some concentration inequalities for the empirical means, which is sufficient
for the analysis of the algorithm tackling subgaussian distributions in the
literature. However, as the family is defined by infinitely many constraints
(due to the parameter $t$) the function
$\mathcal{K}_{\inf}^{\text{SG}_{1}(\sigma)}$ has a complicated form. Hence, it
may be easier to work with an alternative definition of the following form,

$F\in\mathrm{SG}_{1}(\sigma)\Rightarrow\mathrm{E}_{F}\left[\mathrm{e}^{\frac{(X-\mu_{F})^{2}}{8\sigma^{2}}}\right]\leq
2\;,$

and is expressed exactly as an $h$-moment condition, as introduced in this
paper. Hence, we can consider a subgaussian model through the following family
of distribution

$\displaystyle\mathrm{SG}_{2}(\sigma)$
$\displaystyle=\left\\{F:\mathrm{E}_{F}\left[\mathrm{e}^{\frac{(X-\mu_{F})^{2}}{8\sigma^{2}}}\right]\leq
2,\;\mathrm{E}_{F}\left[\mathrm{e}^{\frac{(X-\mu_{F})^{2}}{3\sigma^{2}}}\right]<+\infty\right\\}\;,$

that fits into the scope of our analysis if we intersect it with the set
$\\{F\in\mathcal{P}:\;\mathbb{E}_{F}[X]\geq\mu^{-}\\}$ for any constant
$\mu^{-}\in\mathbb{R}$. The second moment condition holds for any
$F\in\mathrm{SG}_{1}(\sigma)$ because, even for distributions
$\mathcal{N}(\mu_{F},\sigma^{2})$,
$\mathrm{E}_{F}\left[\exp\left(\lambda\frac{(X-\mu_{F})^{2}}{\sigma^{2}}\right)\right]$
is finite for $|\lambda|<1/2$. Then, we can easily verify that

$\mathrm{SG}_{1}(\sigma)\subset\mathrm{SG}_{2}(\sigma)\subset\mathrm{SG}_{1}(\sqrt{24}\sigma)\;,$

so even though $\mathrm{SG}_{1}$ and $\mathrm{SG}_{2}$ do not exactly match,
we are sure that all distributions in $\mathrm{SG}_{2}(\sigma)$ are still
subgaussian (for a potentially larger parameter). Similar relations can also
be seen in many other characterizations of subgaussian distributions (see
Vershynin, 2018).

##  Appendix E NPTS for centered $h$-moment conditions

In this appendix we prove Lemma 5, with the proof for the upper bound on the
BCP in Appendix E.1 and the proof for the lower bounds in Appendix E.3. In
Appendix E.2 we provide a novel lower bound on the BCP of NPTS, for bounded
distributions, that showcases the general intuition of the proof of Appendix
E.3.

###  E.1 Upper bound

We start by proving the upper bound presented in Lemma 5. To do that, we use
the notation $Z_{i}=h(|X_{i}-\mu|)$ for $i\in[n]$ and
$Z_{n+1}=h(|X_{n+1}-\mu|)-\gamma$, and consider

$\text{(P)}\coloneqq\mathbb{P}\left(\exists
X_{n+1}:\sum_{i=1}^{n+1}w_{i}X_{i}\geq\mu,\sum_{i=1}^{n+1}w_{i}Z_{i}\leq
B\right)\;.$

The general sketch of the proof is the following: (1) we prove an upper bound
for a fixed value of $X_{n+1}$ using Chernoff method, (2) we use a union bound
on a partition of the possible values for $X_{n+1}$, and (3) we show that a
finite number of terms in this union bound are significant. For any cdf $F$,
threshold $\mu$ and random variable $Z_{X}$ determined by $X$ we use the
notation

$\Lambda_{\alpha,\beta}(F,\mu)=\mathbb{E}_{F}\left[\log\left(1-(X-\mu)\alpha-(B-Z_{X})\beta\right)\right]\;,$

where
$\mathcal{R}_{2}\coloneqq(\alpha,\beta)\in\left\\{(\alpha,\beta)\in(\mathbb{R}^{+})^{2}:\forall
y\in\mathbb{R},1-(y-\mu)\alpha-(B-Z_{y})\beta\geq 0\right\\}$. Our first
result is the following upper bound for a given value of $X_{n+1}$.

######  Lemma 16 (Upper bound on the BCP)

Consider fixed observations $X_{1},\dots,X_{n},X_{n+1}$, and denote by
$F_{n+1}$ their empirical cdf. Then, for any
$(\alpha,\beta)\in\mathcal{R}_{2}$ it holds that

$\mathbb{P}\left(\sum_{i=1}^{n+1}w_{i}X_{i}\geq\mu,\sum_{i=1}^{n+1}w_{i}Z_{X_{i}}\leq
B\right)\leq\exp(-(n+1)\Lambda_{\alpha,\beta}(F_{n+1},\mu))\;.$

Proof  We use the notation $\widetilde{X}=(X_{1},\dots,X_{n+1})$ and
$\widetilde{Z}=(Z_{X_{1}},\dots,Z_{X_{n+1}})$ for simplicity, and denote by
$R=(R_{1},\dots,R_{n+1})$ a vector of $n+1$ i.i.d. observations drawn from an
exponential distribution $\operatorname{\varepsilon}(1)$. Using usual
properties of the Dirichlet distribution (see Chapter 23 of MacKay (2003)) and
Chernoff method, for any $\alpha>0$, $\beta>0$ it holds that

$\displaystyle\mathbb{P}_{w\sim\mathcal{D}_{n}}(w^{T}\widetilde{X}\geq\mu,w^{T}\widetilde{Z}\leq
B)$
$\displaystyle=\mathbb{P}_{R\sim\mathcal{E}(1)^{n}}(R^{T}(\widetilde{X}-\mu)\geq
0,R^{T}(B-\widetilde{Z})\geq 0)$
$\displaystyle=\mathbb{P}_{R\sim\mathcal{E}(1)^{n}}\left(e^{\alpha(R^{T}(\widetilde{X}-\mu)}\geq
1,e^{\beta(R^{T}(B-\widetilde{Z}))}\geq 1\right)$
$\displaystyle\leq\mathbb{P}_{R\sim\mathcal{E}(1)^{n}}\left(e^{\alpha(R^{T}(\widetilde{X}-\mu)+\beta(R^{T}(B-\widetilde{Z}))}\geq
1\right)$
$\displaystyle\leq\mathbb{E}_{R\sim\mathcal{E}(1)^{n}}\left[e^{\alpha(R^{T}(\widetilde{X}-\mu)+\beta(R^{T}(B-\widetilde{Z}))}\right]\;.$

Then, if $(\alpha,\beta)\in\mathcal{R}_{2}$, it holds that

$\displaystyle\mathbb{P}_{w\sim\mathcal{D}_{n}}(w^{T}\widetilde{X}\geq\mu,w^{T}\widetilde{Z}\leq
B)$
$\displaystyle\leq\prod_{i=1}^{n+1}\mathbb{E}_{R_{i}\sim\mathcal{E}(1)}\left[e^{\alpha(R_{i}(X_{i}-\mu)+\beta(R_{i}(B-Z_{X_{i}}))}\right]$
$\displaystyle=\prod_{i=1}^{n+1}\frac{1}{1-\left(\alpha(X_{i}-\mu)+\beta(B-Z_{X_{i}})\right)}$
$\displaystyle=\exp\left(-(n+1)\mathbb{E}_{F_{n+1}}\left[\log\left(1-\left(\alpha(X-\mu)+\beta(B-Z_{X_{i}})\right)\right)\right]\right)$
$\displaystyle=\exp\left(-(n+1)\Lambda_{\alpha,\beta}(F_{n+1},\mu)\right)\;.$

We use this result by partitioning the set of possible values for $X_{n+1}$.
Consider a step $\eta>0$, and the sequence $(x_{j})_{j\in\mathbb{N}}$ defined
by $\forall j\in\mathbb{N}$, $x_{j}=\mu+j\eta$.

(P) $\displaystyle\leq\sum_{j=0}^{+\infty}\mathbb{P}\left(\exists
X_{n+1}\in[x_{j},x_{j+1}]:\sum_{i=1}^{n+1}w_{i}X_{i}\geq\mu,\sum_{i=1}^{n+1}w_{i}Z_{X_{i}}\leq
B\right)$
$\displaystyle\leq\sum_{j=0}^{+\infty}\mathbb{P}\left(\sum_{i=1}^{n}w_{i}X_{i}+w_{n+1}x_{j+1}\geq\mu,\sum_{i=1}^{n}w_{i}Z_{X_{i}}+w_{n+1}Z_{x_{j}}\leq
B\right)$
$\displaystyle\leq\sum_{j=0}^{+\infty}\underbrace{\mathbb{P}\left(\sum_{i=1}^{n}w_{i}X_{i}+w_{n+1}(x_{j}+\eta)\geq\mu,\sum_{i=1}^{n}w_{i}Z_{X_{i}}+w_{n+1}Z_{x_{j}}\leq
B\right)}_{P_{j}}\;.$

At this step we can use Lemma 16, and isolating the term depending on $x_{j}$
from the others we obtain

$\sum_{j=0}^{+\infty}P_{j}\leq\sum_{j=0}^{+\infty}\frac{\exp\left(-n\Lambda_{\alpha(j),\beta(j)}(F_{n},\mu-\eta)\right)}{1-\alpha(j)(x_{j}+\eta-\mu)-\beta(j)(B+\gamma-h(x_{j}-\mu))}\;,$
(33)

for any choice of sequence $(\alpha(j),\beta(j))_{j\in\mathbb{N}}$. In the
rest of the proof we use that when $x_{j}$ is large enough the denominator
becomes small, and so only a finite number of terms are dominant in the sum.

