SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Algorithm 10 Anytime Soft-Bayes [Orseau et al., 2017]
Input: Parameter  $\lambda_{n}$ , sequence of learning rates  $(\eta_{n})_{n\in \mathbb{N}}$ , initial parameter  $\lambda_{1} = 1 / 2$   
75 Return:  $\lambda_{n + 1} = \lambda_n\times \left(1 - \eta_n + \eta_n\frac{Y_n}{1 - \lambda_n(1 - Y_n)}\right) + \left(1 - \frac{\eta_{n + 1}}{\eta_n}\right)\lambda_1$
Orseau et al. [2017] prove that defining the learning rate as  $\eta_{n} = \sqrt{\frac{\log(2)}{4n}}$  ensures an upper bound on the portfolio regret
$$
\mathcal {R} _ {N} = 4 \sqrt {\log (2) N} + (1 + \log (2)) \log (N + 1) + \log (2).
$$
However, they do not provide a lower bound on the regret.
F Supplementary experimental results
F.1 Detail of the costs of each algorithm
We start this section by detailing the computation of the costs of the algorithms presented in Table 2. We first recall the results that we want to prove.
Algorithm	Run time	Memory	Optimality
KL-UCB [Cappé et al., 2013]	O(n log(n)2)	n	Opt.
kl-UCB	O(log(n))	O(1)	Sub-opt. (kl)
UCB [Auer et al., 2002]	O(1)	O(1)	Sub-opt. (2Δk2)
NPTS [Riou and Honda, 2020]	O(n)	n	Opt.
MED/IMED [Honda and Takemura, 2010, 2015]	O(n log(n))	n	Opt.
Mult. MED/IMED (M items)	O(M log(n))	O(M)	Sub-opt. (Kinf mult.)
FMED/FIMED (this paper)	O(n log(n)) if pulled, O(1) otherwise.	n	Opt. (Theorem 1)
OMED/OIMED (this paper)	O(1)	O(K)	Opt. w. assumptions of Theorem 2
Table 3: Comparison of memory and run time needed per step and for an arm  $k$  with  $n$  observations
In the table,  $n$  is a number of observations for 1 arm, and the costs are computed for this arm only for simplicity.
Memory cost We start with the memory, which is easier to estimate. Most optimal algorithms (KL-UCB, NPTS, MED/IMED, FMED/FIMED) require to store all the observations, for a cost of  $n$ . On the contrary, UCB and kl-UCB require constant memory, since they just need to store the number of pulls and the empirical means of each arm. Finally, the multinomial version of the algorithms have a memory cost of  $M$ : they need to store the total number of pulls, the frequency of each item  $(x_{1},\ldots ,x_{M})$  on the chosen grid, and the empirical means.
Finally,  $K$  appears in the memory cost of  $\mathsf{OIMED}$  because for each arm  $k$  we need to store (at most)  $K$  online estimates of  $\mathcal{K}_{\mathrm{inf}}^{\mathcal{F}}$  and pseudo-counts, in addition to the total number of pulls and empirical mean of the arm. This is a worst-case memory, in case each arm have been leader at least once. The total memory cost of  $\mathsf{OIMED}$  (summing over all arms) is then at most  $\mathcal{O}(K^2)$ , while it is  $\mathcal{O}(T)$  for the optimal algorithms that store all observations.
Computation costs First, we assume that any algorithm that use only quantities that can be updated sequentially has a cost of  $\mathcal{O}(1)$ . We include in this category updates of means, number of pulls, or the output of a portfolio algorithm with a running time independent of  $n$ . We insist that in practice the cost of these algorithms is not exactly the same, but it can be verified that they remain constant over time.
We now consider NPTS: the main cost comes from the sampling of  $n + 1$  weights from a Dirichlet distribution of parameter  $(1, \ldots, 1)$  ( $n$  ones). This sampling is linear in  $n$ , since it can be done by sampling  $n + 1$  i.i.d. samples  $R_1, \ldots, R_{n+1}$  from the exponential distribution  $\mathcal{E}(1)$ , and then defining  $w_i = \frac{R_i}{\sum_{j=1}^{n+1} R_j}$  for any  $i \in [n+1]$ . This justifies the cost in  $\mathcal{O}(n)$  in the table.
