that inf =E[[A]bu [kl(z, ()) e(z (8b) =E(())))))
dt (2,
inf
[kl(2, e(z [X]))
k)((,,))= 1 - (orel=-Mi 2113
(A,H,) inf =E((A)ou))
we obtain
d+().p) S 25t (A, log(1 2er) +1
r
54

---

This concludes the proof of the first result. Using Lemmas 23 and 27 yields the second result.
Suppose that À [0. 1. hence r > 0 and all 2 E (1)):
[N]o- Using Lemmas
28 and 29 and
O, obtain, for all
I c(= (A log 1 -h(re(z A1) < A)+ log 4 - log(1 + 2e(2 - A)r)
Adding kl(z, 14) on both sides and taking the infimum over 2 E ([A) (4) on both sides of both
inequalities yields the proof of third result. Using Lemmas 23 and 27 yields the forth result.
Lemma 31 gathers regularity properties on the modified divergences d In particular, it gives closed-form solution based on an implicit solution of fixed-point equation. This is key property used in our implementation to reduce the computational cost.
Lemma 31, Let dt as Eq. have
and g as in Eq. 30
For all μ E 10, A E RR and > we
at (1.piT)
if E (0, [N]]]
[la(at (A, H, + 9t (1), - (,(rr(( (1.1,) + gt (²( X))
if E ([N)) 1)
where xt (2, v) E (max{0, A gf (w) gt (u)) is the unique solution for (max(0,
gt (W)) ge (11) of the equation
re(x gt
(p) -
log 1+
ge (n)(1 x ge (4))
te
1 0. V(re(x + gt ge )))2 +1+1
For all (J,rr) E (0.1) x Ri, the function X + Ji (2, (,, is positive, twice continuously differ- entiable, decreasing and strictly convex on (-o0, p); satisfies lima- de (N, = 0 and limx dt (2, to0.
For all (X,r) E TR i the function μ ~ (X,,,,)) is positive, twice continuously differentiable, increasing and strictly convex ([*]), 1). Moreover, have
dd (X.
gt (1) x. (X,Hrr)
4p ([A]) 1),
0u
M(1 - 1))
For all (>, 1) E (0,1) 1) such that μ E (0, [A]b], the function + Tt (>, Hr,r) is the zero function.
For all (A,p) E R x (0,1) 1) such that j E ([A],, 1), the function r + dt (A, (2,5r) is positive, continuously differentiable and increasing on R.
Proof. By definition of the indicator function, we have dē (A, [4, r) = Of (.. Let (A, 11) such that μ & (0, [N] i.e. ([>],, μ) is non-empty. Since μ E (0, 1), this implies that (-o0, 1) necessarily, i.e., [A]i = max A).
Recall that d. (A, r) (μ > (11)) inf inf F (A. where F. (A, k)(2,
[h(re(2 A)). Using Lemma 28 direct computations yield that, for (A))..)).
0
x(11 (x(1-1) - 11)
(A, 2) = log
+ -ch((r((=- (r((==11))
dz
(1 - 2)11
xx(1 - 11)
+ re(z - A)
= =log
(1 - (1=2)pp
V(re(z 2))2 +1+1"
87
1
8=2 (A,
+ re (re(( - 1)) 0.
=(1 -2) z)
55

---

Therefore, 2 2 ftr (A, 2, is twice continuously differentiable, positive and strietly convex on (IAb..)) p)) Moreover, we have
0ft =eh(ee) 00,
lim 2 z- Ōz
a5²
re(2 - 2)
= (),,,
<0,
Oz
V(re(z - 21)² + +1+1
oft
(,, = log (1(1 (1)
When (Nlo > g: (u),
<0.
dz
(1 - A\p
2. Note that max (X1b (. (n))
max g2 (n)} since ja E (0,1). Using that aft (A, ur, r,) 0=
continuously differentiable and increasing on (IA])= ()), with negative value max X, 9. (1)} and finite positive limit at μ, 2 2+ fet (A, p.r, admit a unique minimizer on (max A, g. (4) n). Let -. (A, (,, (max(), A. (w)}.H) be defined as this unique minimizer, defined implicitly as solution for 2 E (max(A, (A, I (4) of the equation
2(1 (1-1) 11) (1 2)
re(z - 2)
log
+
V(re(z - 2))2 +1+11+1 =0.
Then, we have af_ (A, M, 2) 0 if and only if = (A, p4, H.r)). Morcover, + (), Øx decreasing (1>b: Te (), ()) and increasing (g. (2, 1e, r),
Let us define = - g." (n)) + x where x E (max ge (1) w- - ge (4)). Then, we have
0f. (A, H. Ti ge (m) xx)
Oz
= llog 1+
x
(ireoo+ (N) (1) 2))2 +1+1
r + g1)
ge (1)(1 - x 9t (w)))
+
Therefore, we have get (.. gt (u) + xt H.T) where x (), p) (max A gt (4)} gt. (+)) is the solution for x E (max(0, - ge (1) 11 - ge (4)) of the equation
log (1+ ge (1)(1 x ge (11)
re(x + gt (1) - A)
+
Vire(x+ (free(++ot
0.
(n) - A))3 +1+1
When X E (0,1) 1) and f + x = [A]p- it is direct to see that Dt (, + X. Then, we have
lim dt (A, (A, lim (kl (A, k)) + -+
lim r rot (A
(h(re( (A, (,r) )))} = 0.
-++
Direct computation yields that, for E ([A], 1)) c 10.
0f: (N,
4-2 > 0,
0p
1(1 - 11)
8Jt (μ (І-=2² (N,1,T,))=
+ =(1 -xx)
> 0,
0p2
1=211 -11/2
FzFt
1
0z (A,14,1,2)
+ (reh"(re(2 - A)) > 0,
2(1 =2)
8
1 <0.
Tudz (A,1,
M(1 j4)
of 85 (A. (A. = 0, the implicit function theorem yields that
Since
825 (A, (N,prr.g (A.p.r)) 0.
0gt
(X, (1,
8pos
Op
82
(A, A.P,T.. (A,
56

---

Moreover, for μ E ([>lg- 1),
ddt
Op (,, = (A, (A, + Og.
a5tt
8qu (). r) 01=
(2,16, ((,1,))) 0z
oft (A,r))) 1 ge¹ (w) x (A, 20,
du
pal - p4)
d 8n2 (,,
Je 0n2
0ge (2, > 0.
1,, (2,
Therefore, for all (A,r) E R x R the function -+ dt (A, 14, is positive, twice continuously differentiable, increasing and strictly convex on ([))),1) 1).
Let (4,r) E (0, 1) x Et. Direct computation yields that, for E ([1): p)) (0,
0f:- (1, = -eh((re(z - A)) <0,
0x
0
A)) < <0
0A0z
0ft g, 82
Since
0, the implicit function theorem yields that
Jg.t
8²54 JAOz
(A, ur, (t (A, 22 05
(A, M.T) 2)) re (2
ax (N.
(. ge (A, r)) =(1=5) (relg (A, -)) <1.
Direct computation yields that, for X (-o0,
0dt 0ftt
0g.t
0f.t
(A,
(,, (A,
(A, H.r)
Oz (,, (A.
0x
OD
Oftt
(A, (A,1,))) - ch'(re (2,11,)) - -)) <0,
0x
j... ax2 (1, = re?
Oge
(x (1 A)) >0.
1 OX
Therefore, for all (H,r) E (0. 1) Ri. the function A dt, n,, r) is positive, twice continuously differentiable, decreasing and strictly convex on (-00, 11). Similarly as above, direct to see that limx-1- dt (A, v, = 0 and -im (A, r) +00.
(A,
Let (A, μ) E R x (0. 1). When μ E (0, [X]]] we have d (A, r) 0 for all r e (o,+ hence
dt (A, μ, r) is non-decreasing. Let x as in Lemma 29
Using Lemma 29 we have
oft (A, = Ox (r. (r,c(z= A)) > 0.
V X,
Or
Or
When μ (D1b, 1), we have の. (A, (max(A, gt (1)}, 1) and, for all 0.
oht
05+
ag. Ost Or p, Oz (A, (,,
dr (A, Or (2, (A,p,)))
0f+
dr (N,prri (N,r)))
where we used that T.t (2, > X. This concludes the last part of the proof.
Lemma 32 gathers regularity properties the modified divergences d. In particular, it gives closed-form solution based on an implicit solution of a fixed-point equation. This is key property used in our implementation to reduce the computational cost.
57

---

Lemma 32. Let di as in Eq. 32 and ge as in Eq. 30 For all E (0, X E IR and r > 0, we
have
dz (,,
if E [[A] 1) (II(OE -INIGE (1)) -5 (4,14, ( + 9. (T())))
0
if E (0, (Al))
where I (, z1 - X.1 (,,r) (max(g (1) X.0), (1) (1) is the solution for x E (max (11) 0E (1) 14) of the equation
ge (g)) (w) - m) + V(re(or - ge (1) + X1)2 +1+1
re(u Te (1) + X)
log 1 91
= 0.
For all (1,r) E (0,1) x RRi, the function X + (), is positive, twice continuously dif- ferentiable, increasing and strictly convex on to0); it satisfies limx-pp dt (A, = 0 and o de p.) +o0.
For all (2, E R E the function μ p- (2, is positive, twice continuously differentiable, decreasing and strictly convex (0, (X]))- Moreover, have
ad= dy (2,
Dt (1)) + xe (,,,,,,)))
Aμ E (A])),
M(1 14)
For all (>, 11) E R x (0,1) 1) such that μe E (0, [A]o] the function r + J. Hyr) is the zero function.
For all ()) E R (0, 1) such that μ ([A]à, 1), the function + de (,, is positive, continuously differentiable and increasing on 14..
Proof. Using Lemmas 27 and 24
we have
(,, r) = d' (1 - X, - p, and
ge (A) = 1- ge (1-1),,
dd. (X, H,r) 0μ
0d.t
dd
0d
X, 1 -. and
(N, p.r)
X, - wr,)
Onp
Let (1 X, r), E (max (0, g. (w)
DE (1) 1d) be the unique solution for
(0, ge (1) - -A.ge (p) μ) of the equation
log (1+
1+ 91 (n))(ge (1) x)) ) +
x
re(x ge (4) + A)
= 0,
(re((x - ge (4) + A))² +1+1
where we used g - pu) = -1 ge (44) to simplify the formula given Lemma 31 r),
--
Therefore, we
define 25 (,, = 2 - >2 -1-,,rr) Then, we have
kl(g. (14) - x u1 (,, + [n(r(( (2. -- =
kl(2 (1 X, v,r) + ge 1), (1) Mrreft(11 -1-1, 111-1,777 r) + g: (1) 1 A))
where we used that kl(g, (w) x (A, H4, r), p)) kl(1 - Df (n) + o5 (N, pi). Combining the above with the properties on dt in Lemma 31 concludes the proof.
Lemma 33 shows that we can invert dz with respect to their first argument, which is key property used in Appendix F
Lemma 33. Let d as in Eq. 32 For all (n, E (0. 1) x R x R there exists > 0 such that dt (μ F, r) e. For all (p.r,c) (0, 1) R R there exists x such that di (n x, r)
58

---

Proof. Let us define f(x) d (I I, M.,)) for all x > 0. Using Lemma 31 we know that f is continuous and increasing on i and satisfies lim f(z) 0 and lim, tro f(x) +00.
Therefore, there exists unique 0 such that d (u 2, p4 r) Using Lemma 32 we can conclude similarly for d;
Lemma 33 shows that J. is non-decreasing with respect to their first argument, which is a key property used in Appendix
Lemma 34. Let di as in Eq. 32 For all (1,r) E (0, x i and all (2), E R x (-o0,
at ()
dt (22,
\ S A2
For all (p,r) E (0,1) x R and all (A1, 22) E x (M, +0o),
(Ai, te (A2:
Proof. Using Lemma 31 we known that X + ote (A, is decreasing on (-o0, Let (A1, A2) R x p-) Then, we have
1
(An. < dt (22,
which is equivalent to the statement of the lemma by contraposition. Using Lemma 32 we can conclude similarly for d;
G.2 Transportation Cost
Recall that We,a. defined in Eq. (50), i.e., for all (M, w) TRK x TR
V(a, b) E [KP, We,abuu, w) = (Iual > [pbl]) inf {wad (Ma: u) + woot u))
wE[u,1]
where d are defined in Eq. G3.
Lemma 35 gathers regularity properties on the transportation costs.
Lemma 35. Let dt as in Eq. G3. For all (A, E (0, such that X 2 μ and w E R??.
The function +
de (A, u) + w2dt (M,, u) is strictly convex on [p, X] when
max(w1, W1, W2) > 0 and on [0, when min(w;, w2) > 0. Then,
inf (wid] (A,u) + wid (4, u)} inf twide (A, u) war u), wElmA)
wE(0.1]
The function (A, u. w) + inf infre (A, u) + wydt (H, u}} is continuous on (0, 1) x (0,1) x R.
max W1, w2] > 0, u, w) arg min [e (wi (A, + wyat (n,, u)} is unique and continuous on (0, 1) x (0, 1) T
If min W2) min(d, (A, u. (A, n. w)), dt (M, u, (), p. w))} > 0.
and
n
w,,HH, e (n₂2) and
Moreover,
inf (lidj (A, u) + wad (p, u) -Efo.1]
inf (o1 (0. S01
(wid, (A, wi) + wads (u, u2))
Proof. These results are obtained by leveraging Lemmas 25 and 26 at each step.
For u I μ4, the function is equal to Wid, (A, u), which is decreasing and strictly convex on [0. A) unless wi = since u S μ S X. Therefore, the minimum over that interval is attained at μ.. For 2 X, the function is equal to w2d+ (n, u). which is increasing and strictly convex on (n, 1] unless U12 since u 2 2 μ. Therefore, the minimum over that interval is attained X. On the interval (n. A), the function is equal to 11d5 (A, u) wyd (μ, u). hence it is the of two convex functions,
59

---

one of which is strictly convex. Furthermore, the function is continuous at f and X. This concludes the first part of the proof.
As we have just shown, we can restrict the infimum to [a, Aj. We apply Berge's Maximum theorem page 116]. Let
BBerge 1997
ouu, A, ue w) -w1d, (A, u) w2d (M,
[( μ4, w) |u,
M(A, 14 w) max( X, [
(A, μ., w) aarg max( A. [(),
We verify the hypotheses of the theorem:
á is continuous on [e, A] x (0, (0,1) R2 by using the properties in Lemmas 25
and 26 (A, μ1) E (0. 1)?.
i' nonempty, compact-valued and continuous (since constant).
We obtain that .M continuous on (0, 1) x (0, 1) x R and that upper hemicontinuous. This concludes the second part of the proof.
When max w1, w2} > we have just shown that Φ is strictly concave function of Combining this with the fact that I' is convex, we can argue as in [Sundaram, 1996, Theorem 9.17] to prove that & is a single-valued upper hemicontinuous correspondence, hence continuous function. This concludes the third part of the proof.
Suppose that min (wi, w2} 0 and X > μ.. Using Lemmas 25 and 26 the function
wid, (A, u) + w2d (H, u) is continuously differentiable on (1, A) with derivative w1 Du (A, u)
ad_
ad_ w22 du (p, u) where
Od
if E (gr (4), 1] if E (1, (n)
Yu E (n, 1].
(m, u)
a( u-p p
du
a(1 ul
adt
if E [0. (A)) if E [gt (A);
Vu E (0,
(2,
Itules-1)
Ou
Since
0dz Tu
(A, u) tu-t 0 and
ad Fu
(1, u)
Ju-upt O, we obtain
Od=
+ w2 dd
Odt w2 Ou (M, A) > 0,
lim
w1 Ou
(>,
(H,, u)
Ou
lim
dd=
(A, u) + Ou
Odt
(1,, u)
}= wi Ou
ad (A, 11) <0
Du
Therefore, the infimum is attained inside the open interval. Using Lemmas conclude the proof of the first part of the fourth property.
and 26 we can
Using the strict convexity w1 + wi (A, u1) and u2 + w2d (M, u2) on (v, A), we obtain that inf inf (lid (>, u) + w2dt u)] (w1.02) : Sen (wid) (A, u1) + w2d (a, u2)
mE(KA))
Re-using the same arguments above, we obtain that inf (lid) (h, w1) + wart (m, u2)}
(M1,w2) : Sao
inf (w (2) Siz
fp1d, (A, u1) west (n, (z))
This concludes the proof of the second part of the fourth property.
Lemma 36 relates the transportation costs We with the transportation costs used in Eq. define the characteristic time. Crucially, this shows the equivalence with the definitions in Eq.
2 to
35
60