#### Upper bound for large $x_{j}$

We consider
$(\alpha_{n}^{\star},\beta_{n}^{\star})=\text{argmax}_{\alpha,\beta}\Lambda_{\alpha,\beta}(F_{n},\mu-\eta)$.
We first use that $\beta_{n}^{\star}>0$, since
$\beta_{n}^{\star}\in\mathcal{R}_{2}$, so when $x_{j}$ is large enough the
denominator scales with $\beta_{n}^{\star}h(x_{j}-\mu)$, which is larger than
the other terms since $x^{1+\gamma}=o(h(x))$ for some $\gamma>0$. Hence, there
exists a constant $y_{n}$ such that for any $x_{j}\geq y_{n}$ it holds that

$1-\alpha_{n}^{\star}(x_{j}+\eta-\mu)-\beta_{n}^{\star}(B+\gamma-h(x_{j}-\mu))\geq\frac{\beta_{n}^{\star}}{2}h(x_{j}-\mu)\;,$

and so the part of the sum corresponding to all $x_{j}\geq y_{n}$ satisfies

$\displaystyle\sum_{j\geq 1:x_{j}\geq y_{n}}^{+\infty}P_{j}$
$\displaystyle\leq\frac{2}{\beta_{n}^{\star}}\left(\sum_{j\geq 1:x_{j}\geq
y_{n}}h(j\eta)^{-1}\right)e^{-n\Lambda_{\alpha_{n}^{\star},\beta_{n}^{\star}}(F_{n},\mu-\eta)}\;,$

where we know that $\sum h(j\eta)^{-1}$ converges since
$x^{1+\eta_{0}}=o(h(x))$ for some $\eta_{0}>0$.

#### Upper bound for $x_{j}\leq y_{n}$

In this case, we define $(\alpha_{n}^{\star},\beta_{n}^{\star})$ as

$(\alpha_{n}^{\star},\beta_{n}^{\star})=\max_{(\alpha,\beta)\in\mathcal{R}_{2}^{\gamma,\eta}}\Lambda_{\alpha,\beta}(F,\mu)\;,$

with
$\mathcal{R}_{2}^{\gamma,\eta}=\\{(\alpha,\beta)\in(\mathbb{R}^{+})^{2}:\forall
x\in\mathbb{R},\;\;1-\alpha(x+\eta-\mu)-\beta(B+\gamma-h(|x-\mu|))\geq 0\\}$.
We further define
$x^{\star}=\text{argmin}_{x\in\mathbb{R}}\left\\{1-\alpha_{n}^{\star}(x+\eta-\mu)-\beta_{n}^{\star}(B+\gamma-h(|x-\mu|))\right\\}$.
$x^{\star}$ depends on $F_{n}$, but we omit reference to $F_{n}$ for lighter
notation. We use again the upper bound of $\sum P_{j}$ provided by (33). The
main task, that we tackle in the following, is to lower bound the denominator
by a constant depending on the empirical distribution $F_{n}$ and $\eta$.

Since the partition $(x_{j})_{j\in\mathbb{N}}$ defined at the beginning of the
proof is arbitrary (and only used in the proof but not by the algorithm), we
can slightly modify it to avoid arbitrarily large values when $x_{j}$ is close
to $x^{\star}$. Let $j^{\star}\in\mathbb{N}$ be the index satisfying
$x^{\star}\in[x_{j^{\star}},x_{j^{\star}+1}]$. In order to introduce some
distance between $x^{\star}$ and $x_{j^{\star}}$, $x_{j^{\star}+1}$, we simply
redefine $x_{j^{\star}}=x^{\star}-\frac{\eta}{2}$ and
$x_{j^{\star}+1}=x^{\star}-\frac{\eta}{2}$. We furthermore use that
$d(x_{j})\coloneqq
1-\alpha_{n}^{\star}(x^{\star}+\eta-\mu)-\beta_{n}^{\star}(B-h(x^{\star}-\mu))\geq
0$ and is minimized in $x^{\star}$. By convexity of $h$, this quantity is
minimized either in $x_{j^{\star}}$ or $x_{j^{\star}+1}$. Thanks to the change
made to the discretization scheme, this provides

$\inf_{j\in\mathbb{N}}d(x_{j})\geq\min\left\\{d\left(x^{\star}+\frac{\eta}{2}\right),d\left((x^{\star}-\frac{\eta}{2}\right)\right\\}>0\;.$

Furthermore, the right-hand term only depends on $\eta$ and on the empirical
distribution $F_{n}$ (through $x^{\star}$). Hence, for $x_{j}\leq y_{n}$ we
can state that there exists a non-negative constant $D(F_{n},\eta)$ such that

$\frac{1}{1-\alpha_{n}^{\star}(x_{j}-\mu)-\beta_{n}^{\star}(B-h(x_{j}-\mu))}\leq
D(F_{n},\eta)\;,$

where $D(F_{n},\eta)$ is continuous in $F_{n}$ w.r.t. the Wasserstein metric
since both $(\beta_{n}^{\star},x^{\star})$ are continuous under this metric
(as a consequence of Berge’s maximum theorem, Berge, 1997). Finally, we remark
that there are no more than $y_{n}/\eta$ terms satisfying $x_{j}\leq y_{n}$
and obtain

$\displaystyle\sum_{j=0}^{k\geq 1:x_{j}<y_{n}}P_{j}$
$\displaystyle\leq\sum_{j=0}^{k\geq
1:x_{j}<y_{n}}D(F_{n},\eta)e^{-n\Lambda_{\alpha_{n}^{\star},\beta_{n}^{\star}}(F_{n},\mu)}$
$\displaystyle\leq
y_{n}\frac{D(F_{n},\eta)}{\eta}e^{-n\Lambda_{\alpha_{n}^{\star},\beta_{n}^{\star}}(F_{n},\mu)}.$

#### Summary

Combining our two results, we obtain that

$\text{(P)}\leq
C(\alpha_{n}^{\star},\beta_{n}^{\star},\eta)\exp\left(-n\Lambda_{\alpha_{n}^{\star},\beta_{n}^{\star}}(F_{n},\mu)\right)\;,$
(34)

where

$C(\alpha_{n}^{\star},\beta_{n}^{\star},\eta)=y_{n}\frac{D(F_{n},\eta)}{\eta}+\frac{2}{\beta_{n}^{\star}}\sum_{k=\left\lceil\frac{y_{n}-\mu}{\eta}\right\rceil}^{+\infty}\frac{1}{h(\eta
k)}\;.$

As stated above the constant $C(\alpha_{n}^{\star},\beta_{n}^{\star},\eta)$
depend on $F_{n}$ through the optimizers
$(\alpha_{n}^{\star},\beta_{n}^{\star})$ (as they also determine $y_{n}$ and
$x^{\star}$), that depends on $\gamma$ and $\eta$ since they belong to
$\mathcal{R}_{2}^{\gamma,\eta}$. Finally, with the same arguments as for the
continuity of $D$, it is clear that $C$ is continuous in all parameters, which
concludes the proof of the result.

###  E.2 Lower bound: Bounded case

In this section we present a novel lower bound for the BCP of NPTS in the
bounded case, that can also serve as a warm-up to present the idea for the
slightly more complicated $h$-moment condition setting. We assume that all
observations are bounded by some constant $B>0$, and that the algorithm is
using the additional support $X_{n+1}=B+\gamma$ for some $\gamma>0$.

######  Lemma 17 (Novel lower bound for the BCP of NPTS)

If NPTS is implemented with exploration bonus $B+\gamma$ for some $\gamma>0$,
then for any $\delta>0$ there exists a constant
$n_{\delta,\gamma}\in\mathbb{N}$ such that

$\text{\emph{[BCP]}}\geq\frac{1-e^{-\frac{\delta}{4}}}{2e^{1+\delta}}\times
e^{-n\left(\Lambda_{\alpha_{n}^{\star}}(F_{n},\mu^{\star})+\delta\right)}\;\quad\mbox{if}\quad
n\geq n_{\delta,\gamma}\;,$

and
$\alpha_{n}^{\star}=\text{argmax}_{\alpha\in\left[0,\frac{1}{B+\gamma-\mu}\right]}\Lambda_{\alpha_{n}^{\star}}(F_{n},\mu^{\star})$.