The remaining algorithms (KL-UCB and MED algorithms) require to solve the optimization problem defined by (4) to compute empirical  $\mathcal{K}_{\mathrm{inf}}^{\mathcal{F}}$ . This can be done with any search algorithm, which costs  $n\log(1/\varepsilon)$  where  $\varepsilon$  is the precision asked for the solution. In the implementations we always choose  $\varepsilon = 1/n$ , so that the precision is enough to preserve the theoretical guarantees of the algorithms. This already provides the cost of the MED algorithms, that require to compute exactly one  $\mathcal{K}_{\mathrm{inf}}^{\mathcal{F}}$  for each arm at each time step. For FMED/FIMED, we precise that this computation happens only if the arm wad pulled. Finally, for the discretized version of the algorithms the computation time becomes linear in  $M$  instead of  $n$ , but the  $\log(n)$  is kept since the precision is still  $1/n$ .
For KL-UCB and kl-UCB, another optimization procedure is needed: for a given function  $f$  we solve  $\max \left\{\mu \in [b,B]:N_k(t)\mathcal{K}_{\inf}^{\mathcal{F}}(F_k(t),\mu)\leqslant f(t)\right\}$ . We also set the precision of this search to  $1 / n$ , which gives the supplementary multiplicative  $\log (n)$  factor compared with MED algorithms: each  $\mathcal{K}_{\inf}^{\mathcal{F}}$  computation costs  $\mathcal{O}(n\log (n))$ , and we perform  $\mathcal{O}(\log (n))$  such computations for a step of KL-UCB. This is also the reason why the cost of kl-UCB is not constant: we perform  $\mathcal{O}(\log (n))$  computations of kl.
Remark 3 (Total computation time). We detailed the cost for one arm and one step of the algorithms only for simplicity, however for those depending on  $n$  there are some differences when considering the total computation time. Indeed, for MED algorithms we do not compute  $\mathcal{K}_{inf}^{\mathcal{F}}$  for the best empirical arm. However, this arm is expected to converge to the true best arm and to be sampled linearly. Hence, if there is only one best arm the costs presented in Table 3 will be paid only for  $n = \mathcal{O}(\log(T))$ . On the contrary, KL-UCB and NPTS will pay their cost for all arms, including the best empirical arm, so one of them satisfy  $n = \Omega(T)$ . The analysis of these algorithms suggest that this cost may not be necessary (see Remark 3 of [Riou and Honda, 2020]), nonetheless this is how they are typically implemented in the literature.
F.2 DSSAT distributions
The DSSAT problem was presented in Section 1 and the experimental results in the last Section 4. To give the reader the full picture of the problem, we plot in Figure 5 all the densities of the seven distributions.
Figure 5: DSSAT distributions
This plot visually confirms the multimodality of all distributions and the difficulty to fit this problem into a parametric framework. We complete the Table 1 presented in the introduction by listing in Table 4 all ratios of the form  $\frac{\mathcal{K}_{\mathrm{inf}}^{\mathcal{F}}}{\mathrm{kl}}$  and  $\frac{\mathcal{K}_{\mathrm{inf}}^{\mathcal{F}}}{2\Delta^2}$ , thus confirming the theoretical importance of using the true  $\mathcal{K}_{\mathrm{inf}}^{\mathcal{F}}$  quantities over their usual relaxed versions.
Kinf(Fk,μ*) / kl(μk,μ*)	4.39	5.22	12.04	10.19	11.73	11.04
Kinf(Fk,μ*) / 2Δk2	4.72	5.55	12.73	10.87	12.44	11.63
To ground this theoretical analysis, we plot again the regret curves and average run time of tested algorithms (Figure 6) as well as Table 5 that compiles the average regret and average run time on the DSSAT experiment at the horizon 10 000. Reading this table, it is clear that FIMED improves the run time of IMED without compromising the regret and that OIMED improves the run time even further, close to the one of UCB, but at the cost of loosing a bit in the regret term. However, of the fastest methods (those in  $\mathcal{O}(1)$  run time), OIMED clearly has the smallest regret. If the practitioner is willing to pay a bit more time complexity and space complexity, then our proposed FIMED seems to be the best algorithmic method.