---

Lemma 36. Ler We a and de as in Eq. 4 and T Let (0, such that a" a²()) = {a²} Ler
Alr(p) = {} E 1)x (A) * {a Then,
Vi Ax.
inf
Σ
Wade(pas Aa) min We ((,,,,((1;,0)))
(EA
axo
aE(K]
Proof. is direct to that Alt(p) Uafar Co where Ca {A (0,1)* Aha 2 da Then,
Vw AK,
inf ΛEAL(())
Σ
wade (Paa da) min inf asaa AECa
E
Wede(Hees de)
aE(K]
cE(K]
By non non-negativity of de (Par Aa) for all a E [K], we obtain
inf Σ
wede inf Σ
Wede(Me, Ac)
cE[K]
cE(a,a*)
inf (E..A.-)((0,111
Σ
(ede de)
d. 2A
cE(a,a)
where the two equalities are obtained by choosing M(a) E (0. 1) such that X(a)s μb for all b & fa, a²} with the two other coordinates choosen freely such that M(a) 2 M(a)a.. Using that paa > paa we can partition this set as follows
Ca.a- = {(aa, ) (0, Ae 2 Ao- = E (0, Au 2 Aa
((-aa)da)) E Ma (0, Ha)}
((Mos )a.) E (Mar, 1)21 Aa 2 Aa
[(xr )a:) E (Har, 1) x [a Ma.
{(Aar da-) E [aa:Ma-] Aa 2 da
Using Lemma 22 Pa- > Ma and Lemmas and 26 we obtain
inf (Auidar (anda))(000..)))) 12 {wa de (Mar, )a=) wade (May Aa)}
inf (E Aat (wa de (Mor, da-) + wad (Mar )a)} wa de (Har Ha):
inf (Aaidar De {war (Haa, )ax) + (e da)}
inf (Nadoo))E 1x(0, twa di (Ma >ar) + watt (Mar da)) = Wa de (Mor Ho) inf (Aaidar 16) Aa
(war (pra >ar) + wa Na))
inf (aaida)) (E(p, 1121 2da- (wa-dt (Mar, )a=) wadt (Mar )a)} = Walte (MaxHar) ( de (Par, : )a-) ) + wad, (May >a)}
inf (Aardat (Aai)a))E((u (1(pa=_1)x(xae.....
inf (Naidr )E(P,* 1)x (Ma Maat inf (Aardar Ml A. inf (Nando)) 1Il P.
(wa- de (Har, dai) + wort (Mas da)} wart (Mor (Ma)Haa) Hay) {wa de (Har da)) + wade (Har )a)}
(wa- (Mar )a-) Wa (Maz da)}
Therefore, we obtain
inf (anda JEC.arr
de )ax) + Wade Wade(Hay da)}
inf (Masdas (Mardaa))((ra MEMaa Ao2 (wa (par )a.) + wadt (Mas da)}
inf (w df (Har, u) + wodt (Ha, u]}
u uE(MeiHuur1
inf (wer df (Mar, u) + wadt (Mas w)} = We (H4, w). wE(00,1
61

---

where the second equality is obtained similarly as in Lemma 35 3y by leveraging the strict convexity of d in their second argument (see Lemmas and and 26). We used Lemma 35 and the definition of Weatta w) for the last two equalities. This concludes the proof.
Lemma 37 gathers additional properties on the transportation costs.
Lemma 37. Let de as in Eq. 33.
. Let (2,1) (0. such that X > When w2 > O. the function + niifeeo,1]fuudd (2, u) + wrd u)) increasing on R+. When wr > the function W2 + minueeoo.j (wi (), + wzd? (ju, u)} is increasing on R4.
Let (d,14) E 10, and P monn (A, u) + w2d (H u)} is concave on R². The function w - (KTT) minuejo.n((udd, (M1,2) + wadd (Ma, u)} is concave on R.
(0.1)*.
The function
Proof. Let w2 > 0 and w/ > wi 2 0. Using Lemma 35 since w/ > 0, there exists u' E [0, 1] with di (A, n') > 0 such that
min fwidd (A, u) + w2d7 (H, u}} w/d5 (A, n') + m2d (H, u') wE|o,1]
wnde (A, u/) + w2d (u, u') min flidz (A, u) + w2d (1, u]} =E(0.1]
Let W1 > and w/ > w/ 2 O. Then, we can show similarly by using Lemma 35 that
min (wid, (A, u) + whd, T (H,4)) u}} min fwid, (A, u) + w2d, (1, u)} =E(o.1]
=
This concludes the first part of the proof. The proof of the second part direct since those functions are minimum of linear functions, hence concave.
Lemma 38 gives a closed-form solution for the transportation costs. This a key property used in our implementation to reduce the computational cost.
Lemma 38. Let dt and ge as in Eq. (3) and (30p. For all (a, c) E R and b E R let ni, (a, b, c) = /ba 4ac-b 2a
For all (), 1)) E (0, 1)² and w E IR2, such that min(wi, w22 > and μ..
When (1) 95 (p) 2 X, or (2) g (1) < X gt (A) S ge (1) and wap-widd E [g (A), g (1)), we wy and
w2twi
have u, (A, p4, w) w2+wr
14, u (4, min {wi de (X, u) w2d (j,, u)} w wlkklA,, uE(0,1]
wokl( un (2, ,4 w))
When (3) ge (p) < h. gt (A) De (R) and u3, (w) E [g5 (M), gt (A) where
wi (e'e" 1) W2 e 9
u3. (w)
(w2 + wT [11 c -
we have u. (), u1, w) = u3, (w) and
(wid (A, u) + w2d u)) min =]
w1 (- log (1 + (w) - 1)) + Eλ) + wz (-log (1 (w)( -e-1) - -eu)
When (4) ge (1) >, g. (A) S 96 (H) and mamtund E (m,g: (A)), or (5) ge (n) X (1)
watwi
g. (n)) and us. < g: (H)) we have u. (2, w) (4, and
(1, w) = ri,t ((w2 w/ wi)(e - 1), (w2 - (wzz + w/ wi)(e( 1)) w2211)
min (wid] (A, u) + w2dd u) _
62

---

= = (-log (1 + (H, 1)) + eλ) + wykl(H, (1, (H,, w))
When (6) de (1) < X gt (N) S De (4) and wantiidd ge (1) and (w) > g.¹ (2), we have u. (2, p1, w) = u2 w) and
E (ge (w), A), or (7) ge (N) < A (t((())
u2 (A, w) = 1 - +,+ ((w2 + 1)), -
(wi(11 A) + w2) - 1)) w1 - -1))
min {wide (A, u) + w2d u)) ve(0,1]
w] U2 (), w)) + wy (- log (1
(1m0)(1155)) E-9) =4L)
Proof. Suppose that g. (1) 2 X. Using Lemma f 2 g: (A). Therefore, for all u E (H,, 2), we have
24 we know that Jr (1) 2 X if and only if
adz W1 Ou
(2, + w2 Du (1, u) = 10 u(1 - u)
ddtt
+ w2 u(1 u)
1
(W2 + w1 )u (WzHt + w1
u(1 - u)
Therefore, we have
Waff + wi
u. (A, Hl, w)
(H,
U2 + w1
Suppose that ge (u) < A. Using Lemma 24
we know that ge (p) < A if and only if μ < ge (A).
Using strict convexity of the function on (e, A), is enough to exhibit one local minimum obtain global minimum on (p, A).
Suppose that 9," (A) S 9e (H). Similarly above, obtain, for all [ (A). 97 (4)),
dd w1
(A, u, + w2 ddt Ou
(H, u) (w2 + wi)u - (w2 + A1
Ou
u(1 u)
Suppose that Wapttrn W2 +w
gst (A). gr (w)]. Then, we can conclude as above that
u. (A, H, w) WzH Vin U2 + wi
lg (A). (m)].
since it is a local minimum of a strictly convex function.
Suppose that wautwrd < g. (A). Since the gradient is positive on lg (A), g. (n)]. we know that tig-- the minimum on (H, A) is achieved on (H. (A)), i.e., (X, w) E (4, gt (A)). Then, for all E (H.9, (A)).
dd wi Ou
(A, u) + w2 Odt
-
4
(H, u) = - 1 u(e" - 1)
+ W12 u(1 - u)
Ou
Using Lemma 24 direct computation yields
Wi Od=
Od Ou
u1
ge (u)
(A, u) + W2
(M, u) > 0
< u
1
du
W2
1+ wi
1 -e (w) (w) ))
1 wi (1
ge
(H)
lim u u-ppett
< u
W2
(1+
W1
(1 ge (w) )) = g.t (A)
Wi
lim
u.
ge (A)
n,
11 40 (2)
wi
W2
where the second result uses that < ge (u) and the last result is obtained by continuity of the differentials (Lemmas 25 and (6) and the positivity on [ot (A), (4)). For all (a, E R and
b E IR, we define (a,
d Therefore, we have 2a
dut (2, u) + W22 Dr (14, u) =0 w1 Ou
adt
(w2 + wi)(e" -1)=² + (w2 (22H + wi (e) - 1)) u w2/u = 0
u. (A, u1 w) =ri+ ((w2 + wi (e - 1), (w2 (w2n + wi)(e" - 1)), w2/h E (u. (A))
63

---

where we used that u. (A, w) E (1, 9. (A)) is unique for the last equivalence, and that the second root of the second order polynomial equation is negative. Notice that uc (A, f4, w) is independent of X.
Suppose that Marttand #22--001 9e (n). Since the gradient is negative on [gc (A), SE (4)], know that the minimum on (H,, A) is achieved on (ge (H)) A), i.c, u, (A, p4, w) E (gr (H), 2). Then, for all (ge A),
0d Ou
dd du
+ u u(1
c u(1 e
(A, u) +
(H, u) = W1
Using Lemma 24
direct computation yields
ddt (A, u) + W1
Odt Du
W2
ge (w)
w2
(M, u) < 50
A >
1 +
1
Du
W1
W2
-- ge" (u)
(2)
)) >
W2
get
(A)
lim
1+
1+
1
A,
a-
w1
w1
lim u-+3. (1)+
1
U22
1
ge (u) )) g. (1)
1 +
W2
1
/ <,,
W1
wi
ge (w)
where the second result uses that > 9. (u) and the last result obtained by continuity of the
differentials (Lemmas 25 and 26 and the negativity on (g, (A), (n)].
Using Lemma
23 obtain
arg min W1 d (A, u) + w2dt (M, u}} arg mig dt (1 X, u) - -2dd (1 - μi u)} wE(0.1]
wE[0.
Using Lemma 24 we obtain
9e (p) <
ge - <1 14,
W2/4 + w1d
W2(1 11) + w1 2)
E (gr (n),
E (1- X, ge -1))
W2 + W1
W2 + W1
Therefore, we can leverage the above case to obtain u. (A, u1, w) 122 (A, w) where
(2, (A, w) = 1 r1,+ ((wy + wn (e' - 1), (wi - (w1 (1 - A) + uz)(e' - 1)) uy (1 A))
Notice that (), ,1, is independent of n.
Suppose that ge (4) < X and g (A) > ge (n). Similarly as above, we obtain, for all a [g (4),gt (A)).
Od c (A, u) + we Ou w1 8u
ddt
c² --1
1 e-e
(M, u) = W1 1 u(E - 1) + u2
1 u E-c)
Therefore, obtain
Od, W1 Du (), u) + +w² du (H, u) >
ddt
w2( (1 - "1 + u(e" - 1)) - wi (e" - 1)(1 w(1 - - >0
u3, (w) = wi(e (e" - 1) uz (1 - (
(wi + wi) (1 e-=) 1)
Suppose that ug,+ (w) E lg. (1), gt (A)]. Then, we can conclude as above that
u. (A, pz w) = (w)) [g_ (#),9t (A)]
since it is a local minimum of a strictly convex function. Notice that (w) is independent of ())
Suppose that ua,, (w)) > (2). Since the gradient is negative (o. (M), gt (2)], know that the minimum on (1,, A) is achieved on (o (A), A), i.e., (2, u4, w) (a= A).
E (g A). Then, for all
0d_ W1
Odt
-W A 4-2 v
1 -
(A, u) + w2
Du (M, u) (p,
W2
Du
u(1 4)
(1 e
64

---

This recovers the condition solved above. As we know that u. (A, w) E (g." (A), A), we obtain u. (A. H, (2, w) = ri+ ((u2 + wi)(e" - 1); - (wi X) + w2)(e 1)) wi (1 A))
w) 22, w) where
Suppose that (3,)(w) < De (u). Since the gradient is positive on [ (4); 9e (2)], we know that the minimum on (1,, A) achieved on (μ. 9e (μ)), i.e. u. (A, w) (Hi 9e (n)). Then, for all E (e (4)), 0dg (X, u)) + Ou Ou
ddt (1,, = un 1 + u(cs - -) + W2
e
u μ
u u) This recovers the condition solved above. As know that u, (A, H4, w) (H.g, (m)), obtain (A, pr, w) = (H, w) where
U1,- (u, w) r1,+ ((w2 + wI)(E - 1), (w2 - (WaH + w/ (E - 1)) w2H)
This concludes the proof.
G.2.1 Modified Transportation Cost
Let > 0 be the geometric parameter used for the geometric grid update of our private empirical mean estimator. Let us define
x
Vx 2 1, r(x) = + 10gi
(33)
which is increasing and only if that a * b, we define
e For all (4, w) E IR x R and all (a, b) [K]² such
I+n
Weab(H, = (Iuald (uold] inf wE(0,1)
{ (adee (Mar u, r(wa)) + wast (mo, u, r(wb))} (34)
where /े are defined in Eq. 332
Lemma 39 gathers regularity properties the function r defined in Eq. 33
Lemma 39. Let r as in Eq. 33). Then,
Vx 2 1, r((x) = log(x(1 + n)/ee)
log(1 7)( log x)2
log( 1))
r((x) = - z(log(1 + +1) (1 + logi x)3
On [1, +00), the function r is twice continuously differentiable. It is decreasing on e/ +n) and increasing on (c/(1 + 11), +o0); its minium is r(e/(1 + 7)) (0, 1). It is strictly convex on [1, e² /1 1)) and strictly concave on (e2/(1 ( ()),
Proof. The proof is obtained by direct differentiation and manipulation. We have
An > 0, +(e/(1+m) eloo(1 (1 + 7)
1+ E (0,
Lemma 40 shows that the modified transportation costs can be rewritten differently, which is key property used in Appendix F
Lemma 40. Let d as in Eq. 33 and in Eq. 33 For all (A, R² such that [사]] > [u]] and (wi, (o0). Then,
inf (wid, (A, u, ((w)) + wed (m, u, r(w2)) wE(0.1)
inf "E((A))-(A)))
fwid (A, u, r(w)) + wed (n, u, r(w2))
inf
( (A, us u..r((w))) wrd un u2 r(w2)))
65

---

Proof. These results are obtained by leveraging Lemmas 31 and 32
Note that the condition [A] [u] implies that ja (-o0, 1) and E (0, (0,+ i.e., 4 max (0, j.} and [A]b- = min (in 2}-
Suppose that μ S 0 and X > 1. Then, we have lulo = 0 and [N] 1. Therefore, the first part of the result holds by definition.
Suppose that S 0 and À E (). Then, we have [u]] = and [t] = A. For a E [A, the function is equal to wyd (i, u, u,z()))). which is increasing and strictly convex on (0, 1). There- fore, the minimum over that interval is attained at A. For u E (0, A), the function is equal to wid7 (X, u, r(wi)) + w2d, (js, u, r(w))). Since it is the sum of two strictly convex function, the minimum over that interval achieved (0, A). This concludes the proof of the first part of the result for this case.
Suppose that μ E (0, 1) and À 2 I. Then, have [u]] = μ and [A]d function is equal to wid, (A, u, r(w2),, which decreasing and strictly convex on (0, 1). There- fore, the minimum over that interval is attained μ. For u (H, 1), the function is equal to w/d, (A, u, r(w1)) + wyd," (js, u, r(w2)).. Since it is the sum of two strictly convex function, the minimum over that interval is achieved in (1,, 1). This concludes the proof of the first part of the result for this case.
For u E (0, μl), the
Suppose that (H, X) E 1)². Then, we have bn]i = and [А = A. For u E [A, 1), the function is equal to w2d (M, u. ((w2)) which is increasing and strictly convex on (0, 1). Therefore, the minimum over that interval is attained at A. For u E (0, p1]. the function is equal w " (A, u, r(w2)). which is decreasing and strictly convex on (0, 1). Therefore, the minimum over that interval is attained at μ. For A), the function is equal to d (A, u, r(w))) wyd, (μ, u, r(w2)). Since it is the sum of two strictly convex function, the minimum over that interval is achieved in (j,, A). This concludes the proof of the first part of the result for this case.
In summary, we have shown that
inf Awid (A, u, r(wi)) + wedt (m, u, r(w2)) wE(0,1)
inf (widz (A, u, r(w))) + wrat (H, r(w2)))
=E((A(b-111))
Using the strict convexity of u1 + wid (A, un rww)) and u2 + w2dt (4, u2, r(w2)) on (Iu]b (Alb), we obtain that
inf "E((M()((())
(wid (A, r(w1)) + wndt (H, u. r(w2))}
inf (11) ) [alp <m Sug
(wid (A. u1, r(w)) + ayd (1, u2 r(w2)))
Re-using the same arguments as above, we obtain that
inf (u1) Suldsmssa<ap (wide (A, wi, r(wi)) + w2d? (1, u2, r(w2)) =
inf fwido (A, u1, r(w))) + wrpt (M, U2 r(w2)))
(E) un
This concludes the proof.
Lemma 41
gives a closed-form solution for the modified transportation costs based on an implicit solution of a fixed-point equation. This is a key property used in our implementation to reduce the computational cost.
xt as in Lemmas 31 and (X, p) E ?B such that NN1 > [Plo and w E [1, +00)2. Then.
Lemma 41. Let dt as in Eq. (32
32 and r as in Eq. 33
For all
inf (wid, (A. u, r(w))())))))))))))))))))) + w2d, (n, u, (,r(w))) wE(0,1)
=wid (A, u² (,, w) w.), + weet (n, nt (A, ul, w), r(w2))
66