Proof  Recall that

$\text{[BCP]}\geq\mathbb{P}_{w\sim\mathcal{D}_{n+1}}\left(\sum_{i=1}^{n+1}w_{i}X_{i}\geq\mu^{\star}\right)\;,$

with $X_{n+1}=B+\gamma$. For any cdf
$F_{n+1}:x\mapsto\frac{1}{n+1}\sum_{i=1}^{n+1}\text{1}(X_{i}\leq x)$ we define

$\Lambda_{\alpha}(F_{n},\mu)=\mathbb{E}_{F_{n+1}}\left[\log\left(1-\alpha(X-\mu)\right)\right]\;.$

We recall that any vector $w=(w_{1},\dots,w_{n+1})\sim\mathcal{D}_{n+1}$ can
be re-written as a normalized sum of $\operatorname{\varepsilon}(1)$
exponential random variables. Hence, we can write the BCP as follows:

$\text{[BCP]}=\mathbb{P}_{R_{1}\sim\operatorname{\varepsilon}(1),\dots,R_{n+1}\sim\operatorname{\varepsilon}(1)}\left(\sum_{i=1}^{n+1}R_{i}(X_{i}-\mu^{\star})\geq
0\right)\;,$

where $(R_{i})_{i\geq 1}$ are i.i.d. and follow an exponential distribution
$\mathcal{E}(1)$. To lower bound this quantity, we will perform a change of
measure for each variable $R_{i}$. In particular, we propose to consider a
variable that has expectation $G_{i}$, for some $(G_{i})_{i=1,\dots,n+1}$. We
can do that by choosing exponential random variables
$\operatorname{\varepsilon}\left(\frac{1}{G_{i}}\right)$: this technique is
known as exponential tilting. If $\widetilde{f}_{i}$ denotes the density of
this random variable and $f$ denotes the density of the distribution
$\operatorname{\varepsilon}(1)$, it holds that

$\forall x\geq
0:\;\frac{f(x)}{\widetilde{f}_{i}(x)}=G_{i}e^{-\left(1-\frac{1}{G_{i}}\right)x}$

We further denote by and $\mathbb{P}$ and $\widetilde{\mathbb{P}}$ their
respective product. Using $R^{T}U=\sum_{i=1}^{n+1}R_{i}(X_{i}-\mu)$, we now
express the BCP using the distributions of densities $\widetilde{f}_{i}$,

$\displaystyle\text{[BCP]}=$
$\displaystyle\mathbb{E}_{\mathbb{P}}\left[\text{1}\left(R^{T}U\geq
0\right)\right]$ $\displaystyle=$
$\displaystyle\mathbb{E}_{\widetilde{\mathbb{P}}}\left[\text{1}\left(R^{T}U\geq
0\right)\prod_{i=1}^{n+1}\frac{f(R_{i})}{\widetilde{f}_{i}(R_{i})}\right]$
$\displaystyle=$
$\displaystyle\prod_{i=1}^{n+1}G_{i}\times\mathbb{E}_{\widetilde{\mathbb{P}}}\left[\text{1}\left(R^{T}U\geq
0\right)e^{-\sum_{i=1}^{n+1}\left(1-\frac{1}{G_{i}}\right)R_{i}}\right]$
$\displaystyle=$ $\displaystyle
e^{-\sum_{i=1}^{n+1}\log\left(\frac{1}{G_{i}}\right)}\mathbb{E}_{\widetilde{\mathbb{P}}}\left[\text{1}\left(R^{T}U\geq
0\right)e^{-\sum_{i=1}^{n+1}\left(1-\frac{1}{G_{i}}\right)R_{i}}\right]\;.$

Then, our proof will consists in: first, deriving the term
$\max_{\alpha}\Lambda_{\alpha}(F_{n},\mu)+\delta$ from the exponential term,
and then showing that the expectation can be lower bounded by a constant. To
do that, for any arbitrary constant $\eta_{1}>0$ and for $i\leq n+1$ we
consider

$G_{i}=\frac{1}{1-\alpha(X_{i}-\mu-\eta_{1})}\quad\text{ for some
}\alpha>0\;,$

so that
$\frac{1}{n}\sum_{i=1}^{n}\log\left(\frac{1}{G_{i}}\right)=\Lambda_{\alpha}(F_{n},\mu+\eta_{1})$.
Now, let us choose

$\alpha=\alpha^{\star}\coloneqq\text{argmax}\,\Lambda_{\alpha}(F_{n+1},\mu+\eta_{1})\;,$

and then
$\Lambda_{\alpha^{\star}}(F_{n+1},\mu+\eta_{1})=\mathcal{K}_{\inf}^{\mathcal{F}}(F_{n+1},\mu+\eta_{1})$,
where $\mathcal{F}$ is the family of distributions bounded by $B+\gamma$, and
the weights $(G_{i})_{i\in[n+1]}$ associated with support
$(X_{1},\dots,X_{n})$ define a probability distribution of mean equal to
$\mu+\eta_{1}$. Furthermore, it holds that

$\displaystyle\alpha\leq\alpha_{M}\coloneqq\frac{1}{B+\gamma-\mu-\eta_{1}},$
(35)

but also that

$\forall i\in[n]\;,\;X_{i}\leq
B\Rightarrow\frac{1}{1+\alpha_{M}(\mu+\eta_{1})}\leq
G_{i}\leq\frac{B+\gamma-\mu-\eta_{1}}{\gamma}\;.$

This property is crucial for the rest of our proof, and motivates the choice
of $X_{n+1}$ as $B+\gamma$ instead of $B$: it ensures that all the exponential
variables $R_{1},\dots,R_{n}$ have a reasonable expectation, that does not
depend on $n$: only $G_{n+1}$ can potentially be large (in the case when the
empirical distribution is very bad, e.g. $X_{1}=X_{2}=\dots=X_{n}=0$).

We now write that

$\text{[BCP]}\geq
e^{-(n+1)\Lambda_{\alpha}(F_{n+1}\mu+\eta_{1})}\times\mathbb{E}_{\widetilde{\mathbb{P}}}\left[\text{1}\left(R^{T}U\geq
0\right)e^{-\alpha R^{T}\widetilde{U}}\right]\;,$

with $\widetilde{U}=(X_{1}-\mu-\eta_{1},\dots,X_{n+1}-\mu-\eta_{1})$, and
after remarking that for all $i$,

$1-\frac{1}{G_{i}}=\alpha(X_{i}-\mu-\eta_{1}).$

We now use that $\alpha\leq\alpha_{M}\coloneqq\frac{1}{B+\gamma-\mu-\eta_{1}}$
to handle the exponential term inside the expectation. Then, for any
$\delta>0$ we write that

$\text{[BCP]}\geq
e^{-(n+1)\left(\Lambda_{\alpha,\beta}(F_{n+1},\mu+\eta_{1})+\frac{\delta}{2}\right)}\times
P_{\alpha}\;,$ (36)

where we defined

$P_{\alpha}=\widetilde{\mathbb{P}}\left(R^{T}U\geq
0,R^{T}\widetilde{U}\leq(n+1)\frac{\delta}{2\alpha_{M}}\right)\;.$ (37)

In some cases, it may happen that $G_{n+1}$ is large (at most $n+1$), and in
that case the exponential variable as variance $n^{2}$. Hence, we would like
to control very precisely the deviations of $R_{n+1}$. For this reason, for
some constant $c>0$ we consider the event

$E_{n+1}\coloneqq\left\\{R_{n+1}\in\left[G_{n+1},\;(1+c)G_{n+1}\right]\right\\}\;.$

Using the cdf of exponential variables, this event has exactly a probability
of

$\mathbb{P}_{\alpha}(R_{n+1}\in\left[G_{n+1},\;(1+c)G_{n+1}\right])=e^{-\frac{G_{n+1}}{G_{n+1}}}-e^{-\frac{G_{n+1}(1+c)}{G_{n+1}}}=e^{-1}(1-e^{-c})\;.$

We now re-write $P_{\alpha}$ as

$\displaystyle P_{\alpha}$
$\displaystyle\geq\widetilde{\mathbb{P}}\left(R^{T}U\geq
0,R^{T}\widetilde{U}\leq(n+1)\frac{\delta}{2\alpha_{M}},E_{n+1}\right)$
$\displaystyle\geq\mathbb{P}(E_{n+1})\mathbb{P}\left(R^{T}U\geq
0,R^{T}\widetilde{U}\leq(n+1)\frac{\delta}{2\alpha_{M}}|E_{n+1}\right)$
$\displaystyle=e^{-1}(1-e^{-c})\left(1-\underbrace{\mathbb{P}_{\alpha}(R^{T}U\leq
0,R^{T}\widetilde{U}\geq
n\frac{\delta}{2\alpha_{M}}|E_{n+1})}_{p_{1}}\right)\;.$

The rest of the proof consists in showing that for $n$ large enough the terms
$p_{1}$ and $p_{2}$ are small. We consider the first event in $p_{1}$. Using
that $\mathbb{E}_{\alpha}[R^{T}U]=(n+1)\eta$

$\displaystyle\left\\{R^{T}U\leq 0\right\\}$
$\displaystyle\subset\left\\{\sum_{i=1}^{n}R_{i}(X_{i}-\mu)+G_{n+1}(X_{n+1}-\mu)\leq
0\right\\}$
$\displaystyle=\left\\{\sum_{i=1}^{n}R_{i}(X_{i}-\mu)-\mathbb{E}\left[\sum_{i=1}^{n}R_{i}(X_{i}-\mu)\right]\leq-(n+1)\eta\right\\}\;,$

and so under $E_{n+1}$ the first event implies a deviation of
$\sum_{i=1}^{n}R_{i}(X_{i}-\mu)$ from its mean. Next, considering the second
event it holds that

$\displaystyle\left\\{R^{T}\widetilde{U}\geq(n+1)\frac{\delta}{2\alpha_{M}}\right\\}\subset\left\\{\sum_{i=1}^{n}R_{i}(X_{i}-\mu)+(1+c)G_{n+1}(X_{n+1}-\mu)\geq(n+1)\frac{\delta}{2\alpha_{M}}\right\\}$
$\displaystyle=\left\\{\sum_{i=1}^{n}R_{i}(X_{i}-\mu)-\mathbb{E}\left[\sum_{i=1}^{n}R_{i}(X_{i}-\mu)\right]\leq(n+1)\frac{\delta}{2\alpha_{M}}-cG_{n+1}(B+\gamma-\mu)\right\\}\;.$

Then, we use that $G_{n+1}\leq n+1$ to tune $c$ in order to obtain something
similar to the first event.