Figure 6: Average regret (left) and run time (right) of the algorithms on the DSSAT bandit problem
Table 4: List of all ratios for the DSSAT bandit problem
Algorithm	UCB	NPTS	IMED	IMED-k1	OIMED	FIMED	kl-UCB	KL-UCB
Regret	252	49	41	148	100	40	74	60
Run time (sec.)	0.38	14	61	0.51	0.47	6	6.8	226
Table 5: Average regret and run time at horizon 10 000 on DSSAT
F.3 Additional bandit experiments
Here, we confirm the analysis of the previous section. First, we benchmark our FIMED and OIMED algorithms against IMED using the experimental setting of the original paper by Honda and Takemura [2015] in which the IMED algorithm is introduced. Then, we run two sets of experiments: one with Bernoulli distributions to complete our main experiment presented in Section 4, Figure 3, and another one with Beta distributions.
F.3.1 Original testbed of IMED
Negative exponential We consider the negative exponential setting in which the law of an arm  $k$  is such that  $X_{k} = 1 - X_{k}^{\prime}$  where  $X_{k}^{\prime}$  is an exponential random variable with parameter  $\mu_k^\prime$ . In that case, the expected reward of arm  $k$  is  $\mu_{k} = 1 - \mu_{k}^{\prime}$  and its support is in  $(- \infty, 1]$ . In Figure 7, we show simulation result of an experiment on a 5-armed negative exponential bandit problem
where, following the setting of Honda and Takemura [2015],  $\{\mu_k^{\prime}\}_k = \{1 / 5,1 / 4,1 / 3,1 / 2,1\}$ , i.e.  $\{\mu_k\}_k = \{4 / 5,3 / 4,2 / 3,1 / 2,0\}$ .
Figure 7: Average regret (left) and run time (right) of the algorithms on a 5-arms negative exponential bandit problem with means  $\{4/5, 3/4, 2/3, 1/2, 0\}$ .
On this setting, the benefit of the FIMED algorithm over the original IMED can easily be read from Figure 7. FIMED enjoys the same regret as IMED while having an order of magnitude smaller computational time, confirming our theoretical analysis of FIMED. OIMED further reduces the computational time (and space) complexity, however at the cost of a larger experimental regret. The regret still exhibits the logarithmic shape in a reasonable experimental time and the choice is left to the practitioner whether to trade regret against time and space complexity or not.
Truncated Gaussian We continue benchmarking our algorithms using the original testbed of Honda and Takemura [2015], we run two experiments where arms are truncated Gaussian distributions. In the first experiment, Figure 8, the distributions are truncated on  $[0,1]$ . In the second experiment, Figure 9, the distributions are truncated on  $(-\infty,1]$ . In both experiments, we use the same set of Gaussian distributions, of means (before truncation)  $\{0.6,0.5,0.5,0.4,0.4\}$  and variances  $\{0.4,0.2,0.4,0.2,0.4\}$ . After truncation on  $[0,1]$ , the expected values are  $\{0.519,0.5,0.5,0.465,0.481\}$ , and after truncation on  $(-\infty,1]$ , the expected values are  $\{0.319,0.390,0.265,0.320,0.206\}$ .
Figure 8: Average regret (left) and run time (right) of the algorithms on a 5-arms [0,1]-truncated Gaussian bandit problem with means  $\{0.519, 0.5, 0.5, 0.465, 0.481\}$ .
Quite accordingly to the intuition, the regret of all algorithms are better on the  $[0,1]$ -truncated bandit problem than on the  $(-\infty,1]$ -truncated one. Indeed, the variance of the former is smaller than the variance on the latter and an increased variance, all things being equal, tend to decrease the divergence between distributions, increasing the need for exploration. Both on Figure 8 and Figure 9, the regret pf FIMED is on par with that of IMED while being much faster. Following our analysis of running times, OIMED indeed is experimentally the fastest but does seem to trade this speed for a larger regret. However, one can see that in Figure 9 that the regret gap is smaller than in Figure 9, probably because the variance is larger. This variance interpretation is confirmed by the experiment on a Bernoulli bandit that is presented in Honda and Takemura [2015].
Figure 9: Average regret (left) and run time (right) of the algorithms on a 6-arms  $(- \infty, 1]$ -truncated Gaussian bandit problem with means  $\{0.319, 0.390, 0.265, 0.320, 0.206\}$ .