---

where u u²(A,,,,,, (A. p4, w) E (14l), (Al)) is the unique solution for u E (144ás (Al)) of the equation
u(wi + W2) wig. (u) w2g"" (u) + wIF, (A, r(w1)) w2xt (u, u, r(w2)) = 0.
Proof. Using Lemma 40 we have
inf {widh (A, u, r(w1)) w2dt (r,, u, r(w2))) we(0,1)
inf (widz (A, u, r(ww)) + w2dt (us r(w2)))
wE(Imlg.1AI))
Using Lemmas 32 and
31
we obtain
ai
9 (u) x2 (A, u, r(wi))
Vu E (0, (Ab)),
(A, u, r(w)))
Du
w(1 u) ge (u) x (H, r(w2) u(1 u)
ddt
Vu (b44) 1),
(1, (((2))
ou
Therefore, for all u E ([u]o: (N])).
dd= (A, u, (w]) w2 dd W1
(M, u, r(w2))
w1 wi(u 9r (u) + x (A, u, r(( (w)) ))) + w2l (u) I (H, u. r(w2)))
u(1 u) u(w1 + w2) (wige (u) + W2g (u)) + w1 (A, (),,,r(w))) w2zt (M, u, r(w2)
u(1 u)
For u E ([m]d (A])), let us define
(u( == u(w1 + w2) wig. (u) w29. (u) + 1i (A, u, r(w))) waa ((,,,)
Using the proof of Lemmas 32 and 31 we know that
lim adit (M, u, (u,(( =0 and lim w-ludd
dे. du (A, u, ((w1))= =0,
a (A
od= du (A,,,)(u)))
ddt Ou
Vu (0, (A]H),
and Vu ([w) 1).
(n, u, r(w1)( 0.
Combined with the strict convexity of /i in their second argument, the equation g1 (u) 0 admits a unique solution on ([r]), (Al)). Since u(1 - u) > O, we obtain the implicit equation defining u* (>, pt, w) as above.
G.3 Characteristic Time
Let v be a Bernoulli instance with means j E (0, 1)2 and unique best arm a" E [K]. i.e., arg max Pa {a*}. For all (0, 1). we define
(T) sup min Wear (o,, w) and ((()) arg max min Wear (1,, w) wEAR axta"
(35)
wE ayaa
TH (V)(v)-1=
sup min W ((,,,, and (V)(v)) = arg max min W. We,e,,bp, w)
where W are defined in Eq.
Lemma 42 gathers regularity properties on the characteristic times and their optimal allocations. Lemma 42. Let W. in Eq. ( Let (Ir (Tr,Tī,) and (w, w in Eq. (5).. The function (n,, w) + minage (u) We (p, w) continuous on (0, 1)k x Ar. The functions + T (v)-1 and v + (T)(v) are continuous on FK. The correspondences v + w (v) and v + w² upper hemicontinuous FK with compact convex values.
67

---

Proof. Let Fo = { E FK I (aEa"(()); Since UaE[k] For = FK it is enough to show the property for all FK for a E [K]. Let a* [K].
First, the function (we v) + minaya- inf uej. (wa- de (Mar, u) + wadt (Mar, u)} is continuous on Ax x &FK by Lemma 35 and the fact that minimum of continuous functions is continuous. It is concave in w by Lemma 33
The correspondence (w, p) + 4 K is nonempty compact-valued and continuous (since constant). By Berge's maximum theorem, get that + T2 (p) - is continuous F& and that v w (v) is upper hemicontinuous with compact values. By [Sundaram, 1996 Theorem 9.17], the concavity of the function being maximized implies that + w (V) is convex-valued.
The correspondence (w, v + A n (wa- = B} is nonempty compact-valued and continuous (since constant). By Berge's 's maximum theorem, get that + T2 (2) is continuous on FK and that w3 (v) is upper hemicontinuous with compact values. By (Sundaram 1996 Theorem 9.17], the concavity of the function being maximized implies that + (V)) convex-valued.
Lemma 4 provides additional properties on the characteristic times and their optimal allocations. In particular, this results show that the (3-)optimal allocations unique, has positive allocation for each arm and that the transportation costs equal at equilibrium. Those properties are key in the analysis of a sampling rule.
Lemma 43, Let We.,,b in Eq. 4 Let (T2, T7,)) and (wi w(,)) in Eq. 35 Let 3 (0,
and v E JKK such that a" (v) {a²} is singleton.
(v)- > 0 and (v)) 00
minae[K] w > and min w3 > 0 for all w² E wz((v) and wg w
the (ß-joptimal allocations unique and the transportation costs are all equals equilibrium
w (v) [w'}} and Ya # a", inf [wi (par, u) + wa,nd (Ho = (v) uE[0,1]
w { and Va * a", inf (wi di (Mar, u) + w (Mai u}} (v)) wE[0.1]
Proof. Using the definition of the supremum with Lk/K E AK and Lemma 35
we obtain
(v)=)= = sup min inf {wa- di (Ma., u) + wa (Mo. u}}
wEAK afa wE[0.1]
min inf {dr (Mor u) + dt (Ha, u)} 0, K axta uE(U,1]
where the last inequality strict uses Lemma 35 and Ha < par for all a + a". Similarly, we can prove that (v) > 0. This concludes the first part of the proof.
We proceed towards contradiction. Suppose that there exists w w (v) and b with wg = 0. Then, we will show T (v) = 0, which is a contradiction with the above result. If b = a" we have (1) min inf w (May u) S min w (Ma: Pa) = 0. uc(0.1]
asa-
If b + a", we have
(1 min inf (wa-de (Mar, u) + w (Mas u)} aza" uE(o,1]
< inf {w de (Mar, u) + w (1b, u)} = inf w de (Mar, u) 0. ac(0.1]
we(0,
similar proof allows to show the result for (v) by reasoning on T (2) This concludes the second part of the proof.
For notational simplicity, we assume without loss of generality that a* = 1 is the best arm. At the optimal allocations, all wa are positive. Let us define Gy(z)) inf inf {d_ (H1 u) + æd (1u, u) for all b 1. Let w E w (v). Then, have
Whb W1
Wh W1
max W1 min Gb WEAK by1
and wt E arg max Wh min Go
WEAN 691
68

---

Introducing oi
mi for all b of 1, using that DBE(K] wi 1, one has wi
25
w 2of1" 2
and vb + 1, w
1 Eef1 22
If x= is unique, then so is w*, Since it is optimal, fat 1K ERK-I belongs to
min631 (1b)
arg max (2s) [zs) ERK--
(36)
1 Ee=2 Fe
Let's show that all the Go (2z) have to be equal.
Let O
{a E [K] \ {1} | G. (xt) = min Gb (xz)) and A [K] [[1] ((1) UO). Assume that A & 9. For
all E A and b O, one has Go (xz) Ga (xz). Using the continuity of the G3 functions and the
fact that they are increasing (Lemma 37 there exists > 0 such that
vb MEEAEEC Go (a)
e/| Go (r) + +/|01) > Ga (x2)
K We introduce Ty = = - e/|AA| for all b E A and Ta = x + + for all a E O, hence Σb=2 Tb
Eiz zi. There exists E O such that min6 Go (it) Go (.zz + c/|O1), hence
mingz1 Gb (Tx) 1+ 82 + ....K 1+ 2
G. (x² c/101)
Go (1²²)
min Go (1/))
x + x2
+ x 1+ 2σ + ... + T
This is contradiction with the fact that x belongs (36). Therefore, we have A D.
We have proved that there is a unique value by y² E R4, such that for all b * 1, Go (2t) = y² Now since G3 is increasing, this defines a unique value for zig, equal to G₁ (v²).
For in the intersection of the ranges of all Gb, let I& (v) = G (v). Then, y² belongs to
y
arg max ye(0,minng1 oi Gx(w)) 1 Enes xs(y)
(37)
For B E (0, 1), the same results (and proof) hold for w (V)) by noting that
(w) = max
3 min Go (wb/B)
wEAx =s by1
Let wh, E (v), since we have equality at the equilibrium, we obtain 3G, (w w/,3,///33
Th (v)-1 for all b b 1. Using the inverse mapping Ha, we obtain w = -306 (v) for all b # This coneludes the third part of the proof.
Lemma 44 shows that an asymptotically 1 1/2-optimal algorithm has asymptotic expected sample complexity which is at worse twice the asymptotic expected sample complexity of an asymptotically optimal algorithm. This result motivates the recommendation to the practitioner of using 3 1/2 when no prior information is available on the true instance V.
Th. w Lemma 44. Let ,Tr,Ti.e a singleton. Then,
as in Eq. (35 Let B E (0, 1) and v E FK such that a* (v) = {a*}
T (v) (v) and (v)
8* 1 3-
1-11-
(v)( I max
with 8r = w
B B-B
Proof. Define for each non- negaaiete vector ub E RK
f(tb) == min inf aza uE(0,1]
(b (Mar, u) + bodt (Na, u)}
(v) is the maximum of (*b over probability vectors sb E Ag. Here, we instead define f for all non-negative vectors, and proceed by varying the total budget of measurement effort available
69

---

EnE[K] Va. Using Lemma 37 f is non-decreasing in th for all a. f is homogeneous of degree 1.
That is f(eap) cf (+b) for all 2 1. For each c1, C2 0 define
(c, (2) max
{
((() up E RRK (a) = Cn Σ
th
C2
(f)
The function inherits key properties of f: it is also non-decreasing and homogeneous of degree 1. We have
(T,(1) max f(() 2 E R Voa 3,
Σ
4 =
KE(K]
[(bb) t tha B, Σ Va S 3 -9(311-9
max
afta"
where the second equality uses that is non nen-decreasing. Similarly, (2(1) = ( - 3r) where
8+ w e.a Setting := max
8" 1-0'
implies rB 2 3 and r(1 3) 2 1 - 1- Therefore
3 1-3
[T, = - g) B)) 2 2 (3" 1- 8*) - -T
Taking 3 = 3. yields that (z) 2mx - -T, s 27 (2)
3
Lemma 45 gives sufficient conditions on the means and allocations in order for the transportation costs to equals to the non-private transportation costs. Moreover, it gives sufficient conditions on the means in order for this equality to hold inrespective of the considered allocation. Taken together, this result allows to have fine and coarse understanding of the separation between the high privacy regime and the low privacy regime for e-global DP BAI.
Lemma 45, Let We,, as in Eq. (4. Let J E (0, 10, such that a* = arg max aa Ha is unique. Let E (R Let E > O. For all E (0, we define fe(x) (1 x) Tta(r--T) (1- x)95 (1)(1 t). Let us define pwr.a Wad Mut to for all + at. For all a", have
Waa tw0
Mar Ma &min {(1 Wo-
Sc(1 par); (1 Wa-
Se(val)
Wa
We, (a (H, war wor kl Mo (a a + w. Walkl(Htssta Ha.
Moreover, we have
max e'E[K], &) (a)) WE(R))K min {1+ and, for all + a", we have
Wa Sc( - (a+); 1
Wa
s.luma) s se/2
Wa
War
[aa (1 - Maa)
Okl
0kl Ox (Mas Har)
log
(Mar Ma)
Ma(1 Max)
Ox
Vw (RT)),
W We,ar, w) wa kl aa) w wakl(pa, 1 aa)
Proof. Let define f((x) = (11x) (1- - for all x E (0, 1). Then, we have (1-Ha)
Pa(1 Ma) -
1 (1-1+)n(e-T) = fe(Ha),
(1- Ha) ua(e" - I1-I+
1 (e 11
Po (1 Ha* )(e- -
(I-T+(I-1u)(ee- -o0 -1) =fell (aa))
6t Ma (et 1) Pa. I+ Mar (6
Using Lemma 24 direct manipulation yields that
fe(1 Mar) < Hat Ha + g.t (Mai) pa = felUa) < pa- Pa.
70

---

De (Haa) < Para
Sc(pa) < War + Wa
Wa.
(Nar Ma))
Wa
get (Har) > Ma*a
fe(1 - (aa)) <
(Mar (aa)
War + wa
Using that max { wen was +wa
S 1, obtain that }
Wa
(9. (Ma) < Hor A Dr (Pa) < (ar,a)
1 +
fc(Ht)) < Mo Ma:
Wa-
(g (Ma) < pat A gt (Ha)) > Par
1 +
Wa
Sc(1 - Haa) < Ma Ma
Wa
(g (Ma) 2 por v (ge (Ha) pe A Mara lot (Ma 2, (Ha))))
(g (ma) 2 par Mar ( 2, (Ho)))
(mintse(aa) Se(1 pa Ha paa
Wa pa- pa S min {11 (1+
Wa-
Sc( par) ))
11
Wa war
Selumd)
Ma Ma S max {min(fe(Ma)) fe(1
1aa' )a
2
Wa*
Wa fc(Mua)
min
fe Mor);
11
We
Wor
Na. Ma < min
1+
Waa
fe(1 Ma-); 1
Wa
fe((oo)
Wa
Wa
Combining those conditions with Lemma
38 concludes the first part of the proof.
For all r E (0, 1). we have
f((x)) (1 (e - 1) x(e² 1)(1 + +afe - -1) -1) -2)=-11) 11) 2x -
-16
(1+ x(e" - 1))²
(1+ (1+x(e" 1))²
6/22-1
[((x) = 0
1 6
(1 + xcee 1))2 - (et 1) (x²(et - 1)+ 2x
(x) -2 -1)
(1 + x(e - 1))3
2e (6- - (1 + a(et 1))3 50.
As fe is strictly concave, the maximum is achieved at 21/3_ er-I with value
(c/2__) 60-11)
(c/2-11 -1 et - 6212 (1- (1-0
62/2
(ee/2 - 1)2
max fe(x) fc =E(U,1)
Let (z) =1(6" - 1) 4(e=12 - 1)2 for all O. Then, we have
(ee/2 - 1)2
c S 5x/4 = <c(c) 2 0.
Then, we have ki (0) = and
(x) 2/2 3E 1 1+ xe" and x[(=)= e² -2)
Using that e=/2 > -2/2, obtain <²(x) O. Using that n² (0) 0, obtain ki (xx 2 0. Using that Ki (0) = 0, we obtain xi (x) 2 0. Therefore, we have shown that
Vc > O, max fc(z)= s c/4.
(E(1))
Direct manipulation yields that
min (1+
Wa.
fe( - Ha-), 1 Wa
fc fc(Ha)
wa
War
71

---

<(1+ min
Wa
Wa
})
fe(x) S c/22 zE(0.1)
max
Wa
Wa.
Taking the supremum over w E (R, E (0, 1) such that a* = a" ax) and over a" E [K] concludes the second part of the proof.
Let a # a". Direct manipulations yield that
Nar - Ma S mmnn - Ho* (a-),fe((aa)
{(++
Vw E (Rt)*, Har - Ma S min
Wa.
fc( Hai), 1 1+ Wp
fc fe(Ma))
W/o
War
Vw E (Rt)*, We (H,, w) = wa kl(Hta , ,a) + w klkk( par,a).
Recall that fe(x) (1 - 2) Itz(e--T)
Then, we have directly that
1 - x -
(y + x1 - x)
fx(x) 2 y
ee
1
1 x(c -)
x(1 y)
Plugging this result, we obtain
c Ma (1 - -Maa
Pa. pa S min(Je(1 - Ma+) Sc(Ha))
Na |4a0
M/. (1 (ba)
2 log
Ma Ma*
Recall that
0kl (Ja- Ma)
0kl (Jay Maa)
Man (1 Ma)
log
Ma(1 fl.
This concludes the proof of the last part of the result.
Lemma 46 shows that our lower bound is larger (hence better) than the one derived in Azize al. 2024
Lemma 46. Let (M, E) as in Theorem 13 in Azize et al. (2024), and T (v) as in Eq. 35 Then, we have (v, _ (v).
Proof. Let T (v,,) in Theorem 13 in Azize et al. [2024]. A sufficient condition obtain (v, < T2t T2(1)) (v) is show that, for all X Alt(ju), have
dal }
Saade(Ho Aa) S min KE[K]
wakl(pa Aa), 6e KE[K]
Σ Wa Walka Ma RE[K]
since we can conclude by taking the infimum over X E Alt(p) and the supremum over w E A on both sides of the inequalities. By definition of d, and evaluation the function at = a and z respectively, we obtain
de(A inf (k)(2, + c(> =1} s min (), - =E(0.1)
By summing those inequalities over [K], obtain
Σ K
Wode (Ma, >a) <
Wa min (kl(pa, Xa), clpa dall
aE(K]
< min
wy wakk(par DadsE aE[K]
Σ
WalHo dul J
[
Using that Ene(K] WalMa >al 2 0 and 6e 2 this concludes the proof.
72

---

In Garivier and Kaufmann [016], the authors show how to rewrite the optimization problem underlying the characteristic time and its optimal allocation as simpler optimization problem. Lemma 47 shows that similar properties holds for e-global DP BAL. In particular, it shows that computing the characteristic time T (V) and their optimal allocation w: (v) can be done explicitly based on solving nested fixed-point equations. This result is key to implement computationally tractable Track-and-Stop algorithms. Additionally, Lemma 47 an explicit lower bound on the characteristic time (v).
Lemma 47. Let di as in Eq. 3). and (I² wx)) as in Eq. (35. Let a o at. For x E [0, +o0), let inf {dr (Mar u) + zd (Mayu), and (x) == arg min {dz (Mar, u) + +2 (Mas u)}. we[o,1]
(z)z)=
wE(Mastar)
The function Go an increasing and strictly concave one-to-one mapping from [0, +00) to [0, dē (Par, Pa)); ); it satisfies that (o = 0 and lim Ga dz (1aa*) 1a)-
The function ua is a decreasing one-to-one mapping [0, +o0) to (Pas Ma- it satisfies that (0) Po and lim Ug (x)
PAa-
Let valy)) be defined the unique solution of Ga(z) y for all y [0, (Jarr (Har, The function Ta is an increasing and strictly convex one-to-one mapping from [0, de (Har Ha)) [0, +o0;; it satisfies that No (0) 0 and y-m (Ma- Mec)
kaly) +00.
For all y E [0, minafoo de (Mar Pa) let us define
de (Ma Ua (a(a dt (Has un (a(aa(a))
Gly) 1 I+E Eupor xaly)
y
and Fly): = E
axa"
The function F an increasing one-to-one mapping from [0, minaza di (Ha),Ha)) to [0, +oo); it satisfies that F(0)= = 0 and lim y-smnnaaaa- di (Pa+.Ba)) F(y) +
On [0, minaya" df (pat, Ha)), the function G is maximized at the unique y² solution in (0, milazaa di (Har,, [a)) of the fixed-point equation Fly) = Moreover, we have w (v)a = (a( for all a + a²,
w (v)a 1+ Ea za( and (1--- 1+ E Eator Ia(
Moreover, we have
(v) 2 min de (Mor W Ha0
Σ axa"
dt (MarHaa))
If E < log
Pa pa(l- we have dit (po: paa) = - -log (1 - Ma- (1- (1-E9) - Ejta and pa(1-11))
d- (Ha*, [o) = log (1 + Male" - 1)) + Efta
know that G. is concave. Let Ua(x) arg min {d_ (Mar , u) + adtt (May for all x [0, +oc), whose explicit formula is
Proof. Using Lemma 37
given in Lemma 455 It is direct to see that Go = 0 and (i) = Mar. Using the optimality condition of (xx), we obtain, for all x E [0, +oc),
(Odz (Har, ux())) + x dutt du
Gx(()) = (x)
Du (Ma, un (x)) + dt (Mas U (x))
= dt (Ma, ua(x)) > 0,
where the last inequality is obtained by Lemma 35 and using that df (Ma, un (0) = dt (HarPar) Ma-) 0. Therefore, G2 is an increasing one-to-one mapping from [0, +oo) to [0, lim Ga(z)).
Let Pa-.a H. t2pa for all x E [0, +oC). It is easy to see that Go = 0, (0) = Mo. and
1tz
limg 400 ph = Ma. Using Lemma 45 we obtain that
lim min {(1 + 1/2) f.(1 pa (1 + x) (e()))) fc(1 par)
*+000
73