For the above goal, let us consider the case of

$\displaystyle c=\frac{\delta}{4\alpha_{M}(B+\gamma-\mu)}.$ (38)

In this case we obtain that

$\displaystyle\left\\{R^{T}\widetilde{U}\geq(n+1)\frac{\delta}{2\alpha_{M}}\right\\}\subset\left\\{\sum_{i=1}^{n}R_{i}(X_{i}-\mu)-\mathbb{E}\left[\sum_{i=1}^{n}R_{i}(X_{i}-\mu)\right]\leq(n+1)\frac{\delta}{4\alpha_{M}}\right\\}\;.$

Then, using the notation
$\eta^{\prime}=\min\left\\{\eta,\frac{\delta}{4\alpha_{M}}\right\\}$ we obtain
with the Tchebychev inequality that

$\displaystyle p_{1}$
$\displaystyle\leq\mathbb{P}_{\alpha}\left(\left|\sum_{i=1}^{n}R_{i}(X_{i}-\mu)-\sum_{i=1}^{n}G_{i}(X_{i}-\mu)\right|\geq(n+1)\eta^{\prime}\right)$
$\displaystyle\leq\frac{1}{(n+1)^{2}\eta^{\prime
2}}\times\sum_{i=1}^{n}G_{i}^{2}(X_{i}-\mu)^{2}$
$\displaystyle\leq\frac{1}{(n+1)^{2}\eta^{\prime
2}}(B-\mu)^{2}\max_{i=1,\dots,n}G_{i}\times\sum_{i=1}^{n}G_{i}$
$\displaystyle\leq\frac{(B-\mu)^{2}}{(n+1)\eta^{\prime
2}}\frac{B+\gamma-\mu-\eta_{1}}{\gamma}$
$\displaystyle\leq\frac{1}{n+1}\times\frac{(B-\mu)^{2}}{\eta^{\prime
2}}\left(1+\frac{B-\mu}{\gamma}\right)\;.$ (39)

As this bound is independent of $F_{n}$, we obtain that for any $\eta>0$,
there exists $n_{\eta}\in\mathbb{N}$ satisfying $p_{1}\leq\frac{1}{2}$ for
$n\geq n_{\eta}$, under the case of (38).

Here note that $c$ in (38) goes to $\frac{\delta}{4}$ as $\eta\to 0$ since
$\alpha_{M}(B+\gamma-\mu)\to 1$ holds from (35). For this reason, we now take
$c=\delta/4$ instead of (38), under which a result similar to (39) holds and
there still exists $n_{\eta}\in\mathbb{N}$ satisfying $p_{1}\leq\frac{1}{2}$
for $n\geq n_{\eta}$ if we choose $\eta$ small enough. For such $n\geq
n_{\eta}$, it holds that
$P_{\alpha}\geq\frac{e^{-1}(1-e^{-c})}{2}=\frac{e^{-1}(1-e^{-\frac{\delta}{4}})}{2}$
(plugging our result in Equation (37)) and so using (36) we obtain that

$\text{[BCP]}\geq\frac{e^{-1}(1-e^{-\frac{\delta}{4}})}{2}e^{-(n+1)\left(\Lambda_{\alpha}(F_{n+1},\mu+\eta)+\frac{\delta}{2}\right)}\;.$

We now tune $\eta$ as

$\eta=\max\left\\{\eta>0:\;\forall
F_{n}\in\mathcal{F}\;,\;\Lambda_{\alpha^{\star}}(F_{n+1},\mu+\eta)\leq\mathcal{K}_{\inf}^{\mathcal{F}^{\gamma}}(F_{n+1},\mu)+\frac{\delta}{2}\right\\}\;.$

Due to the compactness of $\mathcal{F}$, this quantity exists, is strictly
positive, and depends on $\delta$, and goes to $0$ as $\delta$ goes to $0$.
Finally, we remark that since $X_{n+1}$ has a negative contribution to
$\mathcal{K}_{\inf}(F_{n+1},\mu)$ (expressed as a sum of $n+1$ logarithmic
terms, the $(n+1)$-th one being the minimum), then it holds that

$(n+1)\mathcal{K}_{\inf}^{\mathcal{F}^{\gamma}}(F_{n+1},\mu)\leq
n\mathcal{K}_{\inf}^{\mathcal{F}^{\gamma}}(F_{n},\mu)\;$

and consequently, we conclude that for any $\delta>0$, there exists a constant
$n_{\delta}$ such that for all $n\geq n_{\delta}$ it holds that

$\text{[BCP]}\geq\frac{1-e^{-\frac{\delta}{4}}}{2e^{1+\delta}}e^{-n\left(\mathcal{K}_{\inf}^{\mathcal{F}^{\gamma}}(F_{n},\mu)+\delta\right)}\;,$

which concludes our proof for this result.

###### Remark 18

Chernoff method would lead to a tighter upper bound for $p_{1}$, however we
choose Tchebychev inequality for simplicity since knowing the right value of
$n_{\delta}$ is not necessary for our algorithm.

###  E.3 Lower bound: $h$-moment condition

In this section we prove the lower bounds on the BCP in Lemma 5. The proof
follows the exact same steps as the proof of Lemma 17 presented in Section
E.2, with some additional technicality due to the slightly more complicated
definition of the family $\mathcal{F}$. We still write all the details for
completeness. We first recall that for observations $X_{1},\dots,X_{n}$ the
BCP is expressed as

[BCP] $\displaystyle\geq\mathbb{P}_{w\sim\mathcal{D}_{n+1}}\left(\exists
X_{n+1}\geq\mu^{\star}:\;\sum_{i=1}^{n+1}w_{i}X_{i}\geq\mu^{\star},\sum_{i=1}^{n+1}w_{i}Z_{X_{i}}\leq
B\right)$
$\displaystyle\geq\mathbb{P}_{w\sim\mathcal{D}_{n+1}}\left(\sum_{i=1}^{n+1}w_{i}X_{i}\geq\mu^{\star},\sum_{i=1}^{n+1}w_{i}Z_{X_{i}}\leq
B\right)\quad\mbox{for arbitrary $X_{n+1}\geq\mu^{\star}$}\;,$

with $Z_{X_{i}}=h(|X_{i}-\mu|)$ for $i\in[n]$ and
$Z_{X_{n+1}}=h(X_{n+1}-\mu)-\gamma$. To provide a lower bound on this quantity
we will choose $X_{n+1}$ so that the bound is expressed in terms of the
quantity

$\Lambda_{\alpha,\beta}(F,\mu,B)=\mathbb{E}_{F}\left[\log\left(1-\alpha(X-\mu)-\beta(B-Z_{X})\right)\right]\;,$

given some parameters $\alpha,\beta,\mu,B$ and a cdf $F$.