Bernoulli In Figure 10, we show simulation results for a 10-arms bandit with Bernoulli rewards with expected values  $\{0.1, 0.05, 0.05, 0.05, 0.02, 0.02, 0.02, 0.01, 0.01, 0.01\}$ . It can be seen that the performance gap between OIMED and IMED is larger than in any of the previous experiments. It can be attributed to the fact that all the rewards are all very close to zero and the variance of the distributions are small. However, the learning rate of the portfolio algorithm does not take those information into account, and it takes some time for the portfolio algorithm to converge, a time that can be seen in Figure 10 where the regret of OIMED is close to linear with no variance (pure exploration) for the 3500 first time steps. In contrast, IMED and OIMED, which do not rely on a learning rate that is independent of the variance, have a much smaller regret.
Figure 10: Average regret (left) and run time (right) of the algorithms on a 10-arms Bernoulli bandit problem with means  $\{0.1, 0.05, 0.05, 0.05, 0.02, 0.02, 0.02, 0.01, 0.01, 0.01\}$ .
Reproducing the experiments of the original IMED Honda and Takemura [2015], we confirmed that our theoretical analysis is experimentally supported. FIMED always seem to be on par with the original IMED algorithm while enjoying an order of magnitude faster numerical complexity, as proven in the analysis of running times. OIMED enjoys competitive regret, albeit experimentally larger than the original IMED. However, it has the smallest computational time complexity and its space complexity also is minimal. Therefore, it seems like there is currently a trade-off between achieving the smallest empirical computational complexity (and space complexity) and achieving the smallest cumulative regret. The original IMED lies at one extreme, OIMED at the other extreme and FIMED sits in-between. The algorithms studied in this paper allows the practitioner to choose the trade-off that suit her the best. Whether one can find an algorithm as fast as OIMED with experimental regret guarantees that exactly match IMED is a question for future work.
F.3.2 Bernoulli bandits
In Section 4, we studied a Bernoulli bandit problem where all the means are close to 0.5 (Figure 3), identified as a "difficult" problem because the variance of the distributions in maximized among those supported in [0, 1]. Here, we explore two Bernoulli bandit problems, one where the means are located close to 1 (Figure 11, Table 6) and another hereafter where means are located close to 0 (Figure 12, Table 7). For each experiment, we plot the regret curves, the run time curves, as well as a table compiling the average regrets and run times of tested algorithms at the time horizon  $T = 10000$ .
Figure 11: Average regret (left) and run time (right) of the algorithms on a 6-arms Bernoulli bandit problem with means  $\{0.4, 0.6, 0.7, 0.85, 0.9, 0.95\}$ .
Algorithm	UCB	NPTS	IMED/IMED-k1	OIMED	FIMED	k1-UCB/KL-UCB
Regret	289	33	30	43	30	95
Run time (sec.)	0.51	15	0.74	0.67	4.2	7
Table 6: Average regret and run time at horizon 10 000 on on a 6-arms Bernoulli bandit problem with means  $\{0.4, 0.6, 0.7, 0.85, 0.9, 0.95\}$
We recall that for Bernoulli bandit, IMED and k1-IMED are the same algorithms (we assume that IMED is implemented as k1-IMED while only 0s and 1s are observed). For FIMED we could have done the same thing to improve its run time without changing its regret in order to emphasize that FIMED is always faster (or as fast as) k1-UCB, even when using the Bernoulli kl. While the regret and time values change from one experiment to another, the conclusions that can be drawn from them do not, especially when considering the confidence intervals represented by the  $10\% -90\%$  quantiles. Experimentally, 0IMED (or 0MED, see Section F.4) should be the preferred  $\mathcal{O}(1)$  method and FIMED (or FMED, see Section F.4) should be preferred if we really target an empirically optimal regret without compromising too much on the running time.
Figure 12: Average regret (left) and run time (right) of the algorithms on a 6-arms Bernoulli bandit problem with means  $\{0.05, 0.1, 0.15, 0.2, 0.22, 0.25\}$
Algorithm	UCB	NPTS	IMED/IMED-k1	OIMED	FIMED	kl-UCB/KL-UCB
Regret	328	87	82	116	81	98
Run time (sec.)	0.51	15.4	0.75	0.69	5.6	5.9
Table 7: Average regret and run time at horizon 10 000 on on a 6-arms Bernoulli bandit problem with means  $\{0.05, 0.1, 0.15, 0.2, 0.22, 0.25\}$
F.3.3 Beta bandits
Beta distribution of a given mean can have different shapes. In particular, a Beta distribution can be close to a Bernoulli distribution (shape parameter close to zero) with most of the density located around 0 and 1, close to a Dirac distribution (shape parameter significantly larger than one) with most of the density located around the mean, and close to a truncated Gaussian distribution (shape parameter larger than one) with the characteristic bell shape distribution around the mean.