---

When Har Pa S fe(1 - Ma+). we obtain
lim
(a(z)=
lim {kl(Ha Maa ,a = skl(Moo Ma. [a) a+t00
2 2++000
=kl(far Ha) lim x+0000
fakl(Ha 1 aa) kl(pa Ha)
where we used that
zkl(Ma: (a) = x Ma log
1- Mo" Mo
+ log
1
Ma. Ha
1
Na Ma)
Ma* + x
Ho.
1 1- Mo
+
(Ba: Ma)² lim
lim {akl(HHas x ++00
0,
Ma(1 )aaa )²
x +00
Ca + Ma
1- +
where we used that log(1 1 + x) =x-0 (z²)). Using Lemma 26 and the proof of Lemma 45 we know that pa- - Pl fe(1- (1-1))a) and only if ua E [g. (Nar) ); (o ) hence we have kl(jnn,, (a)
d7 (Mar,Pat This concludes the proof in the first case.
When Pa-
Pa > fe(1 (a*). obtain
lim Ga(x) lim { log (1 + u1,, (Ma; ()= - 1)) + ejaa + ækl(Ma (Ha 2-++00
+00
where
u1, (M,a =
(x(1 Males 1)) (E 1))² + 4(1 x)(e" 1)zpaa (x(1 Male" - 1)) - (e 1))
2(1 + x) (e=
Direct manipulation yields that lim U1 "1, (Ma, = Mas hence
lim Gz(z)= -log (1 Ma(e" 1)) Efta x+-00
lim
fakl(ua U1 (Ma:
+10000
Let us denote 21,- (Ma, = "1, x)) Ma 2 0, i.e. lim on (Ma, x) = 0. Direct manipula- tion yields that
U1, U1, 22)
= 1 Male" -1)
(1 1
2( 11
+
2x(1 Male" + +)){EE 1) - (e- 1)2
-1 (c (c-- - 1)(1 2(aa)
1² + (M+Maa ( (ct 1]
xx1 Ma(e - 1))
2x(1 Malet + 1))(e - (er -1)2
11 (e 1)(1 24pa)
21 + Hale - 1))2
x(1 + Hale - 1))
(e 1)(1 - 24pa)
2x(1
pa Paae" < + 1))((e 1) (e -1)2 + 0(1/a²)
xx1 + Ma - 1])
2x (1 + Ma(e" 1))²
2x 1)2(1 Ma)
+ (e" - 112
+ +((122))
2x (1 + MalE 11))2
2(1 -Ma))ale - 1)2
hence
U1 (14a; 2) =20 x(1 + pa(e - 1))²
+ (11/2²)
where we used that VI- - -1= -x 1 + ((2²) to obtain the last result. Similarly as before, we derive
UI (Pa: x) ækl(Ha, U1 (Ma, x²)) = (pe (oo (1 1- Ma(1 Ha) + U1 (Ma: x3/ H/a
1
11,1 11,1 (Haa
+log
1 1- pq
(Ma x)),,)))) x))(1- 14a)
74

---

(Ma,,
lim Mar
lim
400000
pa(1 Maa2 )²
+00000
Ni (He + 21 (1
1-Ma
I11,, (Haa; x)
0,
Ma(1 1aa)
where we used that U (Mo, 2) =2-+000 (1)/x) to conclude. Therefore, we have shown that
1imo (x) log (1 Ma Ma 1)) + Efaar Using Lemma 26 and the proof of Lemma we know that M4a - Ma > fe(1 )aa) if and only if μa E [0, gt (pa-)) hence we have llog (1 Ma(e" -)) eHaa de (Har.Ha)) This concludes the proof in the second case.
Therefore, G. a strictly increasing one-to-one mapping from [0, +00) to [0, de (Ma- Pa)). Using the implicit function theorem, obtain
dd The (Ma: Ua (a(x)
Vr [0, +00),
(1((x)=
<0,
ou (Ma-, (x)) + x afd (Mas Up (z))
where the strict incquality is obtained by using properties in Lemmas 25 and 26 (Par Ha.) by Lemmas 35 and 25 Similarly, obtain
since un (x) e
Odt (Na; un (x)) 0,
Ve [0, + oo), (²(=)= (x))
Ou
Therefore, have shown that G. is strictly concave and that ua is decreasing.
Let us define (y) as the unique solution of Ga(x) y. which well-defined based on our above computations. Therefore, we have
y = di (Mar, ua(a (Ia (v))) + xx(ydd (Mar uo (x)())))
Using the derivative the inverse function, we obtain
1 =1(0)) = G² G²((a(y)) dt (par uo ualar (x)()))) > 0,
Vy E [0, de hence Fa is increasing [0, dē (Har, pa)). Moreover, we have
(((x dct (Mas un (Ia( 0,
Vy [0, di (a)), (n dt (Ht Un (z (z))) Ou
hence Ta is strictly convex on [0, de (Mar Ma)).
For all E [0, minaza di
let us define G(yy = I+ zo(y) and F(y)
Eayar 45 (amns Using the above results, direct manipulations yield that, for all y (F.,...(..( [0, minazaa dr (Mar (H)),
G'v)) 1 -afao Faly) Eafar xv))
1+ Easa Pa(y) - Eupa' at (t
(1 Eator 2y))
(1 Lafa xx()))²
F-F(y)
(1 + Eujea La (y)))²
hence obtain that G'(y) 0 if and only if F(y) = 1. Using that xa(()) O. (0) di (Bar, ) = O, we obtain that F(0)= =
Mo and
Using that lim (Mo-.Ma) (2) = +o0, lim (xx) = Mar df (Mar, Ma) dt (Has Ma) we obtain that lim lim g-mmingga di (Mansaa)) F(y) +oc.
0 and
Let Hw)) Eatar di (aa for all E [0, minaga df Then, have
Σ xa(2) -1)) F(y)
Σ
2'(2) H(y).
azta
axa
75

---

Od:
dctt
We conclude this part of the proof by using the (Mar, Un ua(x)) du (Me: Uo [a(x)) =0, de (Ma- ua (a(aa()))) + Ia (v) (Ma, ua (a(xa(())) =
Du
Then, for all y E [0, minafa de Ha)) we have
u' (1)) (3) dd,
H1()) = E
(de (Mai (a (1) (y))))2 du (Ma, Uu (x(a)
afa
U/ (Fa
adtt Ou (Mar un (x(rll)))).
(de (Ma, Uha (Fa
aza
ul ('(
Fl)) = Σ
(dt (Mas Ua (aa(aa)))))))
aFt'
Out dt (Ja, un (a(xa(V)))
(Nar, uh (xI()))) - die (Mar, uo (mn (1)))) Odt du
(Ma, un (Ira (((())))
Ou
( (dt (Har un (x (x)a())) Du
ddt
=- =- Σ
(Mar un (z)()))
axa
(Ia (v) (Ma: th (a di (Mar, Uha (F
y Σ (F (v) ayta (dz (Mar u (Fa du (Pas (x)( yH'
dct
Therefore, showing that H increasing a sufficient condition to show that F is increasing. Using the above results, we have, for all # a* and all [0, minaşoo de (Bar Pa)), we have
1
dd
(d (Ma, ua (v)))) du (Ma: u, (x))) 0 and ('((Iyy)) O.
Therefore, H is increasing as summation of increasing function, hence F is increasing. Let y* such that Fly") 1. Reusing the above manipulation, obtain
Gyvy) (1() + Eupor Ta (v)) + 2(1 - F(U)) Entar (V) (1+ Layar Za 2z))3
(V)(1 + yH - F(u)) + 2(1 - F()))(())
(1 + yH F()))3
GO( H'
Hy))² <0,
Therefore, y* is the unique maximum of G. intermediate results in the proof of Lemma 43
By strict convexity of and using its properties proven above, obtain
F. (v) 2 (0) + (0) d (Has Ho.)
Summing those inequalities, we obtain
Vy E [0, min de (Ma*, Ha)), G(y) 1 Enpar La Kaly)
afk"
Σ Eafa di (MarPar)
Using that + 1/(1/ (1/y /vv + a) is increasing for > O. we obtain that
T (v)
max yc (0 d. (Mar .n.)))
G(y)
minaza* di (Ha)) + Easar at (pasltar)
This concludes the proof of the second last result. The last result is obtained by combining Lemmas 26 and 25 and the derivation in the proof of Lemma 45
76

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Lemma i8 is technical result used in the proof of sufficient exploration of our sampling rule.
Lemma 48. Let de as in Eq. G3. Ler (0,1)* There exists a > O such that
Cp = min (a,b):Ha Pp
inf Au:As:
{dr (Na: u) + dī (A6, u1} 0. (38) wE[0,1]
inf
max lMfe Aol
Proof. Using Lemma 35 for w1 = W2 = 1, the function j + inf fdr (Pa: u) + d (Po. u)} is continuous on J.K Since it has strictly positive values when Maa 146 (Lemma 35). there exists or such that
inf Ast ma [Mfe Ael
inf { (Na: u) dē (Ab, u)} > 0.
=E[u,1]
Further lower bounding by finite number of strictly positive constants yields the result.
Lemma 48 is technical result used in the proof of convergence towards the optimal allocation of our sampling rule.
Lemma 49. Let de as in Eq. 33. Let (01, 02) E (0,1)² Let = {uE (0, |a E a" (1)) for
all E [K]. For all a" E [K], all Ia,, all (a, E ([K) \ {a²))² such that a # b, and all
e [0, 1), define
Ga,(u1 = inf {.d (Mar, u) + did5 (Pay u)} inf {Ad, (Mar, u) + dad (16, u)} uE(0.1]
wE(0,1)
The function (1,, B) + Ga, (3) is continuous on (0, 1) [0, 1]. For all & 0, the function (), inf 3:13 Ga,b(H, 1,,,,,,,,, 33 continuous on (0, 1)
Proof. Since UvE Kj Is = (0, 1) is enough to show the property for all [K]. Let at E μ E In., (a. b) ([K] {a*) })² such that j b. As done in Lemma 42 by using Lemma 35
[
we obtain that the function (1,,8) + Ga. (4, 3) is continuous on I.. x [0, 1] for all a" E [K].
hence on (0, 1) x [0, 1]. Let o : 3: 1 3 < { it is continuous (constant), compact
valued and non-empty correspondence. Using the above continuity, Berge's theorem yields that j ++ inf 8 --Alsc Ga,([,, B) is continuous on (0, 1)x
H Asymptotic Upper Bound on the Expected Sample complexity
Let v be a Bernoulli instance with means / E (0, 1)2 and unique best arm a* E [K]. i.e., arg max,e(K] Ma {a*}. Let B 1). Let w "t,a(() = {wi,e) be the unique B-optimal al-
location defined in Eq. (35). which satisfies minae[K] W 7,3. > 0 by Lemma 43 At equilibrium, we
have equality of the transportation costs by Lemma 43 namely
Ya + a",
We,t . (1,, w.,a) (1)-1
(39)
where We,,,b is defined in Eq.
④ and Th is defined in Eq. (35
Let > 0. Let w E Ag be any allocation over arms such that min, Wa > O. We denote by T, (w) the convergence time towards w, which is random variable quantifying the number of samples required for the global empirical allocations Nn/(n 1) to be y-close to w for any subsequent time, namely
(w( = inf { T 2
N n-1
vn T,
SNg
(40)
The proof of Theorem 7 follows the same analysis as the unified analysis of Top Two algorithms, see, e.g., Jourdan al. (2022). Appendix is organised as follows. After recalling some technical results (Appendix H. we prove sufficient exploration of our sampling rule (Appendix H.2). Second, we prove that convergence time towards the 3-optimal allocation of our sampling rule (Appendix H.. has finite expectation. Finally, we conclude the proof of Theorems (Appendix H.4
77

---

H.1 Technical Results from the Literature
Lemma 50 relates the global counts (Ne,)) and the local counts (Nn,a)ae(K)-
Lemma 50. Let 1j > 0 be the geometric parameter used for the geometric grid update of our private empirical mean estimator. For all (a, k) [K] N s.I. [ (a)] +00, NTa(a), NEa [(1 + ++)))=111 For all a E [K] and all n E N, Nn,a 2 Nn, 2 a / (1/ 11)
Proof. Let E [K]. After initialisation, we have k = 1, Ti(aa = K + 1 and (N, Using the definition of the phase switch, it is direct to see that NT,(0), = N2. = [1 + when Epx [Tx[]] +oo. Similarly, we obtain (Tx( NN [(1+n)** " when En- (a) < +00.
The last result is a direct consequence of the definition of the per-arm geometric update grid.
Lemma 51
controls the deviation Nº,a BLn,a enforced by the tracking procedure.
Lemma 51 (Lemma 2.2 in Jourdan and Degenne (2024)) For all K and all [K], -1/2 S No.0 - BLm,a 1.
Lemma 52 gathers properties the W- function used in the stopping threshold.
Lemma 52 (Jourdan et al. 2.23)). Let W -1())= == W (-(-e-e- (()=eeeefaaa for all 1, where W-1 the
negative branch of the Lambert function. The function W -1 increusing on (1,+00) and stricily
concave on (1,+ In particular, W -_1xx)- = (1 (+() for all 1. Then, for all y 2 1 and x 2 1,
W W-1ly) Sx
y S x log(z))
Moreover, for all 1,
+ log(s)
< W (1) log(s) min
2 v
Lemma 53 gives an upper bound on a time define implicit as a function of W -1. namely it is an inversion result.
Lemma 53 (Lemma 32 in Azize et al. 2024). Let W defined in Lemma 52 Let A > 0, B> Sscch that B/A + log A and C(A, B) sup {ar x < logg x + B). Then, C(A, B) < (A, with h (2, 9) zW (y/z 2 + log
Lemma 54 shows that upon correction the supremum of sub-exponential random variables is also sub-exponential random variable.
Lemma 54 (Lemma 72 in Jourdan et al. 2022 ). Suppose that (Xn)n21 are sub-exponential random variables with constants (C,). such that inf, C. > 0. Then sup,, (Xn./ log(e + n)) is sub- exponential.
Lemma 55 gives coarse convergence rate of the private empirical estimators of the means towards their true means.
Lemma 55. There exist sub-exponential random variable Wi, such that almost surely, for all a E [K] and all n such that Nn. 2 1,
Nn.alpn,a - Pal W₂ log(e + Nn.a).
In particular, any random variable which polynomial in W has finite expectation.
Proof. Let define
No. Pinaa pal
W
max sup a K] nEN log(e + Nn.a)
Let E [K). Let us define the geometric grid N₁ = [(1 + x7) for all E N, on which we effectively need to control the concentration. The maximum of a finite number of sub-exponential
78

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random variables is subbexxoneentil. Therefore, using the geometric update grid, it suffices to show that
No (Zn. + +k)//kk Ma
sup
KEN
log(e + N2k)
aub-exxponental, where 2N, the cumulative sum N₂ i.i.d. observations from Ber( and Sk the cumulative sum of k i.id. observations from Lap(1/e).
Using that ZN2 Nu sub-Gaussian and Sy is sub-exponential, for fixed k. IZN, Natta Sk is sub-exponential. Applying Lemma 54
we obtain that
No (Z.. + S4)/Nk Pa
sup
KEN
log(e + N₂)
sub-exponential. We finally obtain that the maximum over the finitely many arms has the same property.
H.2 Sufficient Exploration
The first step of in the generic analysis of Top Two algorithms [Jourdan al. 2022 consists in showing sufficient exploration. The main idea is that, there are still undersampled arms, either the leader the challenger will be among them. Therefore, after a long enough time, be undersampled. We emphasise that there are multiple ways to select the leader/challenger pair in order ensure sufficient exploration. Therefore, other choices of leader/challenger pair would yield similar results.
still
Given arbitrary phase p N. we define the sampled enough set, i.c., the arms having reached phase p. and the arm with highest mean in this set (when not empty) as
Sh {a [K] Nn,a 2 (1 + n))
and
a arg max (oa aes!,
(41)
Since min Ha - mol > 0, a is unique. Let E N such that (p - 1)/4 E N. We define the highly and the mildly
sets
UT (a [K] N,,a (1 (1/(--1/22 and V = fa E [K] Nn, (1 + m7) 3(p 3(p-1)/44
(42)
Those arms have not reached phase (p- 1)/2 2 and phase 3(p 1)/4 4, respectively.
Lemma Lemma556showe shoss that, when the leader is sampled enough, the arm with highest true mean among the sampled enough arms.
Lemma 56. Let Sk and at as in (41. There exists Po with Eun [exp(apo)] < too for all a > 0 such that if 2 for all n such that 5% + 0. we have
For all a E ST. we have in,o (0. and a arg max Pn,a-
If B, E Sg, then Bn = a,
Proof. Let Po be specified later. Let p 2 Po- Let N such that Sp. + 0, where Sp and a as in Equation 4T). Since Nn.a (1 + n)--1 for all Sg. we have Nu. 2 (1 77)--1 Using Lemma 55 and + log(e + r)/x is decreasing, we obtain that
Nn,a; 2 Ma; - W log(e+ (1+(++(++)))) 1)--1)
(1+ 7)-1
Va E Sp. (anh An,a Na + Wr log(e + (1+ (1 + 11)
Let Amin = millag |Ma mal and Ao min min( pa > 0. By assumption on the
considered instances, we know that Amin > 0. Let pi [log (X1 - -e]l + with
X, = sup fx >>>11 1| x S 4(min Amin: Ao)) W, log + +e]
< h1 (4(min Amin.
Wt, e).
79