We recall that any vector $w=(w_{1},\dots,w_{n+1})\sim\mathcal{D}_{n+1}$ can
be re-written as a normalized sum of $\operatorname{\varepsilon}(1)$
exponential random variables. Hence, we can write the BCP as follows:

$\text{[BCP]}\geq\mathbb{P}_{R_{1}\sim\operatorname{\varepsilon}(1),\dots,R_{n+1}\sim\operatorname{\varepsilon}(1)}\left(\sum_{i=1}^{n+1}R_{i}(X_{i}-\mu)\geq
0,\sum_{i=1}^{n+1}R_{i}(B-Z_{X_{i}})\geq 0\right)\;,$

where $(R_{i})_{i\geq 1}$ are i.i.d. and follow an exponential distribution
$\mathcal{E}(1)$. We perform an exponential tilting of the distributions,
considering the exponential variables with expectation
$\mathbb{E}[R_{i}]=G_{i}$, for some sequence $(G_{i})_{i=1,\dots,n+1}$, so
$R_{i}\sim\operatorname{\varepsilon}\left(\frac{1}{G_{i}}\right)$. If
$\widetilde{f}_{i}$ denotes the density of this random variable and $f$
denotes the density of the distribution $\operatorname{\varepsilon}(1)$, it
holds that

$\forall x\geq
0:\;\frac{f_{i}(x)}{\widetilde{f}_{i}(x)}=G_{i}e^{-\left(1-\frac{1}{G_{i}}\right)x}.$

We further denote by and $\mathbb{P}$ and $\widetilde{\mathbb{P}}$ their
respective product. Using the notation
$R^{T}U=\sum_{i=1}^{n+1}R_{i}(X_{i}-\mu)$ and
$R^{T}V=\sum_{i=1}^{n+1}R_{i}(B-Z_{X_{i}})$, we now re-write the BCP using the
distributions of densities $\widetilde{f}_{i}$,

$\displaystyle\text{[BCP]}=$
$\displaystyle\mathbb{E}_{\mathbb{P}}\left[\text{1}\left(R^{T}U\geq
0,R^{T}V\geq 0\right)\right]$ $\displaystyle=$
$\displaystyle\mathbb{E}_{\widetilde{\mathbb{P}}}\left[\text{1}\left(R^{T}U\geq
0,R^{T}V\geq
0\right)\prod_{i=1}^{n+1}\frac{f_{i}(R_{i})}{\widetilde{f}_{i}(R_{i})}\right]$
$\displaystyle=$
$\displaystyle\prod_{i=1}^{n+1}G_{i}\times\mathbb{E}_{\widetilde{\mathbb{P}}}\left[\text{1}\left(R^{T}U\geq
0,R^{T}V\geq
0\right)e^{-\sum_{i=1}^{n+1}\left(1-\frac{1}{G_{i}}\right)R_{i}}\right]$
$\displaystyle=$ $\displaystyle
e^{-\sum_{i=1}^{n+1}\log\left(\frac{1}{G_{i}}\right)}\mathbb{E}_{\widetilde{\mathbb{P}}}\left[\text{1}\left(R^{T}U\geq
0,R^{T}V\geq
0\right)e^{-\sum_{i=1}^{n+1}\left(1-\frac{1}{G_{i}}\right)R_{i}}\right]\;.$

We want to make the term
$\max_{\alpha,\beta}\Lambda_{\alpha,\beta}(F_{n},\mu,B)+\delta$ come from the
exponential term in front of the expectation, and to show that the expectation
can be lower bounded by a constant. To do that, we consider two arbitrary
constants $\eta_{1}>0,\eta_{2}>0$, and for $i\leq n+1$ we define

$G_{i}=\begin{cases}\frac{1}{1-\alpha(X_{i}-\mu-\eta_{1})-\beta(B-\eta_{2}-Z_{X_{i}})},&i\in[n],\\\
\frac{1}{1-\alpha(X_{i}-\mu-\eta_{1})-\beta(B+\gamma-\eta_{2}-Z_{X_{i}})},&i=n+1,\end{cases}$

for some constants $(\alpha,\beta)$. We still center the $h$-moment in $\mu$,
as we want the objective inside the expectation to be strictly easier to reach
with our change of distribution. However, we keep the notation
$\Lambda_{\alpha,\beta}$ with a slight abuse for simplicity.

We then choose $X_{n+1}$ in the BCP as

$X_{n+1}=\text{argmin}_{x\in\mathbb{R}}\\{1-\alpha_{n}^{\star}(x-\mu-\eta_{1})-\beta_{n}^{\star}(B-\eta_{2}-h(|x-\mu|))\\}\;.$
(40)

Then, by definition of $(\alpha_{n}^{\star},\beta_{n}^{\star})$ we obtain that

$\displaystyle 1-\alpha_{n}^{\star}(X_{i}-\mu-\eta_{1})$
$\displaystyle-\beta_{n}^{\star}(B-\eta_{2}-h(|X_{i}-\mu|))$
$\displaystyle\geq
1-\alpha_{n}^{\star}(X_{n+1}-\mu-\eta_{1})-\beta_{n}^{\star}(B-\eta_{2}-h(|X_{n+1}-\mu|))\quad\text{(by
\eqref{eq::X_n+1})}$
$\displaystyle\geq\beta_{n}^{\star}\gamma+\underbrace{1-\alpha_{n}^{\star}(X_{n+1}-\mu-\eta_{1})-\beta_{n}^{\star}(B+\gamma-\eta_{2}-Z_{n+1})}_{\geq
0}$
$\displaystyle\geq\overline{\beta}\gamma\coloneqq\left(\min\limits_{F_{n}\in\mathcal{F}:\mathbb{E}_{F_{n}}[X]\leq\mu}\beta_{n}^{\star}\right)\gamma\;,$

where $\overline{\beta}$ is well-defined and satisfies $\overline{\beta}>0$ by
compactness of $\mathcal{F}$. With the same argument, we can also deduce the
existence of a (universal) upper bound $X^{+}$ of the potential additional
support $X_{n+1}$ (exploiting the continuity of mapping
$F\mapsto\alpha_{F}^{\star}$, $F\mapsto\beta_{F}^{\star}$ in $F$, and that the
additional support satisfies
$\alpha_{F}^{\star}=h^{\prime}(X-\mu)\beta_{F}^{\star}$). Now, given that
$X_{n+1}$ cannot be arbitrarily large, $W(F_{n},F_{n+1})=\mathcal{O}(n^{-1})$,
where $W$ denotes the Wasserstein metric. Furthermore, using the continuity of
the mapping $F\mapsto
1-\alpha_{F}^{\star}(x-\mu-\eta_{1})-\beta_{F}^{\star}(B-\eta_{2}-h(|x-\mu|))$
for any $x\in\mathbb{R}$, we finally conclude that there exists a rank
$n_{0}\in\mathbb{N}$ such that for any $n\geq n_{0}$ it holds that

$\forall i\in[n]\;,\;G_{i}\leq\frac{2}{\gamma\overline{\beta}}\;.$

This property ensures that all the exponential variables $R_{1},\dots,R_{n+1}$
have a reasonable expectation, that does not depend on $n$ but on $\gamma$:
only $G_{n+1}$ can potentially be large (at most $n+1$).

We now get back to lower bounding the BCP, writing that

$\text{[BCP]}\geq
e^{-n\Lambda_{\alpha,\beta}(F_{n},\mu+\eta_{1},B-\eta_{2})}\times\mathbb{E}_{\widetilde{\mathbb{P}}}\left[\text{1}\left(R^{T}U\geq
0,R^{T}V\geq 0\right)e^{-\alpha R^{T}\widetilde{U}-\beta
R^{T}\widetilde{V}}\right]\;,$

with $\widetilde{U}=(X_{1}-\mu-\eta_{1},\dots,X_{n+1}-\mu-\eta_{1})$ and
$\widetilde{V}=(B-\eta_{2}-Z_{X_{1}},\dots,B+\gamma-\eta_{2}-Z_{X_{n+1}}$, and
after remarking that for all $i\leq n$ (and with $B+\gamma$ $i=n+1$),

$1-\frac{1}{G_{i}}=\alpha(X_{i}-\mu-\eta_{1})+\beta(B-\eta_{2}-Z_{X_{i}}).$

Then, we handle the exponential term inside the expectation by first using
from Agrawal et al. (2020) (their proof can be simply adapted to get the
centered condition instead of the uncentered one) that $\alpha$ and $\beta$
are both upper bounded respectively by some constants $\alpha_{M}$ and
$\beta_{M}$ depending only on $\mu+\eta_{1}$ and $B-\eta_{2}$. Then, for any
$\delta>0$ we write that

$\text{[BCP]}\geq
e^{-n\left(\Lambda_{\alpha,\beta}(F_{n},\mu+\eta_{1},B-\eta_{2})+\frac{\delta}{2}\right)}\times
P_{\alpha,\beta}\;,$

where we defined

$P_{\alpha,\beta}=\widetilde{\mathbb{P}}\left(R^{T}U\geq 0,R^{T}V\geq
0,R^{T}\widetilde{U}\leq n\frac{\delta}{4\alpha_{M}},R^{T}\widetilde{V}\leq
n\frac{\delta}{4\beta_{M}}\right)\;.$

As in the bounded case, we control separately the term $R_{n+1}$ since it is
the only term with an expectation that can depend on $n$. Considering a
constant $c>0$, we define the event

$E_{n+1}\coloneqq\left\\{R_{n+1}\in\left[G_{n+1},\;(1+c)G_{n+1}\right]\right\\}\;.$