In Section 4, we studied a Bernoulli bandit problem where all the means are close to 0.5 (Figure 3). Here, using the same set of means,  $\{0.3, 0.4, 0.45, 0.5, 0.52, 0.55\}$ , we run two experiments, one on Beta distributions with an intermediate shape parameter of 5 (Figure 13) and another where Beta distributions have a shape parameter of 50 (Figure 15), hence with highly piked distributions. We do so to illustrate the effect of changing the shape of distributions without changing the set of means.
Figure 13: Beta bandit distributions with means  $\{0.3, 0.4, 0.45, 0.5, 0.52, 0.55\}$  and shape parameter of 5
Figure 14: Average regret (left) and run time (right) of the algorithms on a 6-arms Beta bandit problem with means  $\{0.3, 0.4, 0.45, 0.5, 0.52, 0.55\}$  and shape parameter of 5
Algorithm	UCB	NPTS	IMED	IMED-k1	OIMED	FIMED	kl-UCB	KL-UCB
Regret	322	35	29	103	63	30	100	49
Run time (sec.)	1.2	14.5	53.5	1.5	1.4	5.2	7.7	284
Table 8: Average regret and run time at horizon 10 000 on on a 6-arms Beta bandit problem with means  $\{0.3, 0.4, 0.45, 0.5, 0.52, 0.55\}$  and shape parameter of 5
Comparing the regret curves of Figure 3, Figure 14, and Figure 16, it is clear that the Bernoulli distributions does induce the larger variance on the regret curves as we identified while the Beta distributions with the largest shape parameter (more concentrated around their means) induce the smallest variance on the regret curve. This, once again, confirms our theoretical findings. While the order of regret curves is globally preserved, one can see on Figure 3 that, except for UCB, the Bernoulli experiment makes all curves to be very similar, which is normal since, except for UCB, all algorithms are roughly solving the same problem (all sample are 0s and 1s). When dealing with Beta distributions, IMED is different than k1-IMED and KL-UCB is different than k1-UCB.
Figure 15: Beta bandit distributions with means  $\{0.3, 0.4, 0.45, 0.5, 0.52, 0.55\}$  and shape parameter of 50
Figure 16: Average regret (left) and run time (right) of the algorithms on a 6-arms Beta bandit problem with means  $\{0.3, 0.4, 0.45, 0.5, 0.52, 0.55\}$  and shape parameter of 50
Algorithm	UCB	NPTS	IMED	IMED-k1	OIMED	FIMED	kl-UCB	KL-UCB
Regret	323	20	15.9	103.9	55.8	15.8	95	19.2
Run time (sec.)	1.2	14.4	70.8	1.5	1.4	5	7.8	303
Those experiments confirm our numerical findings that OIMED (or OMED, see Section F.4) should be the preferred method with  $\mathcal{O}(1)$  computation time, and FIMED (or FMED, see Section F.4), should be preferred if we really target an empirically optimal regret.
F.4 Comparison of MED and IMED versions
In this paper, we derived algorithmic and theoretical results for modification of both IMED and MED algorithms. In this section, we compare IMED, its modifications FIMED and OIMED, MED, and its modifications FMED and OMED. The comparison is made on three experimental setting, the DSSAT bandit problem and Bernoulli bandit problem presented in the main experimental Section 4, as well as a Beta bandit problem with an intermediate shape parameter of 5 and centered means equal to the Bernoulli setting.