---

where we used Lemma 53 and lı defined therein. Then, for all p E N such that 2 + 1 and all such that S o 0, we have
Va E Sn Pa - manmin min: Ao3/4 S inaa S Ma + min! Amin 403/4.
Therefore, we have Pn.a (0. for all Sh. Since Pn,a; 2 Maz min f A Au) /4 and Inna S Ma + min (iin: 4 for a E S {a we obtain a arg max Pn.a since arg max Nn,a is unique. The leader is defined as Bn = arg ma] obtain
Bn Sp. we
Bn arg max arg Un.o a.. DESS
aEST
hence En (exp(op1)] +oo by using Lemma 55 and hı (z, e) ~--+00 lloo to obtain that exp(ap1) is at most polynomial in W Taking Po P1 concludes the proof.
For all a E R2, we have exp(op1) d (X1 /a/////// log?, l)a/
Lemma 57 shows that the transportation costs between the sampled enough arms with largest true means and the other sampled enough arms are increasing fast enough.
Lemma 57. Let Sh as in Eq. (4I). There exists p1 with Eyn [exp(apt)] < too for all a > 0 such that if p P1, for all n such that SY, # 0. for all (a, b) (S%)² such that pa > Pb, have
(Pn, Nn) 2 (1 n)²
where C₁ > 0 is problem dependent constant.
Proof. Let Po in Lemma 56
Let p 2 Po- Let N such that S% 0, where St. as in Eq. 41
Since Nn.a (1+ r) for all 5% have Nn. 2 (1 77)--1 by using Lemma 50 Let
(),b) E (Sg)² such that pa ps. Using Lemma 48 there exists a₂ > 0 such that
Cn
min (o,b)-He>14
inf Aa:An:
inf {dr (Aas u) dt (Ao: u)} > 0. wE(0.1]
mAX.E (u.6) Me Ael
Let 17 > 0 s.t. 7 < 1/ min Amins Do, an} where Amin = min MMa - mol and Å = [ik] min(Ma. 1 Po). Similarly in the proof of Lemma [56] we construct p2 with Epa [exp(ap=) < too for all a > O such that if p 2 P2, for all n such that S * 0. we have Iin.a - Mal S 1 for all a E Sg. Therefore, we have An. = [ūn,a]d and [ūmn,e] = Mn,6. Moreover, we have Un,a 2 Ma 7) μbb 2 jnn,s. Then, we obtain
W We,a,b(än (n)
inf aN (Kn,s u) + Nn, (Km,b: u)}
wE(0.1]
2 (1 + 1)) inl {dte (An,as u) + d (ūm.b.u) u)} we(n,1]
2 (1+1))=1
inf
inf {dt (Aa, u) + dt (Ab. u)) 2 (1 + p- me(o.1]
AaiAs: max.e(a.a) life Sel
This concludes the proof.
Lemma 58 shows that the transportation costs between sampled enough arms and undersampled arms are not increasing too fast.
Lemma 58. Let 5% be as Eq. 41 There exists p2 with Euz (oxp((22)) < +o0 for all a > 0 such that if 2 p2, for all n such that S + U, For all p 2 P2 and all such that Sy * 0, for all a E S and b & Sg.
W Wea.slii N,) $ (1 + 1))
where Dp E (0, +00) a problem dependent constant.
Proof. Let n E N such that ST. + 0, where SE as in Eq. (41) Since Nn. 2 (1 + 1)p-1 for all S2, we have Nn. 2 (1 + m- by using Lemma 50 Likewise, Nn. (1 7) for all
80

---

b b 5% we have Nu. < (1 + [1+ Let Sh and b & S. Since the result is direct when [Nnald S plo we assume lin, > Min.ol in the following.
min || - 146 and An construct Pa with E (exp((22)) < +o0 for all a > 0 such that if p 2 P2. for all such that S o 0. we have [in.a - Mal I for all a Sp. Let ge (x) a(I-et)+e" as in Lemma Using Lemma 24 for all
Let > 0 s.t. 7 < mmn Aut where Amin [i] [a 1 pal > 0. Similarly in the proof of Lemma 56
a E 5%. we have
1> Ma min Amiu- Ao)/4 2 Pma > oe (Bm,) 2 ge (Na min ( Ao)/4) > 0.
Taking u = Pn,a E [0, and using that di (An,a (n.a) ) = 0, we obtain
W Weaablin N,) = inf
(Nn,ed (Amaa u) + Nu (Pn...)) u)}
we/u,1]
S &Nnod (Ko,b Ro,,) I 7)-=1d de (z bi (n,
where the last term is positive since in.a > lin, and in,a E (0, by Lemma 25 When An, S O, Lemma 2ilyie that
dtt (Km,br Amaa) = - loog in - (1- Se,
where used that x log (1 x(1 - )) is increasing on (0, 1). When un, (0,ge (Rna)), Lemma 25 yields that
at (fn Pn,a) log (1 Mn,,a1 (1. cfn,bb Se.
When An, E g (Rm,a); an, Lemma 25 yields that de (pn am) = kl an.a) ) S log min - (n,a
- log (ia min min Do / /4, Mo min Amin. /4}/4} where we used the classical result that kl(g.)) S log min (p. 1 - -p). Let us define
Dp e max KE[K]
{- min(ua min Amin A0)/4, 14, Ma minn(min Ao}/4}}
Then, have shown that df (Jin,bs a S D, where D (0, +0o). This yields the result.
Lemma 59 shows that the challenger mildly undersampled if the leader is not mildly undersampled.
There exists ps with Epw (exp((ass) < to0 for all a such that if p 2 ps. for all n such that UR # 0. Bn & Vp implies Cn Vp.
Lemma 59. Let VP be as in Equation 42
Proof. Let ps be specified later. Let P3- Let n N such that U. + 0 and V + [K]. where UP C VP are defined in Eq. (42) Since the statement holds when B, E VP, we suppose that
B, $ VP in the following.
Let Po as in Lemma 56
Let P1 = 1/3), which satisfied that Eyn (exp(аp4)] < +00 for all > by Then, for all p 2 Pa= = max 2pp 1, 4 max p1] - 1/3] and all n such that B, & we have Nn,a E (0, 1) for all & V.. B,, bū = arg max Mai B,, UP and
P1 and C₁ as in Lemma 57 and P2 and D2 as in Lemma 58
max max(2p2 1, max (Po Pt
using Lemmas 56 57 and
58
3(p 1)
vb & (b UVP Webn Na) + log Nu. 2 (1 - 1))
3(p 3(p 1)/1/)/// +
log(1 + m),
4
We o(n Nn) + log Nn. $ (1 + (1- (1 D, + p-1
Vb E Up,
log + +n),
2
where we used Lemmas 56
57 and 58
Direct manipulations yield that
n) 10800-1)/4C,,41
3(p
log(1 + n) 2 (1 (- 1)/2 D,,, P-
log(1 (1 + 7))
81

---

P2Ps = 4 4f1og1+> to (Dw/C,)] +1,
where Evg [exp(aps)] < too for all a 0 since it is deterministic constant. Let max Pr) which satisfies Eug [exp(apa]] < too for all a > O. Then, we have shown that for all p 2 P3. for all such that Bn & VR. we have B, = bn and
min (b;)uve
We,6,,66im N₂) + log Nn.b) > max (We,5, (0,, N₂) + log Nn,6)
DEUR
By definition of the TC challenger, i.e., Cn E arg min (( obtain that CR E VPP. Otherwise, there would be a contradiction since we assumed UR & 0. This concludes the proof.
(P,, NNa) + log Nn.bh. we
Lemma 60 shows that all the arms are sufficient explored for large enough n.
Lemma 60. There exists Ng with Eug [No] < +oo such that, for all n 2 No and all a E [K]. Na. 2 Vn/K.
Proof. Let po and p3 in Lemmas 56 and 59 Combining Lemmas 56 and 59 yields that, for all p 2 P1 = max (P3. 4po/3 - 1/33 and all 7i such that UR # 0, we have B, E V or C₁ E VP. We have (px[(1+m)))
< +oc. We have 11+ 2 K 11 3/)-1//)////4 for all p 2 ps = 4 [1og1t K] +1.
Let 2 max (p5, }. For notational simplicity, we conduct the proof as if that k(1 + n) n) E N for all [K] It direct to adapt the proof by using the operator [.].
Suppose towards contradiction that U ((1+m))> is not empty. Then, for any 1 < < K + 7))= U. and V?" are non empty well. Using the pigeonhole principle, there exists some a E [K] such
that N(1+n))-1. 1. (1 + 8)--1)/4 Thus, have V ((++))=1 K - 1. Our goal is to show
that VP *2(++
S K - 2. A sufficient condition is that one arm in VP (1+1)0-11 is pulled at least
(1 + n) 3/ 1)/4 times between (1 + 77)=-1 1 and 2(1++ 2(1+n)p-111
Case 1. Suppose there exists a E VP (1+m)-1 such that L2(1+r)) - L(2t))-11 2
(3-1 ((1 + 1)3(0-1)/44 11/4 + 3/2). Using Lemma 51 we obtain
Na N2 - No (1+1)0-1,0 2 ((22)1))) L(1+n)--1,a) 3/2 2 (1 +1) 1)3(0-11/44 13/4
hence is sampled (1 7) 3(p 1)/4 times between (1 + 17) and 2(1 1))p-11- 1.
Case 2. Suppose that for all a E VP ((tt))-11 we have L2(14t)0-1,0 L(++p-11 3-1 1 ((1 11]+))3ppm 3(p-11//4 3/2). Then,
Σ
(I2214))-11 aa - 1(14m)-11aa) 2 (1+1) K3-1 (1++))00-1)44 +3/2)
OgVE (1-----1
Using Lemma 51 obtain
(No (1 No 1at -8
Σ
(L2(1+))011, - -(1t))p1aaa)
agVE (1+m)P-1
agVe (1-m)--1
$ -3(K-1)/2
Combining all the above, obtain
Σ
(L2(1t))-1 ((1t))-11a)
Σ
(Na (12 - No -Mitm)o-1aa))
agve (1-11)--2
agvi (1
2 (1 - 3)
Σ
(12(1t))7-1, L(14n)--1,a) - 3(K - 1)/2
ag V (111)2-1
2 (1 -1) 8) (1+ [1=-))=1_ -K31 (1+ 1)/4 + +3/2)) - 3(K - 1)/2 2 K(1 + 300
82

---

where the last inequality is obtained for p 2 P6 + with
P6 sup { PENNEEN (1- (11 1)) 1))-->)--- - K3-1 (1+ 1)/4 + 3/2))
3
2 - 1)
<[1 11)3pp11/44)
The left hand side summation is exactly the number of times where an arm & V (1+g)e-1 was leader but wasn't it sampled, hence we have shown that
2(1+m)---1
Σ
1 1(B, Bu $ VP (1+ a1
= a. K(1 +1) 1))
(1+7)111
For any (1 ()) St 2(1 + +1) 1, Up is non-empty, hence we have Be & Vp
(1+0)² (hence
B, & Vi) implies C E V2 _ CV (1+))-11 Therefore, have shown that
2)
1-1 Σ
2(1+n}² 1-1
1(x+ ((t))o-))
E
EV V2
Σ
V (1+n)--1; ai =c. ((1+n) 3 3(0-1)/4 (B.
(11+-))0-1
Therefore, there is at least one arm in V (1 + (1+n)== and 2(1 + n))=1
((+n)--1 that is sampled (1 + 11) 3))((-1)/4 times between
S K for p 2 Pr max m P4. By 0 for all p 2 p7. Defining Ng = K(1 + 1)-11 we have Eu [o] < +o by using Lemmas 56 and 59 for P4 = max (P3, 4pa/3 1/1// 1/3} and and are deterministie. For all n 2 No. we let (1 (1+1)==+= K
In summary, we have shown VP 1
induction, for any S S K, we have VP
S K and finally UP K(1+a)>
R(1+t))
Then, by applying the above, we have UP K(1+1))-1 Un (olog..((////---__) is empty, which shows that
Nn. 2 Vn/K for all E [K].
H.3
Convergence Towards 3-Optimal Allocation
The second step of in the generic analysis of Top Two algorithms Jourdan et al. [202 is to show the convergence of the empirical proportions towards the 3-optimal allocation. First, we show that the leader coincides with the best arm. Hence, the tracking procedure will ensure that the empirical proportion of time we sample it exactly B. Second, we show that a sub-optimal arm whose empirical proportion overshoots its B-optimal allocation will not be sampled next as challenger. Therefore, this "overshoots implies not sampled" mechanism will ensure the convergence towards the 3-optimal allocation. We emphasise that there multiple ways select the leader/challenger pair in order to ensure convergence towards the 3-optimal allocation. Therefore, other choices of leader/challenger pair would yield similar results. Note that our results heavily rely having obtained sufficient exploration first.
Lemma 61 shows the leader and the candidate answer are equal to the best arm for large enough n.
Lemma 61. Let No be in Lemma 60 There exists N1 2 No with Epr [Na] < +00 such that, for all 2 N₁, we have ünn (0, 1) and an B,
Proof. Let Amin minazo (Ma- Ma) and Do min min(Pa, Ma > 0. Using
Lemma 55 we obtain, for all n No.
an. 2 Pa- W log(e +
(1)
Vn/K/ In/K/(1 /11 +1)
Ya # a", An, S Mo + W log(e + Vn/K/(1+1)
Vn/K/(1+m)
83

---

where we used that r - log(e e + 22/2 is decreasing and Nn,a 2 (1ta/(1m)) 2 Vn/K/ + n).
Let N1 = m [K + 7) ] where
Xi = sup f- > 1 x S m
W,, + + S (4(44 20)-111 Was e),
where used Lemma 53 and hn defined therein. Using Lemmas 55 and 60 obtain Eur [N]
too. Then, we have 0 Ha A0/4 & An, I Ha + Ao/4 < 1 for all E [K] Moreover, for all
Ni. we have an.
2 Ma- A and Un,a S Pa + mmna/ for all o a", hence
a arg max Pn,a arg max ā, B
aE[K]
aE[K]
This concludes the proof.
Lemma 62 shows that that the pulling proportion of the best arm converges towards 3. It is direct consequence of Lemma by using the same proof as Lemma 39 in Azize et al. [2024] hence we
omit the proof.
Lemma 62 (Lemma in Azize al.
Let y > O, and N₁ be as in Lemma 67 2024)
There exists
deterministic constant Co such that, for all n CoN1, have Nu,a"
B S 7
n-1
Lemma 63 shows that if sub-optimal arm overshoots its ß-optimal allocation then it cannot be selected as challenger for large enough
Lemma 63. Let E (u,) where is a problem dependent constant. Let N1 and Co be as in There exists N2 CoN1 with Eys +o0 such that, for all n 2 N2.
Lemma 67 and 62
Nn.a 2 wi
3a + a"
Cn t Fa.
Proof. Let 17 0 and 7 > 0 be small enough, which we will specify below. Let ÿ E (0, Let Ni as in Lemma and Co in Lemma 62 ý. Let n 2 N, Therefore, we have in (0, 1)* and
ä. B, a* and N,,.* B 9. Using the same proof as in Lemma 61 there exists N3 with n-1
Exx [Na]] < too such that, for all N3. we have || pl S Let m
Let # at such that Na.s 2 8 + 7. Suppose towards contradiction that n w2, for all
Nu.
x-1 b & {a*,a). a). Then, for all n 2 N1 have
Nn.a*
Nn
1- +
Σ
7 +
Σ B = 11
n
bydaa
which yields a contradiction since & < 7. Therefore, for all n 2 CoN1, we have
Ea + aa, Nn.a 2 w +
Nn. n n-1 I w.
36 ta aa",a,,
Nu. I w By definition of the TC challenger, we obtain Let & {at,a} a} such that n-1
On # a = We,araaPun No. + log Nn,o > We, olnny N₂) + log Nn.
n (We (pn N₂) - Wear ( N₂)) > log 1
1
W c,
w
B,a
1 n (We,er,ale (P Nn) e ( Nv))
1
w Wr,
max afb
log
n
w 8.0
where we used the positivity of the 3-optimal allocation (Lemma
43 to ensure that
whe maxayth log مa
E (0, +oo). Using that An,a- max (ūn,a: P we obtain
(We,ar (ūn, We ((nn.NN))
Nn.
de (im,a*, w) + (wt + 7)d (Km,e w)
inf 0.(0,1]
n -
84