Using the cdf of exponential variables, this event has exactly a probability
of

$\mathbb{P}_{\alpha,\beta}(R_{n+1}\in\left[G_{n+1},\;(1+c)G_{n+1}\right])=e^{-\frac{G_{n+1}}{G_{n+1}}}-e^{-\frac{G_{n+1}(1+c)}{G_{n+1}}}=e^{-1}(1-e^{-c})\;,$

so we obtain that

$\displaystyle P_{\alpha,\beta}$
$\displaystyle\geq\widetilde{\mathbb{P}}\left(R^{T}U\geq 0,R^{T}V\geq
0R^{T}\widetilde{U}\leq(n+1)\frac{\delta}{4\alpha_{M}},R^{T}\widetilde{V}\geq\frac{\delta}{4\beta_{M}},E_{n+1}\right)$
$\displaystyle\geq\mathbb{P}_{\alpha,\beta}(E_{n+1})\mathbb{P}\left(R^{T}U\geq
0,R^{T}V\geq
0,R^{T}\widetilde{U}\leq(n+1)\frac{\delta}{4\alpha_{M}},R^{T}\widetilde{V}\leq(n+1)\frac{\delta}{4\alpha_{M}}|E_{n+1}\right)$
$\displaystyle=p_{c}\left(1-\underbrace{\mathbb{P}_{\alpha,\beta}(R^{T}U\leq
0,R^{T}\widetilde{U}\geq
n\frac{\delta}{4\alpha_{M}}|E_{n+1})}_{p_{1}}-\underbrace{\mathbb{P}_{\alpha,\beta}(R^{T}V\leq
0,R^{T}\widetilde{V}\geq
n\frac{\delta}{4\beta_{M}}|E_{n+1})}_{p_{2}}\right)\;,$

with $p_{c}=e^{-1}(1-e^{-c})$. The rest of the proof consists in showing that
for $n$ large enough the terms $p_{1}$ and $p_{2}$ are small. We now consider
that
$(\alpha,\beta)=\text{argmax}\Lambda_{\alpha,\beta}(F_{n+1},\mu+\eta_{1},B-\eta_{2})$.
We consider the first event in $p_{1}$. Using that
$\mathbb{E}_{\alpha,\beta}[R^{T}U]=(n+1)\eta$

$\displaystyle\left\\{R^{T}U\leq 0\right\\}$
$\displaystyle\subset\left\\{\sum_{i=1}^{n}R_{i}(X_{i}-\mu)+G_{n+1}(X_{n+1}-\mu)\leq
0\right\\}$
$\displaystyle=\left\\{\sum_{i=1}^{n}R_{i}(X_{i}-\mu)-\mathbb{E}\left[\sum_{i=1}^{n}R_{i}(X_{i}-\mu)\right]\leq-(n+1)\eta_{1}\right\\}\;,$

and so under $E_{n+1}$ the first event implies a deviation of
$\sum_{i=1}^{n}R_{i}(X_{i}-\mu)$ from its mean. Then, to handle the second
event we use that

$\displaystyle\left\\{R^{T}\widetilde{U}\geq(n+1)\frac{\delta}{4\alpha_{M}}\right\\}\subset\left\\{\sum_{i=1}^{n}R_{i}(X_{i}-\mu)+(1+c)G_{n+1}(X_{n+1}-\mu)\geq(n+1)\frac{\delta}{4\alpha_{M}}\right\\}$
$\displaystyle=\left\\{\sum_{i=1}^{n}R_{i}(X_{i}-\mu)-\mathbb{E}\left[\sum_{i=1}^{n}R_{i}(X_{i}-\mu)\right]\leq(n+1)\frac{\delta}{4\alpha_{M}}-cG_{n+1}(X^{+}-\mu)\right\\}\;.$

Then, we use that $G_{n+1}\leq n+1$ to obtain a first condition on $c$. We
arbitrarily consider $c\leq\frac{\delta}{8\alpha_{M}(X^{+}-\mu)}$, to obtain
that

$\displaystyle\left\\{R^{T}\widetilde{U}\geq(n+1)\frac{\delta}{4\alpha_{M}}\right\\}\subset\left\\{\sum_{i=1}^{n}R_{i}(X_{i}-\mu)-\mathbb{E}\left[\sum_{i=1}^{n}R_{i}(X_{i}-\mu)\right]\leq(n+1)\frac{\delta}{8\alpha_{M}}\right\\}\;.$

Hence, using the notation
$\eta_{1}^{\prime}=\min\left\\{\eta,\frac{\delta}{4\alpha_{M}}\right\\}$ we
get

$p_{1}\leq\mathbb{P}_{\alpha,\beta}\left(\left|\sum_{i=1}^{n}R_{i}(X_{i}-\mu)-\sum_{i=1}^{n}G_{i}(X_{i}-\mu)\right|\geq(n+1)\eta_{1}^{\prime}\right)\;.$
(41)

We then perform the exact same steps with $p_{2}$. This time, it will depend
on the sign of $B-h(X_{n+1}-\mu)+\gamma$. If $B-h(X_{n+1}-\mu)+\gamma>0$, we
obtain similarly that

$\displaystyle\left\\{R^{T}V\leq
0\right\\}\subset\left\\{\sum_{i=1}^{n}R_{i}(B-h(|X_{i}-\mu|)-\mathbb{E}\left[\sum_{i=1}^{n}R_{i}(X_{i}-\mu)\right]\leq-(n+1)\eta_{2}\right\\}\;\text{,
and}$
$\displaystyle\left\\{R^{T}\widetilde{V}\leq\frac{\delta}{4\beta_{M}}\right\\}\subset\left\\{\sum_{i=1}^{n}R_{i}(B-Z_{X_{i}})-\mathbb{E}\left[\sum_{i=1}^{n}R_{i}(B-Z_{X_{i}})\right]\geq(n+1)\left(\frac{\delta}{4\beta_{M}}-c(B+\gamma)\right)\right\\}\;.$

So, this time we can require for instance
$c\leq\frac{\delta}{8\beta_{M}(B+\gamma)}$. On the other hand, if
$B-h(X_{n+1}-\mu)+\gamma\leq 0$ this time we have

$\displaystyle\left\\{R^{T}V\leq
0\right\\}\subset\left\\{\sum_{i=1}^{n}R_{i}(B-h(|X_{i}-\mu|)-\mathbb{E}\left[\sum_{i=1}^{n}R_{i}(X_{i}-\mu)\right]\leq-(n+1)\eta_{2,c}\right\\}\;,$
$\displaystyle\left\\{R^{T}\widetilde{V}\leq\frac{\delta}{4\beta_{M}}\right\\}$
$\displaystyle\subset\left\\{\sum_{i=1}^{n}R_{i}(B-Z_{X_{i}})-\mathbb{E}\left[\sum_{i=1}^{n}R_{i}(B-h(|X_{i}-\mu|))\right]\geq(n+1)\frac{\delta}{4\beta_{M}}\right\\}\;,$

with $\eta_{2,c}=\eta_{2}-c(h(X^{+}-\mu)-B-\gamma)$, and so this time we can
ask that $c\leq\frac{\eta_{2}}{2(h(X^{+}-\mu)-B-\gamma)}$. Hence, if all
requirements on $c$ are satisfied we can define
$\eta_{2}^{\prime}=\min\left\\{\frac{\eta_{2}}{2(h(X^{+}-\mu)-B-\gamma)},\frac{\delta}{8\beta_{M}(B+\gamma)},\eta_{2}\right\\}$,
and we proved that

$p_{2}\leq\mathbb{P}_{\alpha,\beta}\left(\left|\sum_{i=1}^{n}R_{i}(B-h(X_{i}-\mu))-\sum_{i=1}^{n}G_{i}(B-h(|X_{i}-\mu|)\right|\geq(n+1)\eta_{2}^{\prime}\right)\;.$
(42)

Then, just as in the proof for the bounded case we can use the Tchebychev
inequality to further upper bound $p_{1}$ and $p_{2}$ from (41) and (42), and
obtain that

$\displaystyle p_{1}\leq\frac{1}{(n+1)^{2}\eta_{1}^{\prime
2}}\times\sum_{i=1}^{n}G_{i}^{2}(X_{i}-\mu)^{2}\;,$ and $\displaystyle
p_{2}\leq\frac{1}{(n+1)^{2}\eta_{2}^{\prime
2}}\times\sum_{i=1}^{n}G_{i}^{2}(B-h(|X_{i}-\mu|))^{2}\;.$

This time the quadratic terms depending on $X_{i}$ cannot be discarded as
easily as in the bounded case. However, we can use a similar trick to the one
we used to prove the upper bound in Lemma 5, remarking that if $X_{i}$ is too
large then $G_{i}$ is necessarily very small. Indeed, using that
$\beta\geq\beta_{m}$ and that when $X_{i}$ is large then
$G_{i}^{2}\sim\frac{1}{\beta(B-h(|X_{i}-\mu|))^{2}}$, we obtain that there
exists a constant $C>0$ such that for $|X_{i}-\mu|\geq C$ it holds that

$G_{i}^{2}\leq\frac{2}{\beta_{m}(B-h(|X_{i}-\mu|))^{2}}\;.$

Using this result and the notation $A_{C}=\max_{|x-\mu|\geq
C}\frac{(x-\mu)^{2}}{\beta_{m}(B-h(|x-\mu|))^{2}}$, we first obtain that

$\displaystyle p_{1}$ $\displaystyle\leq\frac{1}{(n+1)^{2}\eta_{1}^{\prime
2}}\left(C^{2}\sum_{i=1}^{n}G_{i}^{2}+2nA_{C}\right)$
$\displaystyle\leq\frac{1}{(n+1)^{2}\eta_{1}^{\prime
2}}\left(C^{2}\max_{i=1,\dots,n}G_{i}\times\sum_{i=1}^{n}G_{i}+2nA_{C}\right)$
$\displaystyle\leq\frac{1}{(n+1)^{2}\eta_{1}^{\prime
2}}\left(C^{2}(n+1)\max_{i=1,\dots,n}G_{i}+2nA_{C}\right)$
$\displaystyle\leq\frac{1}{(n+1)\eta_{1}^{\prime
2}}\left(\frac{2C^{2}}{\gamma\beta_{m}}+2A_{C}\right)=\mathcal{O}(n^{-1})\;.$