Figure 17: Average regret (left) and run time (right) of the algorithms on DSSAT
Table 9: Average regret and run time at horizon 10 000 on on a 6-arms Beta bandit problem with means  $\{0.3, 0.4, 0.45, 0.5, 0.52, 0.55\}$  and shape parameter of 50
Algorithm	IMED	FIMED	OIMED	MED	FMED	OMED
Regret	41.7	40.4	100.5	55.1	55.6	87
Run time (sec.)	60.8	5.9	0.5	60.8	6.6	0.6
Table 10: Average regret and run time at horizon 10 000 of the algorithms on DSSAT
In Figure 17 and Table 10, we observe that IMED slightly overperforms MED. Consequently, FIMED also slightly overperforms FMED for the regret metric. Regarding the running time, the MED versions of algorithms are a bit slower than the IMED versions due to the numerical complexity of sampling, larger than that of computing the argmin of IMED indexes. Interestingly, Omed overperforms OIMED in terms of average regret performance. As we can see in Figure 18 and Table 11 presenting the results of the Bernoulli experiment, this is not a general rule.
Figure 18: Average regret (left) and run time (right) of the algorithms on a 6-arms Bernoulli bandit problem with means  $\{0.3, 0.4, 0.45, 0.5, 0.52, 0.55\}$
Algorithm	IMED	FIMED	OIMED	MED	FMED	OMED
Regret	105.8	106.6	123.3	141.5	145.2	140.1
Run time (sec.)	39	6	0.7	39.3	6.8	0.8
Figure 19: Average regret (left) and run time (right) of the algorithms on a 6-arms Beta bandit problem with means  $\{0.3, 0.4, 0.45, 0.5, 0.52, 0.55\}$  and shape parameter of 5
Table 11: Average regret and run time at horizon 10 000 of the algorithms on a 6-arms Bernoulli bandit problem with means  $\{0.3, 0.4, 0.45, 0.5, 0.52, 0.55\}$
Algorithm	IMED	FIMED	OIMED	MED	FMED	OMED
Regret	30	29.7	63.7	40	41.4	64.7
Run time (sec.)	53	5.2	1.4	54	5.4	1.6
Table 12: Average regret and run time at horizon 10 000 of the algorithms on a 6-arms Beta bandit problem with means  $\{0.3, 0.4, 0.45, 0.5, 0.52, 0.55\}$  and shape parameter of 5
The experiment on Beta distribution, Figure 19 and Table 12, confirms that the regret loss incurred by the transition from MED to OMED is smaller than the one of transitioning from IMED to OIMED. It may
be because the sampling strategy of MED is better to handle the statistical approximation of portfolio algorithms.
F.5 Stability of OIMED with respect to the learning rate
The portfolio algorithm, Soft Bayes, behind OIMED and OMED depends on a hyperparameter,  $\eta$ , the learning rate. A learning rate scheme is prescribed in Theorem 3 of [Orseau et al., 2017] with
$$
\eta (n) = \sqrt {\frac {\log (2)}{4 n}}
$$
where  $n$  is the number of collected samples. In section, we test the numerical stability of our OIMED algorithm with respect to the learning rate. To do so, we will replace this original  $\eta$  by
$$
\eta_ {r} (n) = \sqrt {\frac {r \log (2)}{4 n}}
$$
where  $r$  will range from 0.01 to 100. We illustrate the stability of 0IMED on three bandit settings: the DSSAT bandit problem and Bernoulli problem that were introduced in the main Section 4 and a Beta bandit problem where all the means are centered around 0.5 and the same as in the Bernoulli bandit.
First, we test the stability of our 0IMED algorithm on the DSSAT bandit problem that we already used in the main experimental Section 4 and Appendix F. On Figure F.5, the average regret ranges from 103.5 for the smallest values of the parameter  $r$  to 88.5 for the largest value of  $r$  with an intermediate regret of 99.5 for our original value  $r = 1$ . On this same plot, it is interesting to see that despite the large range of tested parameter size, the quantile tubes remain of roughly the same size with no evident bad behavior. While the effect of changing  $r$  may seem important, in the range of tested parameter, the effect of changing  $r$  would not have affected the ranking of algorithms in the original experiment, Figure 2 and Table 5. This experiment shows the stability of 0IMED with respect to the learning rate, in the sense that a multiplicative constant does not seem to be able to dramatically deteriorate its performance.
Figure 20: Average regret of the algorithms on DSSAT
Those findings are confirmed on a second experiment performed on a Bernoulli bandit problem with means  $\{0.3, 0.4, 0.45, 0.5, 0.52, 0.55\}$ . The regret changes from 107 to 124 when varying the parameter  $r$  from 0.01 to 100, and, while the upper quantile is larger for small value of  $r$ , we still observe a logarithmic curve which confirms the stability of OIMED.