---

inf { Nn.a uE[0,1] n n
de (An,ar, u) ai (Am, u)
2 inf 3-3-8155
Ga, B) B) 2
inf
inf Ga.(A,
> Sn 3
where, for all (a, b) E ([K) \ {a² such that + b,
9) inf wE[U.1]
{ [d u)) + (w + +dd (Aa u)
inf we(0.1]
{ Bdd (Nar, u) + wi aod (Ao. un}
(do
Using the equality at equilibrium from (39 (see Lemma 43) and the fact that the transportation
costs are increasing in their allocation argument (see Lemma 37 we obtain Ga,(U, B) > 0 for all (a, b) E ([K) {a²})² such that a * since
inf {3d (Mar, u) + (w a + 7d (Ha, > We, aa, wE[0,1]
(p, w2,z) = We, b(1, w7,a)
By Lemma 49. functions (2, - Ga, (A, 3) and + inf B:0-0153 Gap(A, (A, 3) are continuous. Therefore, there exists The and N/ small enough such that
inf Ga,(A,,3) 2 Ga,lu, (a,,(1,,))22 X:I|A x Sn 8-8--155
inf
min Ga (8)
aæb.0 Aak
where the last strict inequality uses that the minimum of finite number of positive constants also positive. Considering such (Nw 7a) at the beginning of the proof and taking N₂ = max{CoN11, N3, K where
maxaşb log 2ee wi
Kp = 2 aiDa aea (H, B)
< +00,
As it satisfies Ex [Na] < +oo, this concludes the proof.
Lemma 64 shows that the convergence time towards the 3-optimal allocation has finite expectation.
It is a direct consequence of Lemmas 61 62 and 633 using the same proof as Lemma 41 in Azize al. 2024), hence we omit the proof.
Lemma 64 (Lemma 41 in Azize al. 2024 ). Let E (0, 7u) where Ti is problem dependent constant, and T, (w) as in Eq. (40). Then, we have Eyr [T, (wi,p)] < t00.
H.4 Asymptotic Upper Bound
The final step of the generic analysis of Top Two algorithms [Jourdan et al. 2022 to invert the GLR stopping rule in Eq. (7) by leveraging the convergence of the empirical proportions towards the 3-optimal allocation. Provided this convergence is shown, the asymptotic upper bound on the expected sample complexity only depends the dependence in log(1 (1))) of the threshold that ensures 5-correctness. Compared to the non-private GLR stopping rule, the GLR stopping rule in Eq. (7) pay an extra cost ensure privacy.
Lemma 65. Let 0, 1) O and (5, E (0,1)2. Let (v,g(v) in Eq. (35) and wi,3 be its associated 3-optimal allocation. Assume that there exists ?ip > 0 such that Evx [T,, (wi +o0 for all y E (0, 7u), where T. (w) defined in Eq. (40). Combining such a sampling rule, using the GPE, update, with the GLR stopping rule as in Eq. 7 and the stopping threshold e as in Eq. (8) yields an c-global DP and 8-correct algorithm which satisfies that, for all with mean p such that la* (1)) (1]]=1, 1,
Epz [T..]] < 2 + (v))
lim sup 8-30
log(110)
Proof. Lemma yyiclds the e-global DP. Theorem 6 yields the 5-correctness.
85

---

Let Ç > 0 and a" be the unique best arm. Using the equality at equilibrium from (39 (see Lemma 43
and the continuity of (1,, w) + minaza (n) W We,a-())- (p, w) (see Lemma 42 there exists 7<
such that n _ S is and Il - all $ Ni implics that
va + a², Wea (Pn, Na/ 1)) 2 (1-5) Tú T2 (v)
We choose such a 7c. Let 7r > 0 be such that for Eva Eun(Ty (TT (w) < too for 7 E (0, 7a), where T, (w) is defined Eq. (40). Let y E (0, min Ys. min Wr, / 4, Amin/ /4, Ao/ 4]) where
Amin = minafo (Har Pa) and Ag = ei] min(pa: 1 Maa. For all n 2 (w r we have
3
Nn. 2 / /1aa/111 + (1
2 (n 1), 7)/(1 2 (n 1)
min w 6,, > 0, aE(K]
4(1 + 77)
where the last inequality used the positivity of the 8-optimal allocation (Lemma 43 ). Since arms sampled linearly, it is direct construct N₂ 2 (w(,),) with Eyn
+o0 such that
Ilān - al S y and
n-1 wh,s S y (as well as minge[k] Nn, > trivially).
ci (n, 8) c2( where + (n 8) and n ei (n, 8) are increasing (see Lemmas 52 and 39 Since Nn, Nn. S n, we obtain
Recall that c(n, e, a)
Σ a(Nn, 6) s 2(c) (n, 8) + c2(m,
bE(a",a}
Using Lemma 37 and Nna 2 / + 1)) for all a E [K] (Lemma
50
we obtain
Nn
W almm: No) 1+ We
Pn. n
Let 10 and T N3/K. k. For all n E [KT, we have ān = a and, for all + a*.
Ted
3a + a",
Σ c(Nn,e,,)
BEfa"ia
3a + a'
n W We,ara
An n2 1
N₂
$ 2(c((n,)) + c2(n)))
1+
n (1 ()
3a a",
S (T)
1 Trr 2. (v)
where we used that + 0 and + cn(n)c) e) are increasing and n S T. Therefore, obtain
min (T. SKI KTT
Σ
1 (78 > n)
T
n1 n- 11-1(1- (1-
<
< 2c, + 202 202(TTe) 2(1 + 1)T
1 Th
n=
< RT +1+
(G +cg(,e)
1 1-
Let (8( defined as
2(1 + m)T
Te(0) = inf
T 1
1
(cr (T,())) + (T) $T
1
Using Lemma
52
we know that W-1(x) =x0200 T + logær
hence we have
lim sumsu0p- C1 (T. /) log (1/8)
< 1.
Since lims- c2(T) (oaecl///ogg//// (1/8
0. we obtain
86

---

lim sup-up log 1/6)
T((8)
1
2 2(1+n)2(g(1)
(1-(I(1-)) For every T 2 max(Tx(6), Ns/k3,, we have Tei S T.
hence Eur [Tr,n] S (6() + Bus (Na) /n < toc. Therefore, for all 6 K > O, we obtain
Eus [T,6o]
Tς(
2(1 + 1 T, (1 () kk)
lim sup
< lim limsup 1og(1/5)
5-+0
log(1 10)
6-0
Letting and go zero concludes the proof.
Proof of Theorem 7
The proof is obtained by combining Theorem and Lemmas
5 60
64 and 65
I Variants of Algorithms
In Appendix ! we propose several variants of the algorithmic components used our algorithm. The objective is to give freedom of choice for the practitioners interested in solving e-global DP BAL Given the rich literature on BAI, it is unreasonable to provide details for the e-global DP version of all the existing BAI algorithms. Therefore, settle for a few instances that has received increased scrutiny in the BAI literature.
First, we adapt the Track-and-Stop sampling rule Gaarivier and Kaufmann, 2016 to solve e-global DP BAI (Appendix [.I]. This leverages the computational tractable procedure compute the optimal allocation wt derived in Lemma 47 Second. we explore some alternative choices of components of the Top Two sampling rule for e-global DP BAI (Appendix [.2p. This includes adaptive choice of target for the leader, hence aiming at achieving T: (1) instead of (t)) Third, we adapt the LUCB
sampling rule Kalyanakrishhan et al. 2012 for e-global DP BAI (Appendix L.3
1.1 Track-and-Stop Sampling Rule
The Track-and -nt-S (TaS) sampling rule was introduced in the seminal paper (Garivier and Kaufmann 2016). At each time n, it solves the optimization problem defining the characteristic time for the current empirical estimator Ωn- When ü.n E (0, 1)K we define W, w (Pn.) where in is the Bernoulli instance with means u., When fi,, & (0, 1),1) [āw,] corresponds to a degenerate Bernoulli instance, hence we define Wn lk/K Since fú, updated on per-arm geometric grid governed by 7, the optimal allocation Wn is updated on the same per-arm geometric grid. Therefore, choosing larger y yields lower computational cost of TaS the cost of larger expected sampled complexity, i.e. asymptotic multiplicative factor 1 + 77 due to the update grid.
Given the vector n E A K, the next arm an to sample obtained by using C-Tracking Garivier and Kaufmann 2016) with forced exploration in order to ensure that sufficient exploration holds. This is done here by projecting on A& (w E [e, 1] ΣeE[K] Wa 1} for well chosen E (0, K.
Let ff. be the lo projection of n, on 4g with e, = (K2 + n) 12122/2 While we consider a projection that changes at each time (due to 6n), wf could also be updated on per-arm geometric
grid, i.e., when Wn is updated itself. For all n 2 K + 1, the TaS sampling rule defines
E arg max
Σ Wi EE
Nn.
(43)
aE[K]
In summary, our proposed Track-and-Stop algorithm is defined as in DP-TT that Lines 13-14 are replaced by the sampling rule defined in Eq. 43
with the sole modification
Optimal Allocation Oracle In Lemma 47 we show that w (v) can be computed explicitly based on the unique fixed-point solution Fz (v) 1 for y [0, miin (u) d, (Ha-ta)) Ma)), where Fp is an increasing one-to-one mapping from [0, minaza (N) de (Ha Ma)) to [0, +0o) defined as
di (Mor(a) Un (w. (x (v)))
Fv(v) = Σ
(44)
at (pas un (xa
(()
The definitions of un and xn defered to Lemma 47 ua is decreasing and va is increasing and strictly convex.
87

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Asymptotic Expected Sample Complexity Combining the TaS sampling rule as in Eq. (43 with the GPE, (e) update and the GLR stopping rule in Eq. 7 for the stopping threshold in Eq. (8) yields d-correct and 6-global DP algorithm (see Lemma and Theorem 6). Moreover, we conjecture that its satisfies that, for all v E JK with unique best arm,
Eys (T, log(1/8)
lim
sup sup 60>
S (1I+))TTT0)
The multiplicative factor 1 comes from the per-arm geometric update grid, and the factor 2 comes from the asymptotic scaling in 2 (65) of the stopping threshold. Using Theorem 7 for 3 1/2 and 1/21/2( (v) S 2T (Lemma 44). proving this conjecture would only yield an asymptotic improvement by factor at most 2. However, this would come at the price of significantly higher computational cost.
Proof Sketch of Conjecture While the detailed proof of this conjecture is beyond the scope of this work, an astute reader could notice that all the necessary steps were proven to derive Theorem for DP-TT At a high level, it is intuitive that the asymptotic analysis of Track-and-Stop is simpler than the one of DP-TT
First, the forced exploration is enforced algorithmically. hence an equivalent of Lemma 60 can be shown for the Track-and-Stor sampling rule. In contrast, the proof of sufficient exploration for DP is more challenging and involves a subbtle reasoning towards contradiction, see Appendix H.2 for more details.
Second, the convergence towards the optimal allocation is also enforced algorithmically. Thanks to the forced exploration and due to the continuity of + w (1) (Lemmaa 42) and the convergence An -n μe, the empirical optimal allocation Wp converges towards the true optimal allocation 64 w (v). Therefore, equivalent of Lemma 64 can be shown for the Track-and-Stop sampling rule. In contrast, the proof of convergence towards 3-optimal allocation for DP-TT is more challenging and leverage subbtle regularity properties of the 3 characteristic time and its optimal allocation, e.g.. the equality at equilibrium of all the transportations costs in Eq. (39). see Appendix H.3 for more details.
Third, the invertion of the GLR stopping rule can be done similarly as for DP-TT The sole modifica- (V), e.g., the equality equilibrium of all the transportations costs in Lemma 43 Therefore, equivalent of Lemma 65 can be shown for the Track-and ann-Sppp sampling rule with 2 + T) instead of 2(1 + T) c,g (V), see Appendix H.4 for more details.
tion lies in using our derived regularity properties for w (v) instead of (,,,((),,
1.2 Top Two Sampling Rule
As detailed in Chapter 2.2 in Jourdan (2024), Top Two sampling rule is defined by four choices:
a leader arm B, [K]. a challenger arm CN E [K] {Bx). target 8, (B,, C.)) [0, 1] and mechanism to reach the target, i.e., On E (Bn, C₂ by using (n,,,, C.). For instance, the sampling rule in DP-TT TP--TTusss uses the EB leader, the TCI challenger, fixed target 3 E (0, 1) and K independent 3-tracking procedures (one per leader). We propose adaptive choice of target (Appendix I.2.1) as well as leader fostering implicit exploration (Appendix 12..2
L.2.1 Adaptive Target
When the target is fixed beforehand, the Top Two sampling rule can achieve (1) best. We propose adaptive choices of the target inspired by the recent literature asymptotically optimal Top Two algorithms You et al. 2023 Bandyopadhyay et al. 2024].
BOLD Target Given the EB-TCI leader/challenger pair (B,, C,)) defined DP-TT we adapt the BOLD target from (Bandyopadhayy et al. (2024). Let us define
Ue (Pn, N₂) = arg min { (Nn.B, d (Bm, u) + Nn.adt (Rm,b, 4)} 0,
(45)
88

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whose closed-fone solution is given in Lemma 45 next arm to pull as
Then, the deterministic BOLD target defines the
de (in, Br. a (Mn, Na))
On Bn if
Σ ar
and On Cn
otherwise.
(46)
di (An,a Ue Be (Jms N₂))
In summary, the sole modification in DP-TT is Line 14 that is replaced by the sampling rule defined in Eq. (46).
For any paggemeareeeter exponential family of distributions, Bannhopudya al. [0224] shows that the BOLD target allows to reach asymptotic optimality. Forced exploration is added by Bandy- opadhyay al. 2024 ensure that sufficient exploration holds. Showing that the BOLD target can achieve asymptotic optimality without forced exploration, i.e., meaning that ensures sufficient exploration its own, is open problem.
IDS Target Given the EB-TCI leader/challenger pair (B,, C₂)) defined in DP-TT we adapt the IDS target from You et al. 2023 Namely, the randomized IDS target defines the next arm to pull from as
Bn Cn
with proba B, Bn, C.) otherwise
where 3n C..) Nn.B, di (Rn.B., B,, Br, (Das Nu) C,, (Dons N₂)
(47)
where M,.B..C, (Jin N,,) is defined Eq. 45 In summary, the sole modification DP-TT 14 that is replaced by the sampling rule defined in Eq. (47).
Line
While we could above for the sake of simplicity. For Gaussian distributions with known variance, You et al. [2023) shows that the IDS target allows reach asymptotic optimality. Showing that the IDS target can achieve optimality for other classes of distributions is open problem.
K(K 1) tracking procedures select an E (Bn, C. use randomization
for the EB-TCI leader/challenger pair (Bn, Cn) defined in DP-TT based on the GPE (e) update and the GLR stopping rule in Eq. (7) for the stopping threshold as Eq. (8) yields d-correct and e-global DP algorithm (see Lemma and Theorem 6).
Asymptotic Expected Sample Complexity Sampling O, as Eq.
While we conjecture that their asymptotic expected sample complexities lim sups-s0 Tog(1/8) are both upper bounded by 2(1 TT (w), we emphasize that analysis doesn't 't provide the necessary steps for this result to hold. This is an interesting research direction left for future work.
Eve I..
I.22 Implicit Exploring Leaders and TC Chalenger
The empirical best (EB) leader greedy choice of leader that doesn't foster implicit exploration. Without additional exploration mechanism, it can suffer from large empirical stopping time despite being enough for asymptotic analysis, the TCI challenger for DP-TT] since it fosters additional implicit exploration by penalizing over sampled challengers with the log Nn,a term. We propose other choices of leaders that foster implicit exploration, and define the TC challenger that removes this penalization.
[Jourdan et al.) 2022]. This motivated the choice of
The UCB leader is defined as
UCB E arg max Un,a where Un.a = max fu uu [0, Nn, (lin,alb u) S log(n)) (48)
aE(K)
By adding bonus to the empirical mean, we are optimistic since we consider that the means are better than suggested by our observations.
The IMED leader builds on the IMED algorithm (Honda and Takemura 2015 is defined as
IIMED
arg min a (lúm,llb (h) + log Nn.) where P max jún,al0 <E(K]
(49)
aE|K]
The TC challenger is defined
CTC arg W WeB.n. Na)) B,
(50)
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where We,,b is defined as in Eq.
In summary, the sole modification in DT Line 13 which be replaced by choosing the leader as in Eq. 48 or Eq. (49). or choosing the challenger as in Eq. (50).
Asymptotic Expected Sample Complexity Choosing the leader as in Eq. 48 or Eq. 49 or the challenger as in Eq. (50) based on the ß-tracking as in DT-TT the GPE
update and the GLR stopping rule as in Eq. 7 for the stopping threshold as in Eq. (8) yields d-correct and e-global DP algorithm (see Lemma5aad Theorem 6). Moreover, we conjecture that satisfies that, for all
JkK with distinct means,
Eyx [Te,]] S 2(1 (v))
lim sup
8-50 log(1/6)
While the detailed proof of this conjecture is beyond the scope of this work, an astute reader could notice that all the necessary steps challenger as Eq. (50), the proofs of Lemmas (59 and 63 can be readily adapted. When using the
proven to derive Theorem 7 for DP-TT When using the TC
UCB leader in Eq. (48) or the IMED leader in Eq. (49 the proofs of Lemmas an and [T could also be adapted.
I.3 LUCB Sampling Rule
While the Top Two terminology was introduced in Russo 2016 the first sampling rule having a Top Two structure the greedy sampling strategy LUCBI introduced by Kalyanakrisshan et al. [2012]
At each time n, it selects the EB leader B ã, and the UCB challenger defined as
CUCB
E arg max Un.a where BF
Un.a as in Eq. 488
(51)
Then, samples both B and CUCB Instead of using the GLR stopping rule as in Eq. LUCBI
stops when the LCB (lower confidence bound) of the leader exceeds the UCB of the challenger, i.c.,
LUCBI inf {n L.,,BE > Un,cum
T,,
(52)
where
In max [u E [0, 1] Nn,ad, ((in,al) u) S log(n))
(53)
In summary, the modifications in DP-TT are: (1) the sampling rule in Lines 13-15 is replaced by sampling both BEB and C and (2) the stopping rule Line 10 is replaced by Eq. (52). While studying this algorithm is beyond the scope of this work, emphasize that LUCB is known to not reach asymptotic (8-)optimality.
Implementation Details and Supplementary Experiments
Appendix J organized as follows. First, we provide additional detail on the implementation details for our algorithm (Appendix J.Tp. Second, provide supplementary experiments to illustrate the good performance of our algorithm (Appendix J.2).
J.1 Implementation Details
We present additional experiments comparing the algorithms in different bandit instances with Bernoulli distributions, as defined by (Sajed and Sheffet (2019), namely
(0.95, 0.9, 0.9, 0.5,,
H2 = (0.75, 0.7, 0.7,0.7,007 0.7),
μ4 (0.1, 0.5, 0.7,
μ4 (0.75, 0.625, 0.5, 0.375, 0.25)
(0.75, 0.53125, 0.375, 0.28125, 0.25), For each Bernoulli instance, implement the algorithms with e {0.001, 0.005, 0.01, 0.05, 0.1, 0.2.0, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1, 10, 100, 125).
A4 = (0.75, 0.778,0, 0.625, 0.46875, 0.25)
The risk level is set at 5 = 0.01. We verify empirically that the algorithms are 8-correct by running each algorithm 1000 times.
We implement all the algorithms in Python (version 3.8) and on an core 64-bits Intel @ GHz CPU.
90