Similarly, we obtain that

$\displaystyle p_{2}$ $\displaystyle\leq\frac{1}{(n+1)^{2}\eta_{2}^{\prime
2}}\left((B-h(C))^{2}\sum_{i=1}^{n}G_{i}^{2}+\frac{2n}{\beta_{m}}\right)$
$\displaystyle\leq\frac{1}{(n+1)\eta_{2}^{\prime
2}}\left(\frac{2(B-h(C))^{2}}{\gamma\beta_{m}}+\frac{2}{\beta_{m}}\right)=\mathcal{O}(n^{-1})\;.$

As this bound is independent of $F_{n}$, we obtain that for any
$\eta_{1}>0,\eta_{2}>0$, there exists a rank $n_{\eta_{1},\eta_{2}}$ such that
$p_{1}\leq\frac{1}{4}$ and $p_{2}\leq\frac{1}{4}$ for $n\geq n_{\eta}$. In
that case, it holds that

$\text{[BCP]}\geq\frac{e^{-1}(1-e^{-c})}{2}e^{-(n+1)\left(\Lambda_{\alpha_{n}^{\star},\beta_{n}^{\star}}(F_{n+1},\mu+\eta_{1},B-\eta_{2})+\frac{\delta}{2}\right)}\;,$

where this bound holds if
$c\leq\min\left\\{\frac{\delta}{8\alpha_{M}(X^{+}-\mu)},\frac{\delta}{8\beta_{M}(B+\gamma)},\frac{\eta_{2}}{2(h(X^{+}-\mu)-B-\gamma)}\right\\}$,
so $c$ is of order $\mathcal{O}(\min\\{\delta,\eta_{2}\\})$. We now tune
$\eta_{1}$ and $\eta_{2}$. For simplicity we consider

$\eta_{1}=\eta_{2}=\max\left\\{\eta>0:\;\forall
F_{n}\in\mathcal{F}\;,\;\max_{\alpha,\beta}\Lambda_{\alpha,\beta}(F_{n},\mu+\eta,B-\eta)\leq\max_{\alpha,\beta}\Lambda_{\alpha,\beta}(F_{n},\mu,B)+\frac{\delta}{2}\right\\}\;.$

Thanks to the compactness of $\mathcal{F}$, this quantity exists, is strictly
positive, and only depends on $\delta$. This allows us to conclude that for
any $\gamma>0$ and $\delta>0$, there exists two constants $n_{\mathcal{F}}$
and $c_{\delta}$ such that for all $n\geq n_{\mathcal{F}}$ it holds that

$\text{[BCP]}\geq
c_{\delta}e^{-n\max_{(\alpha,\beta)\in\mathcal{R}_{2}^{\gamma}}\left(\Lambda_{\alpha,\beta}(F_{n},\mu,B)+\delta\right)}\;,$

which concludes the part of the proof corresponding to $n\geq
n_{\mathcal{F}}$.

We finish the proof by showing that the BCP can also be lower bounded for
$n\leq n_{\mathcal{F}}$. However, this bound does not need to be tight since
it will not impact the validity of the conditions that we want to prove. We
provide a simple lower bound by fixing the value of the additional support to
$X_{n+1}=\frac{\mu+h^{-1}(B)}{2}$. We show that for an appropriate constant
$c$ depending only on $B$ and $\mu$, if the empirical distribution belongs to
$\mathcal{F}$ then it holds for any $n\geq 1$ that

$\text{[BCP]}\geq\mathbb{P}\left(w_{n+1}\geq 1-\frac{1}{cn}\right)\geq
e^{-n\log(nc)}\;.$

Indeed, if $F_{n}\in\mathcal{F}$ then for all $i$, $\mu-h^{-1}(nB)\leq
X_{i}\leq\mu+h^{-1}(nB)$ under the event $\\{w_{n+1}\geq 1-\frac{1}{cn}\\}$ we
obtain

$\displaystyle\sum_{i=1}^{n+1}w_{i}X_{i}$
$\displaystyle\geq\left(1-\frac{1}{cn}\right)\left(\frac{\mu+h^{-1}(B)}{2}\right)-\frac{h^{-1}(nB)-\mu}{cn}$
$\displaystyle=\mu+\frac{h^{-1}(B)-\mu}{2}-\frac{1}{cn}\left(\frac{\mu+h^{-1}(B)}{2}+h^{-1}(nB)-\mu\right)\;,$

and

$\displaystyle\sum_{i=1}^{n+1}$ $\displaystyle w_{i}h(|X_{i}-\mu|)\leq
Z_{X_{n+1}}+\frac{1}{cn}\sum_{i=1}^{n}h(|X_{i}-\mu|)\leq
h\left(\frac{h^{-1}(B)-\mu}{2}\right)+\frac{B}{c}$

Then, the tuning on $c$ is made to ensure that the bounds are satisfying for
all values of $n$. We can verify that this is true if

$c\geq
C_{B,\mu}\coloneqq\max\left\\{\frac{3h^{-1}(B)-\mu}{h^{-1}(B)-\mu},\frac{B}{B-h\left(\frac{h^{-1}(B)-\mu}{2}\right)}\right\\}\;.$

######  Remark 19 (Adaptation for the un-centered condition)

The same proof scheme leads to a similar result for the un-centered condition,
simply with a different constant $C_{B,\mu}$. Indeed, we can fix the same
value for $X_{n+1}$, and write the same results using instead that for all
$i$, $-h^{-1}(nB)\leq X_{i}\leq h^{-1}(nB)$. The constant $C_{B,\mu}$ would
become

$C_{B,\mu}\coloneqq\max\left\\{\frac{3h^{-1}(B)+\mu}{h^{-1}(B)-\mu},\frac{B}{B-h\left(\frac{\mu+h^{-1}(B)}{2}\right)}\right\\}$

###  E.4 Proof of the equivalence of checking that (12) or (13) hold

In this section we prove that for any parameter $\gamma>0$, observations
$X_{1},\dots,X_{n+1}$ and weights $w_{1},\dots,w_{n+1}$ the following
condition

$\exists
x\geq\mu^{\star}:\;\sum_{i=1}^{n}w_{i}X_{i}+w_{n+1}x\geq\mu^{\star}\;,\;\text{and
}\sum_{i=1}^{n}w_{i}h(|X_{i}-\mu^{\star}|)+w_{n+1}(h(|x-\mu^{\star}|)-\gamma)\leq
B\;,$

is equivalent to

$h\left(\left(\frac{1}{w_{n+1}}\sum_{i=1}^{n}w_{i}(\mu^{\star}-X_{i})\right)^{+}\right)\leq
B+\gamma+\frac{1}{w_{n+1}}\sum_{i=1}^{n}w_{i}(B-h(|X_{i}-\mu^{\star}|))\;,$

where $(x)^{+}=\max\\{x,0\\}$.

Assume that $x$ is a solution to the first (mean) constraint. Then, using that
$\sum_{i=1}^{n+1}w_{i}=1$ and that $\\{w_{n+1}=0\\}$ has probability zero it
must hold that

$\displaystyle\sum_{i=1}^{n}w_{i}(X_{i}-\mu^{\star})+w_{n+1}(x-\mu^{\star})\geq
0\Rightarrow
x\geq\mu^{\star}+\frac{1}{w_{n+1}}\sum_{i=1}^{n}w_{i}(\mu^{\star}-X_{i})\;.$

Then, doing the same with the second constraint it must also hold that

$h(x-\mu^{\star})\leq
B+\gamma+\frac{1}{w_{n+1}}\sum_{i=1}^{n}w_{i}(B-h(|X_{i}-\mu^{\star}|))\;.$

We then consider the two cases, depending on the sign of
$\sum_{i=1}^{n}w_{i}(\mu^{\star}-X_{i})$. If it is positive, as $h$ is a
bijection on the positive line we directly get that

$h\left(\frac{1}{w_{n+1}}\sum_{i=1}^{n}w_{i}(\mu^{\star}-X_{i})\right)\leq
h(x-\mu^{\star})\leq
B+\gamma+\frac{1}{w_{n+1}}\sum_{i=1}^{n}w_{i}(B-h(|X_{i}-\mu^{\star}|))\;.$

In the case that $\sum_{i=1}^{n}w_{i}(\mu^{\star}-X_{i})\leq 0$, it is only
necessary to check if the condition holds for $x=0$, that is if

$0=h(0)\leq
B+\gamma+\frac{1}{w_{n+1}}\sum_{i=1}^{n}w_{i}(B-h(|X_{i}-\mu^{\star}|))\;.$

Hence, the formulation of (13) is just a way to summarize the two cases and
(12) $\Rightarrow$ (13). Furthermore, if (13) holds then (13) holds with
either $x=\mu^{\star}$ or
$x=\mu^{\star}+\frac{1}{w_{n+1}}\sum_{i=1}^{n}w_{i}(\mu^{\star}-X_{i})$, which
concludes the proof.