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Choice of Hyperparameters The default choice 3 = 1/2 is motivated by the worst-case inequality T T21/2(v) I (v) proven Lemma 44 It implies that an asymptotically 1/2-optimal algorithm has an expected sample complexity that at worst twice as high as an asymptotically optimal algorithm. Moreover, this choice ensures that sample the leader arm (i.e., the empirical estimate for a*) half the time, and spend the remaining samples to explore all the other suboptimal arms (i.e., the challengers) until
confident enough about a, being a".
The default choice 7) and empirical performance. Based on Theorem 7 smaller yields "better" theoretical asymptotic guarantees, i.e., smaller upper bound on the expected sample complexity of DPPTT However, the asymptotic regime of 8 -> 0 fails to capture the empirical trade-off for moderate values of 8. Choosing arbitrarily close to 0 will result in large stopping threshold as k, scales as 1/ log(1 + 17), see Eq. (8) in Theorem [ Smaller also implies that larger cumulative Laplacian noise is added the cumulative Bernoulli signal. The limit v = 0 coincides with adding one Laplace noise per Bernoulli observation. The concentration results in Appendix F.6 (Lemmas 18 and 19] show that the rate is governed by ded (1 =2 pr 1), which is not equivalent to di (1 + 2, (1) asymptotically.
motivated by the trade-off existing between theoretical guarantees
Remark 2. To implement the thresholds of AdaP-TT, AdaP-TT* and DP-TT we use empirical thresholds that we get by upproximating the theoretical thresholds. The expressions of the empirical thresholds used can be found in the code in the supplementary material.
J.2
Supplementary Experiments
2 confirms our experimental findings from Section DP-IT 5-correct and e-global DP BAI algorithms,, for different values of e and in all the instances tested. two regimes. A high-privacy regime, where the
Figure
outperforms all the other
The empirical performance of [ stopping time depends on the privacy budget e, and low privacy regime, where the performance of DP is independent of c, and requires twice the number of samples used by the non-private EB-TCI-B.
91

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EB-TCI
High Privacy Regime
DPSE
High erivacy Regime
DPSE
AdaPTT
AdaPTT
10a
Ad
Ad
Stopping Time r
10c
Stopping
104
Low Privacy Regimee
Privacy Regime
10a
10a
10
10
10
10-1
10
107
102
10
10
10-1
10
10
102
Privacy Budg et
Privacy Budget
EB-TCI
10²
EB-TCI
Uih Privacy Regime
DP-SE
High Privacy
DP-SE
AdaPTT
AdaRTT
105
A
AT
Stapping Time
DP-TT
Stopping Time
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Low Privacy
10a
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Privacy B
Privacy Budget
Figure 2: Empirical stopping time T.5 (mean +2 std. over 1000 runs) for d = 10 with respect to the privacy budget for e-global DP on Bernoulli instances μ3, μ4, As and ps (top left to bottom right). The shaded vertical line separates the two privacy regimes.
92


Reviews:
[{"review_id": "nt8MrFFRat", "reviewer": "Reviewer_iuVL", "summary": "The paper describes a fixed-confidence Best Arm Identification (BAI) setup under epsilon-global Differential Privacy (DP) for Bernoulli bandits. The paper presents a lower bound and a corresponding algorithm, DP-TT, which closes the gap between the two to within a small constant factor.", "questions": "First, I have to note that I do not work in this field, I tried my best to read around and understand the work.  \n\n* For the constant factor in the upper bound, is there a way to improve the allocation strategy to help the sample complexity further? Can it similarly be problem independent?  \n* Current derivation appears quite dependent on Bernoulli distributions, is there a way to generalise the setup to cover other types of bandits? What would be the main blocker in such extension? \n* It appears that not forgetting samples here leads to improved sample complexity. As the same time, it feels that this statefullness can also be a problem for perhaps slightly different threat models since data keeps on being accumulated. I wonder what other trade-offs appear here e.g. memory?\n* How does one pick (n, b) in practice? What would sample counts actually look like in practical scenarios?", "limitations": "Yes", "rating": 4, "confidence": 1, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n* Adoption of DP framwork to a new domain with signifincally better guarantees\n\nWeaknesses:\n* Still a constant factor in the bound\n* Currently focusing only on Bernoulli", "was_revised": true, "final_justification": "Its a strong paper and the authors response made sense to me.", "rebuttal": "We thank Reviewer iuVL for the time spent reviewing and the effort made despite not working in this field. \nWe address the reviewer's questions below.\n\n**W1/Q1 - Constant factor between the upper and lower bounds.**  \nSee answer to Reviewer YnCC.\n\n**W2/Q2 - Beyond Bernoulli distributions.**  \nSee answer to Reviewer inBP.\n\n**Q3 - Implications of not forgetting.**\n\n(a) *Memory.* DP-TT has the same memory complexity as the forgetting algorithms. The reason is that DP-TT only needs to store one *accumulated* noisy sum of rewards *across phases*, while the forgetting ones store the noisy sum of rewards of *only* the last phase. \n\n(b) *$\\epsilon$-global DP.* For privacy considerations, under our threat model where the adversary only observes the output of the algorithm (sequence of actions, final recommendation and stopping time), both forgetting and non-forgetting algorithms provide the same DP guarantees thanks to post-processing. \n\n(c) *Pan privacy.* One can indeed imagine other threat models where forgetting the rewards provides better privacy guarantees. One possible example of this is the pan-private threat model [32], where the adversary can also *intrude* into the internal states of the algorithm during its execution. In this threat model, if an adversary observes the internal states of the execution of DP-TT at some phase $\\ell > 1$, they can see the full sum of rewards and maybe infer the membership of, say, the first user (up to tradeoffs guaranteed by the $\\epsilon$-DP constraint, since the sum is noisy). However, for the forgetting algorithms, if the adversary observes the internal states of the execution of forgetting algorithms at some phase $\\ell > 1$, the adversary can infer *nothing* about the reward of the first patient, since its reward has been deleted completely.\n\n**Q4.1 - Choice of hyperparameter $(\\eta,\\beta)$.**  \nWithout prior information on the mean vector $\\mu$, we recommend using the default choice of hyperparameters as described in Algorithm 1.\n\n(a) *Leader target $\\beta$.* \nThe default choice $\\beta = 1/2$ is motivated by the worst-case inequality $T^\\star_{\\epsilon, 1/2}(\\nu) \\le 2 T^\\star_{\\epsilon}(\\nu)$ proven in Lemma 43.\nIt implies that an asymptotically $1/2$-optimal algorithm has an expected sample complexity that is at worst twice as high as an asymptotically optimal algorithm.\nMoreover, this choice ensures that we sample the leader arm (i.e., the empirical estimate for $a^\\star$) half of the time, and spend the remaining samples to explore all the other suboptimal arms (i.e., the challengers) until we are confident enough about $\\tilde a_n$ being $a^\\star$.\nWith prior knowledge on the range of the optimal allocation for the best arm, one should choose $\\beta$ accordingly to be closer to asymptotic optimality.\nSee Appendix C.2 for references on an adaptive choice of $\\beta$ to achieve $T^\\star_{\\epsilon}(\\nu)$ instead of $T^\\star_{\\epsilon, \\beta}(\\nu)$.\n\n(b) *Geometric grid $\\eta$.* \nThe default choice $\\eta = 1$ is motivated by the trade-off existing between theoretical guarantees and empirical performance.\nBased on Theorem 6, smaller $\\eta$ yields ``better'' theoretical asymptotic guarantees, i.e., smaller upper bound on the expected sample complexity of DP-TT.\nHowever, the asymptotic regime of $\\delta \\to 0$ fails to capture the empirical trade-off for moderate values of $\\delta$.\nChoosing $\\delta$ arbitrarily close to $0$ will result in a large stopping threshold as $k_{\\eta}$ scales as $1/\\log(1+\\eta)$, see Eq.(8) in Theorem 5.\nSmaller $\\eta$ also implies that a larger cumulative Laplacian noise is added to the cumulative Bernoulli signal.\nThe limit $\\eta = 0$ coincides with adding one Laplace noise per Bernoulli observation.\nThe concentration results in Appendix F.6 (Lemmas 17 and 18) show that the rate is governed by $\\tilde d_{\\epsilon}^{\\pm}(\\mu \\pm  x, \\mu, 1)$, which is not equivalent to $d_{\\epsilon}^{\\pm}(\\mu \\pm  x, \\mu)$ asymptotically.\n\n**Q4.2 - Value of sample counts in practical scenarios.** In practice, the sample counts depend on the hardness of the bandit instance (characterised by $T^\\star_\\epsilon$, the characteristic time of Eq.1 based on $d_\\epsilon$), the choice of the privacy budget $\\epsilon$ and the $\\delta$-correctness parameter. For easy instances (big gaps), $\\delta$ in the order of $0.01$, and $\\epsilon$ in the order of  $0.1$, the sample count can be in the order of the thousands."}, {"review_id": "VqNrFuxPCw", "reviewer": "Reviewer_inBP", "summary": "The authors consider a bandit problem with $K$ arms generating a Bernoulli distributed random variable when picked. The goal is to find the arm with highest corresponding mean $\\mu \\in (0,1)^K$, whilst adhering to an $\\epsilon$-differential privacy guarantee, in the sense that the arm identification algorithm's output should not reveal any information about the underlying sample.\n\nFormally, an algorithm consists of a stopping time $\\tau$, an arm recommendation $\\tilde{a}$ and a sequence of arm recommendations up until $\\tau$; $(a_1,\\dots,a_\\tau)$. Given two sequences of Hamming distance 1-apart arm draws $R,R'$, \n$$P( (T,\\tilde{a},(a_1,\\dots,a_T) \\in A | R) \\leq e^\\epsilon P( (T,\\tilde{a},(a_1,\\dots,a_T) \\in A | R').$$\n\nWithin this setup, the authors:\n* Provide a lower bound on the expected stopping time of a approximately corrected and $\\epsilon$-DP algorithm.\n* They provide a $\\epsilon$-DP stopping rule that matches the lower bound up to a constant factor.", "questions": "Do the authors think that the results hold under $(\\epsilon,\\delta)$-differential privacy as well? I.e. can the lower bound be shown to hold for the more general differential privacy definition?", "limitations": "The main limitation of the paper is its setting; which only covers Bernoulli observations.", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "As a main strength: I believe the authors provide a novel and technically sound contribution to bandit problems under privacy, improving over existing bounds in the literature.\n\nThe main weakness of the paper I see is its setting: considering only Bernoulli samples is quite limiting. I would like to see more discussion on why this limited setting is studied and which parts do no hold if one has for example a Gaussian BAI problem. \n\nMinor comments:\n* The grammar and punctuation could be improved in certain places.\n* I would suggest (1) and (2) not to be given *inside* of Theorem 2 but before it.\n* Similarly, Theorem 3, a lot is defined inside of the theorem statement, unnecessarily so? I would suggest the authors use the additional space to properly introduce and unpack the necessary quantities for Theorem 3 (and 2).\n* I would avoid the name \"global DP\" personally; it seems that the setting is close to what is commonly called central DP in the i.i.d. setting.\n* The choice of the $\\delta$ notation is somewhat unfortunate given the differential privacy definition in full generality involving $\\epsilon$ and $\\delta$.\n* The notation $\\cF^K$ is poorly explained and seems superfluous; as it is a set that is characterized by $\\mu$.", "was_revised": false, "final_justification": "", "rebuttal": "We thank Reviewer inBP for the time spent reviewing and the positive feedback. \nWe address the reviewer's questions below.\n\n**W - Beyond Bernoulli distributions.**\nWe would like to stress the following points about our results and analysis.\n\n(a) *Lower bound.*\nThe non-asymptotic lower bound on the expected sample complexity (Theorem 2) holds for any class of distributions $\\mathcal F$, and thus is already true beyond Bernoullis (e.g. exponential families, sub-Gaussians, etc). However, by going beyond Bernoullis, $d_\\epsilon$ might not admit a closed-form solution, e.g., the TV between Gaussian distributions has no simple formula. We conjecture that most regularity properties used to derive Theorem 3, and needed for the upper bound proofs, also hold for other classes of distributions. From a theoretical perspective, this might render the proof of the sufficient regularity properties more challenging.\nFrom a practical perspective, this increases the computational cost of the stopping rule and the challenger arm, as they require computing an empirical transportation cost based on the $d_\\epsilon$ divergence. Finally, since the objective function inside the $d_\\epsilon$ is continuous and convex over a compact interval, using off-the-shelf convex optimisation solvers to compute $d_\\epsilon$ is still straightforward.\n\n(b) *Privacy analysis.* Our privacy analysis (Lemma 4) holds for any distribution with support in $[0,1]$, and thus is already true beyond Bernoullis. It is straightforward to extend this to any distribution with a support in $[a, b]$, by multiplying the noise terms by the range $(b-a)$. Also, all prior works in bandits with DP are either for Bernoulli or bounded distributions, for the simple reason that this assumption makes estimating the empirical mean privately using the Laplace mechanism straightforward, as the sensitivity is controlled. This helps focus on the more interesting tradeoffs between DP, exploration and exploitation, without any additional technical overheads that may be introduced by estimating privately the mean of unbounded distributions. We believe that considering $(\\epsilon,\\delta)$-DP and unbounded Gaussian rewards may be an interesting direction for future work. \n\n(c) *Concentration inequality.* \nThe concentration results (Theorem 5) used to derive a stopping threshold ensuring $\\delta$-correctness are highly specific to Bernoulli distributions.\nHowever, the proof builds on fine-grained tail bounds of the sum of two independent random variables, i.e., we bound those probabilities by the maximal product between their respective survival functions (Lemma 9). \nTherefore, the technical tools used for this intermediate result can be also used to study other one-parameter exponential families, e.g., Gaussian observation, with other noise mechanisms, e.g., Gaussian noise.\n\n(d) *Expected sample complexity upper bound.*\nThe proof of Theorem 6 (Appendix H) builds on the unified analysis of the Top Two algorithm [50], coping with many classes of distributions, e.g., one-parameter exponential families and non-parametric bounded distributions.\nIt relies heavily on the derived regularity properties (Appendix G) for the signed divergences, transportation costs, characteristic times, and optimal allocations.\nBased on the non-private BAI literature, we conjecture that most regularity properties used to derive Theorem 6 also hold for other classes of distributions.\nDue to the lack of explicit formulae, the proofs are challenging.\nAs mentioned above, the DP-TT algorithm becomes computationally costly due to the intertwined optimisation procedure when computing the transportation costs numerically.\n\n(e) *Bernoulli setting.*\nBernoulli bandits are a fundamental setting found in many applications (clinical trials, ad recommendations through clicks, etc). \nOur intermediate results can improve upon existing state-of-the-art in other settings, e.g., bandits with regret minimisation, sequential hypothesis testing, and DP under continual observations.\n\n**Q - Beyond $\\epsilon$-DP.**  \nSee answer to Reviewer VJtw.\n\n**Minor comments.** We thank the reviewer for the precise recommendations and suggestions. In the revised version, we will use the extra page to polish the presentation, add additional intuition, polish the grammar/punctuation, unpack the definitions of the quantities defined in Theorems 2 and 3 (e.g., $T^\\star_{\\epsilon}(\\nu)$, $g_{\\epsilon}^{\\pm}$, $d_{\\epsilon}$, $d_{\\epsilon}^{\\pm}$, $W_{\\epsilon,a,b}$, $\\Delta_{\\epsilon,a}$, etc) outside the theorem statements, clarify the notation $\\mathcal F^K$ by introducing a notation $\\mathcal M$ for the set of mean instances with unique best arm, add the meaning of the abbreviation ``GLR'' before its first use. \n\n- *Notation $\\delta$.* To the best of our knowledge, the notation $\\delta$ is used in almost all the papers that study pure exploration problems for stochastic multi-armed bandits in the fixed-confidence setting.\nWhile it is also standard to use this notation for $(\\epsilon,\\delta)$-DP, we chose to be consistent with the BAI literature when studying $(\\epsilon,0)$-DP.\nSee answer to Reviewer VJtw for comments on the presentation.\n\n- *$\\epsilon$-global DP.* \nWe chose the name \"global DP\" as it is the name used in prior works for bandits under central DP, in both regret and BAI settings. \nIt is indeed exactly the central DP model with a trusted central decision maker. \nWe add a comment about this in the Background section."}, {"review_id": "OL4OrX7wCF", "reviewer": "Reviewer_VJtw", "summary": "This paper studies fixed-confidence best-arm identification under global differential privacy for Bernoulli bandits. It develops a new information-theoretic lower bound on sample complexity, refining previous KL/TV-based transportation cost characterizations, and proposes an algorithm that achieves this lower bound to within a small constant factor.", "questions": "1. Theorem 6 relies on a unique best arm. In practice, many arms could have nearly identical means. How does DP-TT behave in these near-tie scenarios, and would the stopping rule or estimator become unstable?  \n2. Experiment section describes the experiments on Bernoulli means but does not explore systematically how the sample complexity changes for a wider range of K, δ, or ε. Could the authors expand on how to design experiments to stress-test DP-TT, and share best practices for tuning these hyperparameters?  \n3. The conclusion mentions a possible extension to other DP frameworks, but no technical discussion is provided. Could the authors outline what would need to change in DP-TT or the lower bound if moving to (ε,δ)-DP or Rényi DP?", "limitations": "Yes.", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths:**\n\n- **Strong theoretical contributions**: Developing novel concentration inequalities for Laplace distributions and mixed Bernoulli-Laplace sums, the `d_epsilon` divergence is elegant and leads to provably tighter lower bounds.\n- **Near-optimal performance**: The elimination of \"observation forgetting\" is clever, while it introduces more Laplace noise, the logarithmic growth is well-controlled and preserves more signal. The geometric batching approach is elegant.\n- **Empirical support**: Good experimental results showing DP-TT outperforms existing methods across different privacy regimes.\n\n**Weaknesses:**\n\n- **Limited scope**: Focused only on Bernoulli rewards; no extension to other sub-Gaussian families.\n- **Presentation density**: Some sections (e.g., signed `d_epsilon` definitions) are hard to parse; clearer intuition would help.\n- **Practical considerations**: Little discussion on how to set ε or interpret privacy-utility tradeoffs in real applications.", "was_revised": false, "final_justification": "", "rebuttal": "We thank Reviewer VJtw for the time spent and the encouraging feedback. \nWe address the reviewer's questions below.\n\n**W1 - Beyond Bernoulli distributions.**  \nSee answer to Reviewer inBP.\n\n**W2 - Presentation.**\nIn the revised version, we will use the extra page to polish the presentation, add additional intuition, polish the grammar/punctuation, unpack the definitions of the quantities defined in Theorems 2 and 3 (e.g., $T^\\star_{\\epsilon}(\\nu)$, $g_{\\epsilon}^{\\pm}$, $d_{\\epsilon}$, $d_{\\epsilon}^{\\pm}$, $W_{\\epsilon,a,b}$, $\\Delta_{\\epsilon,a}$, etc) outside the theorem statements, clarify the notation $\\mathcal F^K$ by introducing a notation $\\mathcal M$ for the set of mean instances with unique best arm, add the meaning of the abbreviation ``GLR'' before its first use.\n\n**W3 - Privacy-utility tradeoffs and choice of $\\epsilon$.** \nOur matching upper and lower bounds suggest the existence of two privacy regimes: (i) a low privacy regime (high values of $\\epsilon$) where the hardness of the problem reduces to the non-private KL-based characteristic time ($T^\\star_\\mathrm{KL}$) and privacy can be achieved for `free' (up to some multiplicative constants). And (ii) a high privacy regime (low values of $\\epsilon$) where privacy has an additional cost characterised by $T^\\star_\\epsilon$, and at the limit when $\\epsilon \\rightarrow 0$, reduces to a TV-based characteristic time. We also provide exact characterisations of the value of $\\epsilon$ at which the change of regimes happens. Please refer to the ``Allocation Dependent Low Privacy Regime'' paragraph (Lines 184-191) where we discuss in depth both an allocation-dependent and a weaker allocation-independent value of $\\epsilon$ of change of regimes. In light of this discussion, the best choice of $\\epsilon$ that provides the highest privacy protection while maintaining a great sample complexity is exactly the change-of-regime $\\epsilon$. We will add a discussion about this at the end of Line 191.\n\n**Q1 - Behavior with near-optimal arms.**\nThe assumption in Theorem 6 that the means are distinct is used to prove sufficient exploration; it can be removed by using forced exploration or a fine-grained analysis [49, 52].\nThe unique best arm assumption is a standard assumption in the fixed-confidence BAI setting.\nEven in the non-private setting, this assumption is necessary to obtain a $\\delta$-correct algorithm with finite expected stopping time, as the characteristic time becomes infinite otherwise.\nIn the private setting, this phenomenon persists as can be seen by the lower bound in Eq.(5), i.e., $T_{\\epsilon}^{\\star}(\\nu) \\to +\\infty$ when $\\Delta_{\\epsilon,a^\\star} \\to 0$.\nIn near-tie scenarios, DP-TT will perform as badly as any other $\\delta$-correct and $\\epsilon$-global DP algorithms due to the fundamental lower bound.\nEmpirically, this will be reflected by empirical transportation costs that are too small for the stopping condition to be met.\nIn applications where the near-tie scenario is common, practitioners consider $\\gamma$-Best Arm Identification, i.e., identify an arm that is $\\gamma$-close to the best arm.\nFor the non-private fixed-confidence $\\gamma$-BAI setting, we refer to [52] for references and guarantees satisfied by Top Two algorithms.\nTo tackle $\\gamma$-BAI, DP-TT should be modified by using the appropriate transportation costs in both the stopping rule and the definition of the challenger, i.e., $W_{\\epsilon,\\gamma,a,b}$ instead of $W_{\\epsilon,a,b}$.\nStudying this setting with multiple correct answers is left for future work.\n\n**Q2 - Additional experiments.**\nIn Figures 1 and 2, we study the impact of $\\epsilon$ with $17$ distinct values (see Appendix J.1), ranging from $0.001$ to $125$.\nBased on our upper and lower bounds, the $\\delta$-dependent dominating term of the expected sample complexity of DP-TT scales as $\\Theta(T^\\star_{\\epsilon}(\\nu)\\log(1/\\delta))$.\nOn the “equal means” instance (one optimal arm, and other arms with the same strictly lower mean), Eq.(5) suggests that $T^\\star_{\\epsilon}(\\nu)$ scales linearly with $K$.\nEmpirically, this linear scaling has been observed for Top Two algorithms in the non-private regime (e.g., Figure 1 in [49] for EB-TCI); hence, we conjecture that it would hold for DP-TT.\nSee answer to Reviewer YnCC for the impact of varying $\\delta$.\nSee answer to Reviewer iuVL for the choice of hyperparameters.\nAs figures are prohibited in the rebuttal, we will include supporting evidence in the revised version.\n\n**Q3 - Beyond $\\epsilon$-DP.** \n\n(a) *Lower bound.* Our lower bound technique can be seen as a generalisation of the packing argument, and thus depends on the group privacy property. Specifically, in Appendix D, just before Equation 11, we used the group privacy of $\\epsilon$-DP to bound $\\text{KL}(M_{D}, M_{D'}) \\leq \\epsilon \\text{dham}(D, D')$. \n\n- For $\\rho$-zCDP, there is a similar group privacy property that can directly be plugged here, stating that $\\text{KL}(M_{D}, M_{D'}) \\leq \\rho \\text{dham}(D, D')^2$. This means that for $\\rho$-zCDP, $\\epsilon \\times \\mathrm{dham}$ is replaced by $\\rho  \\times \\mathrm{dham}^2$\nin the fundamental optimal transport inequality of Equation 11. We leave solving tightly this new optimal transport for future work, and add a detailed discussion about this in an extended Future extensions paragraph in the appendix.\n\n- For $(\\epsilon, \\delta)$-DP, the group privacy property is not tight. This is a classic issue in $(\\epsilon, \\delta)$-DP lower bounds, where other techniques are used (e.g., fingerprinting). Adapting these techniques for bandits is an interesting open problem (even for bandits with regret minimisation). \n\n(b) *Privacy analysis.* It is straightforward to make DP-TT achieve either $\\rho$-zCDP or $(\\epsilon, \\delta)$, by replacing the Laplace mechanism with the Gaussian mechanism.\n\n(c) *Concentration inequality.* To design a correct stopping rule with the Gaussian mechanism, it is possible to use the same fine-grained tail bounds of the sum of two independent random variables (Lemma 9) by only replacing the concentration of Laplace random variables with the concentration of Gaussian random variables.\n\n(d) *Algorithm design.* In the Top Two family of algorithms, an important design choice is the transportation cost, used for stopping and sampling the challenger. DP-TT uses a $d_\\epsilon$ based transport, inspired by the lower bound of Theorem 1. Thus, to go beyond $\\epsilon$-DP for algorithm design, it is important to derive a tight lower bound that suggests the use of a new transportation cost.\n\nTo sum up, there are multiple technical challenges to have a complete solution with matching upper and lower bounds, up to a constant."}, {"review_id": "88cZNZRcWl", "reviewer": "Reviewer_YnCC", "summary": "The authors study the problem of Best Arm Identification (BAI) with fixed confidence $\\delta$ under global $\\epsilon$-DP, assuming Bernoulli bandits. In this setting, they improve upon a previously established lower bound and propose an asymptotically optimal algorithm that matches the new lower bound, leaving only a small multiplicative factor gap between the bounds. The proposed algorithm (DP Top-Two Sampling with new sampling rule) achieves better sample complexity than previously established algorithms, requiring only about twice as many samples as the non-private BAI in a low privacy regime.", "questions": "I have a few questions regarding numerical experiments:\n\n1) How would Figure 1 change for different values of $\\delta$? \n\n2) Could you quantify the boundary between the low- and high-privacy regimes shown in Figure 1?", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "The paper is technically solid, well written, and builds upon a series of previous works. The Related Work section in the appendix provides a comprehensive overview of the literature on private bandits.\n\nThe paper proves a new lower bound for the BAI problem. Even though the tradeoff between KL and TV distances has already been introduced for regret minimization, the BAI setting required a novel proof technique. The authors propose a new sampling rule for the DP Top-Two algorithm, achieving provably asymptotically optimal sample complexity and demonstrating better numerical results than alternative methods.\n\nEven though this might be of limited interest to the broader DP community, I believe it will be a valuable contribution to DP bandit research. Therefore, I support the acceptance of the paper.\n\n**Weaknesses:**\n\n1) The setting is quite limited, as it focuses on BAI with a fixed confidence level, \\epsilon-DP, and only Bernoulli bandits.\n\n2) The problem is only solved asymptotically; a gap still remains, and it does not shrink in the limit to the non-private case.\n\n3) I do not find the paper’s motivation for private BAI convincing. While I do not doubt the importance of the problem, I believe it could be presented better, possibly with more examples.\n\n4) The presentation would benefit from polishing. For instance, the abbreviation \"GLR\" is used in the contributions section but is only defined in full in Section 4.", "was_revised": true, "final_justification": "The paper is technically solid, well written, and builds upon a series of previous works. The Related Work section in the appendix provides a comprehensive overview of the literature on private bandits.\n\nThe paper proves a new lower bound for the BAI problem. Even though the tradeoff between KL and TV distances has already been introduced for regret minimization, the BAI setting required a novel proof technique. The authors propose a new sampling rule for the DP Top-Two algorithm, achieving provably asymptotically optimal sample complexity and demonstrating better numerical results than alternative methods.\n\n\nThe limitations of the paper remain: it focuses on best-arm identification with a fixed confidence level, $\\epsilon$-DP, only Bernoulli bandits, and provides only asymptotic bounds. However, I find the contributions of the paper to outweigh these limitations.\n\nMy concerns regarding the paper’s motivation and presentation have been addressed by the authors and are promised to be further resolved.\n\nTherefore, I continue to support the acceptance of the paper.", "rebuttal": "We thank Reviewer YnCC for the time spent and the support given towards acceptance. \nWe address the reviewer's questions below.\n\n**W1 - Limited Setting.** Here, we provide some comments on going beyond our setting.\n\n(a) *Beyond Bernoulli distributions.* See the answer to W for Reviewer inBP.\n\n(b) *Beyond $\\epsilon$-DP.* See answer to Q3 for Reviewer VJtw.\n\n(c) *Beyond fixed-confidence.*\nThe extension of our results to the fixed-budget and the anytime settings is an ongoing research direction.\nSee paragraph ``Performance Metrics'' in Appendix C.1 for detailed references.\nBased on the guarantees achieved by Top Two algorithms in the non-private BAI setting [52], similar results might hold for DP-TT, or some variant of it.\nHowever, this requires a non-asymptotic understanding of DP-TT.\nAs described below, this raises new technical challenges.\n\n(d) *Beyond BAI*  \nThe technical arguments used to show Theorem 2 allow obtaining a similar lower bound for (i) one-parameter exponential families, e.g., Gaussian, (ii) pure exploration problems with a unique correct answer, e.g., top-$m$ best arm identification, and (iii) structured instances without arm correlation, e.g., unimodal bandits.\nThis is straightforwardly done by adapting the definition of Alt$(\\nu)$, $a^\\star(\\nu)$, and $\\mathcal F$ in Equation 1.\nFor settings that do not fall into the above three categories, the characteristic time requires more subtle modifications; yet our intermediate technical results can still be used.\nBased on the non-private fixed-confidence literature, we conjecture the changes of characteristic times described below.\n\n- For bounded distribution (non-parametric) or Gaussian with unknown variance (two-parameters exponential family), the KL should be replaced by an infimum over KL [2,51]. The plurality of distributions having the same mean implies the existence of distribution that is the most confusing given a specific constraint on the mean, which is captured by this ``inf KL'' term.\n\n- For $\\gamma$-Best Arm Identification (multiple correct answers), an outer maximization over all the correct answers should be added [Degenne and Koolen, 2019, Pure exploration with multiple correct answers]. The plurality of correct answers implies the existence of a correct answer that is the easiest to verify, which is captured by this ``max'' over correct answers.\n\n- For linear bandits (correlated means), the $\\inf$ in the definition of $d_{\\epsilon}$ per-arm should be replaced by a joint infimum over all the arms and moved outside the sum over arms [Degenne et al., 2020, Gamification of Pure Exploration for Linear Bandits]. \n\nSince our intermediate technical results jointly and tightly capture the interplay between the $\\delta$-correct and the $\\epsilon$-global DP constraints, it paves the way for numerous studies of private pure exploration settings.\n\n**W2 - Asymptotic analysis, with gap remaining.**\n\n(a) *Beyond asymptotic guarantees.*\nTheorem 2 is already a non-asymptotic lower bound, that holds for any values of $\\epsilon$ and $\\delta)$.\nTheorem 5 shows that DP-TT is $\\delta$-correct and $\\epsilon$-global DP for any $(\\epsilon,\\delta)$.\nWhile Theorem 6 is an asymptotic upper bound for $\\delta \\to 0$, it holds for any $\\epsilon$.\nSince our experiments reveal the good performance of DP-TT for the moderate value of $\\delta=0.01$, deriving a non-asymptotic upper bound on its expected sample complexity is an interesting direction for future work.\nBased on the non-asymptotic guarantees achieved by Top Two algorithms in the non-private BAI setting [52], DP-TT might also enjoy similar properties.\n\n(b) *Constant factor between the upper and lower bounds.*\nAppendix C.2 contains a discussion on how to further reduce the multiplicative problem-independent constant gap between the upper and the lower bound.\nCombined with the appropriate analysis, an adaptive choice of $\\beta$ such as IDS [86] or BOLD [14] would replace $T^\\star_{\\epsilon,\\beta}(\\nu)$ by $T^\\star_{\\epsilon}(\\nu)$, which is the optimal scaling asymptotically.\nBased on finer concentration results, the stopping threshold could scale asymptotically as $\\log(1/\\delta)$ instead of $2\\log(1/\\delta)$.\nProvided these improvements are proven, the multiplicative factor would be reduced to $1+\\eta$, where $\\eta$ is the parameter that controls the geometrically increasing phases. In theory, one can set $\\eta$ arbitrarily close to $0$, thus completely shaving the multiplicative gap in the asymptotic results. However, this may come at the cost of the non-asymptotic performance of DP-TT with this choice of $\\eta$, see the answer to Reviewer iuVL. \n\n**W3 - Motivation for private BAI.** In the updated version, we will add an extended motivation section with multiple examples of privacy constraints in bandits. In addition to clinical trials, two other interesting applications are: hyperparameter tuning when learning under privacy constraints, and ad recommendations through click feedback.\n\n**W4 - Improved presentation.**\nSee answer to Reviewer VJtw.\n\n**Q1 - Varying $\\delta$ in Figure 1.**\nWhile it is not allowed to include figures in the rebuttal, we will add the equivalent of Figure 1 for $\\delta \\in \\{0.1, 0.001\\}$ in the revised version. \nBased on our upper and lower bounds, the $\\delta$-dependent dominating term of the expected sample complexity of DP-TT scales as $\\Theta(T^\\star_{\\epsilon}(\\nu)\\log(1/\\delta))$.\nThis is tight when $\\delta \\to 0$, up to a small instance-independent constant.\nFor moderate values of $\\delta$, we conjecture that $\\delta$-independent additive terms in the upper and lower bounds exist and that they cannot be ignored.\nAs those terms already exist for non-private BAI, additional ones are expected for private BAI.\nFor a study of non-private Top Two algorithms in the non-asymptotic regime, we defer to [49].\nIn Figure 6 in Appendix G.2.1 of [49] for EB-TCI on the “equal means” instance with $(K,\\mu_{a^\\star},\\Delta) = (35,0,0.5)$, we see a ratio of roughly $2$ between the $\\delta \\in \\{0.1, 0.001\\}$, while it should be $3$ when using a cross product solely based on $T^\\star(\\nu)\\log(1/\\delta)$.\n\n**Q2 - Low and high-privacy regimes in Figure 1.**\nIn Figure 1, the vertical lines are at (a) $\\epsilon = 0.45$ and (b) $\\epsilon = 0.45$.\nIn the `Allocation Dependent Low Privacy Regime' paragraph (Lines 184-191), we provide both an allocation-dependent and allocation-independent theoretical values of the change-of-regime $\\epsilon$.\nWe compare with the allocation-independent condition from Line 191, i.e., $\\max_{a \\ne a^\\star} \\epsilon_{a^\\star,a}$ where $\\epsilon_{a,b} = \\log \\left( \\frac{\\mu_{a^\\star}(1 - \\mu_{a})}{\\mu_{a}(1 - \\mu_{a^\\star})} \\right)$.\nFor $\\mu_{2}$, we have $\\epsilon_{1,a} = 0.25$ for all $a \\ne 1$.\nThis separation boundary roughly coincides with the one stipulated by Line 191.\nFor $\\mu_{1}$, we have $\\epsilon_{1,5} = 2.94$ and $\\epsilon_{1,a} = 0.75$ for all $a \\in \\{2,3,4\\}$.\nThe condition of Line 191 is stronger than the empirical separation when there are arms with significantly lower mean (arm $5$).\nThis highlights the need for a ``tighter'' allocation-dependent condition, such as Eq.(6).\nIntuitively, an arm with a significantly lower mean will also be sampled significantly less, hence there is a compounding effect."}]

Author rebuttal:

Meta-review: The authors consider the Best Arm Identification (BAI) problem, where the goal is to find the arm with highest mean, under the constraint of $\epsilon$-differential privacy guarantee. Prior work showed tight upper and lower bounds for this problem for the small $\epsilon$ regime. This work closes the gap for the large $\epsilon$ regime. This is a classic problem with solid theoretical contributions that will likely be of interest to the community. All reviewers support the paper.
Decision: Accept (poster)