###  E.5 Proof of Theorem 6

We already proved in Lemma 1 that (A1)–(A5) hold for families defined by a
$h$-moment condition. Adapting these results for
$D_{\pi}(F,\mu)=\Lambda_{\gamma}^{\star}(F,\mu)\text{1}(F\in\mathcal{F})$
would be direct (we proved them for $\Lambda_{0}^{\star}(F,\mu)$ with this
notation), so we omit the proof for simplicity. The rest of the proof consists
in proving that Assumption B.1 holds.

#### Upper bound

Lemma 5 states that there exists a mapping $C$ such that for any $\eta>0$ and
any $n\in\mathbb{N}$

$\text{[BCP]}\leq C(F_{n},\eta)\times
e^{-n\Lambda_{\eta,\gamma}^{\star}(F_{n},\mu^{\star})}\;,$

where $C$ is continuous w.r.t. the Wasserstein metric in $F_{n}$, is
continuous in $\eta$, and scales in $\eta^{-2}$. For any cdf
$F\in\\{F_{2},\dots,F_{K}\\}$ (sufficient to prove the result) and
$\operatorname{\varepsilon}>0$, we define a neighborhood of $F$ as
$\mathcal{B}=\\{F_{n}\in\mathcal{F}:\;,W(F_{n},F)\leq\operatorname{\varepsilon}\\}$.
If $F_{n}$ denotes the empirical distribution of the observations
$X_{1},\dots,X_{n}$ drawn i.i.d. from $F$, using Theorem 2 of Fournier and
Guillin (2015) and that $\mathbb{E}_{F}[|X|^{2+\xi}]<+\infty$ for some $\xi>0$
(by definition of $\mathcal{F}$) we obtain that

$\sum_{n=1}^{+\infty}\mathbb{P}(W(F_{n},F)\geq\operatorname{\varepsilon})<+\infty\;.$

Using this result, by continuity we can first choose
$\operatorname{\varepsilon}$ small enough so that for instance
$C(F_{n},\eta)\leq 2C(F,\eta)$ for any $\eta$ small enough. Then, we choose
$\eta^{-1}=\mathcal{O}(n^{a})$ for some $a>0$, so that (1) $C(F_{n},\eta)$ is
a polynomial in $n$, and (2)
$e^{-n\Lambda_{\eta,\gamma}^{\star}(F_{n},\mu^{\star})}\leq
2e^{-n\Lambda_{\gamma}^{\star}(F_{n},\mu^{\star})}$. With these choices of
parameters, we satisfy all the conditions of Assumption B.1 regarding the
upper bound on $p_{k}^{\pi}(t)$.

#### Lower bound

The formulation of the lower bound in Lemma 5 already fits Assumption B.1
regarding the lower bound part. Indeed, the lower bound in Lemma 5 becomes
that of Assumption B.1 if we set $(n_{\eta},c_{n},p_{0})$ in Assumption B.1 to
$(n_{\delta,\gamma},c_{\delta}^{-1},e^{-n_{\delta,\gamma}\log(n_{\delta,\gamma}C_{B,\mu})})$.
This concludes the proof.

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as compared to JSON or YAML Schema.

    <?xml version="1.0"?>
    <xs:schema xmlns:xs="http://www.w3.org/2001/XMLSchema">
      <xs:element name="WindTurbine">
        <xs:annotation>
          <xs:documentation>A representation of a Wind Turbine.</xs:documentation>
        </xs:annotation>
        <xs:complexType>
          <xs:attribute name="model" type="xs:string">
            <xs:annotation>
              <xs:documentation>The model of the Wind Turbine.</xs:documentation>
            </xs:annotation>
          </xs:attribute>
          <xs:attribute name="ratedPower" type="xs:float">
            <xs:annotation>
              <xs:documentation>The rated power of the Wind Turbine in megawatts.</xs:documentation>
            </xs:annotation>
          </xs:attribute>
        </xs:complexType>
      </xs:element>
    </xs:schema>

##  Appendix B Serialisation formats

OWL and RDFS ontologies can be serialised for storage using a variety of
formats. These formats enable the representation of knowledge in a machine-
readable and standardized way. RDF, as the foundation of the Semantic Web, can
be serialized in various formats such as RDF/XML, Turtle (Terse RDF Triple
Language), N-Triples, and JSON-LD.

RDF/XML was the first standardized serialisation format for RDF, but its
verbosity and complexity led to the development of other formats such as
Turtle and N-Triples, which offer more human-readable syntax. JSON-LD (JSON
for Linked Data) has gained popularity as it combines the simplicity and
widespread use of JSON with the ability to express RDF data. OWL, being an
extension of RDF, can also be serialised using the aforementioned RDF
serialisation formats. However, OWL has its own serialisation formats as well,
such as OWL/XML and Functional-Style Syntax (also known as OWL Functional
Syntax). OWL/XML is an XML-based syntax specifically designed for expressing
OWL ontologies, while Functional-Style Syntax is a human-readable, text-based
format that closely follows the structure of the OWL 2 specification. The
choice of serialization format depends on factors such as readability,
compatibility with existing tools, and ease of parsing and processing.

#### JavaScript Object Notation for Linked Data (JSON-LD)

JSON-LD is a lightweight data interchange format that extends JSON to provide
a means for encoding Linked Data using standard JSON conventions. JSON-LD is
designed to be easy to read and write by humans, as well as simple to parse
and generate by machines. It is developed by the World Wide Web Consortium
(W3C) and provides a way to represent RDF data model in JSON. JSON-LD is
particularly useful for web developers who want to incorporate structured data
into web applications and APIs while leveraging the existing JSON tools and
libraries. JSON-LD introduces a notion of a context, which allows defining
short aliases for long IRIs (Internationalized Resource Identifiers) used in
RDF, simplifying the representation of RDF triples in JSON. It also supports
the definition of data types, language tags for string values, and nested JSON
objects to represent complex relationships and structures. JSON-LD
serialisation example:

    {
      "@context": {
        "schema": "https://schema.org/",
        "geo": "http://www.w3.org/2003/01/geo/wgs84_pos#"
      },
      "@type": "schema:EducationalOrganization",
      "@id": "https://www.ost.ch",
      "schema:name": "OST - Ostschweizer Fachhochschule",
      "schema:owns": {
        "@type": "schema:Product",
        "@id": "urn:ost:aventa:av-7",
        "schema:name": "Aventa AV-7 Wind Turbine",
        "schema:manufacturer": {
          "@type": "schema:Organization",
          "@id": "https://en.wind-turbine-models.com/manufacturers/316-aventa",
          "schema:name": "Aventa"
        },
        "schema:model": {
          "@type": "schema:ProductModel",
          "@id": "https://en.wind-turbine-models.com/turbines/1529-aventa-av-7",
          "name":"AV-7"
        },
        "geo:location": {
          "@type": "geo:Point",
          "geo:lat": "47.52000",
          "geo:long": "8.68236"
        }
      }
    }

Figure 6: Aventa AV-7 described using JSON-LD

##  Appendix C FAIR Principles and Semantic Web Technologies

FAIR Principle |  Semantic Web Technology Enablement
---|---
Findable |
F1 |  URIs, especially HTTP URIs are globally unique and can be persistent. This requirement is similar to the first LD principle, even though persistence is not specifically addressed by LD.
F2 |  Rich metadata models are developed and adopted by communities, for their specific domain. These models can be published on the web as LD and referred to, when describing the data.
F3 |  RDF triples connect unique identifiers (URIs) with well-defined properties to their respective values.
F4 |  RDF data can be stored in Triplestores which are optimized for semantic queries using SPARQL. This requirement along with F2 and F3 are similar to the third LD principle.
Accessible |
A1.1 |  Semantic web technologies are built on open standards such as HTTP and RDF. This requirement is simmilar to the second linked data principle.
A1.2 |  Web protocols like HTTPS include provisions for authentication and authorization.
A2 |  Persistent URIs and versioning methodologies ensure that metadata remain accessible. Metadata models can be published on third party websites like schema.org, more expressive semantic artefacts can be published on ontology hosting websites like the ones powered by OntoPortal. This requirement has no equivalent in LD principles
Interoperable |
I1 |  RDF, RDFS, and OWL are formal languages that are broadly applicable for knowledge representation.
I2 |  Semantic web enables the use of vocabularies and taxonomies through standard data models such as SKOS.
I3 |  RDF’s linking capability enables metadata to include qualified references to other metadata. This requirement is similar to the fourth LD principle.
Reusable |
R1.1 |  Metadata in RDF can include licensing information using appropriate vocabularies. In general, LD has evolved into the direction of open data, meanwhile FAIR principles can applied to data subject to any explicitly defined license.
R1.2 |  Provenance information can be recorded using cross-domain, community accepted vocabularies such as PROV-O.
R1.3 |  The semantic web supports the use of domain-specific ontologies, which can be developed according to community standards. This requirement is similar to the third LD principle.
Table 7: FAIR Principles and Semantic Web Technology Enablement

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