$\displaystyle\leq-\mu\sum_{i=1}^{N}\langle\nabla_{\pi_{i}}G(\pi_{i}^{t},\sigma_{i}^{\prime})-\nabla_{\pi_{i}}G(\pi_{i}^{\mu,\sigma^{\prime}},\sigma_{i}^{\prime}),\pi_{i}^{t}-\pi_{i}^{\mu,\sigma^{\prime}}\rangle$
$\displaystyle\leq-\mu\left(\min_{i\in[N],~{}j\in[d_{i}]}\sigma_{ij}^{\prime}\right)\|\pi^{t}-\pi^{\mu,\sigma^{\prime}}\|^{2}.$
(39)

On the other hand, by compactness of $\mathcal{X}_{i}$, the second term of
(F.11) is written as:

$\displaystyle\mu\sum_{i=1}^{N}\langle\nabla_{\pi_{i}}G(\pi_{i}^{t},\sigma_{i}^{\prime})-\nabla_{\pi_{i}}G(\pi_{i}^{t},\sigma_{i}),\pi_{i}^{t}-\pi_{i}^{\mu,\sigma^{\prime}}\rangle$
$\displaystyle\leq\mu\cdot\mathrm{diam}(\mathcal{X})\sqrt{\sum_{i=1}^{N}\|\nabla_{\pi_{i}}G(\pi_{i}^{t},\sigma_{i}^{\prime})-\nabla_{\pi_{i}}G(\pi_{i}^{t},\sigma_{i})\|^{2}}$
(40)

By combining (F.11), (F.11), and (40), we get:

$\displaystyle\frac{d}{dt}\mathrm{KL}(\pi^{\mu,\sigma^{\prime}},\pi^{t})$
$\displaystyle\leq-\mu\left(\min_{i\in[N],~{}j\in[d_{i}]}\sigma_{ij}^{\prime}\right)\|\pi^{t}-\pi^{\mu,\sigma^{\prime}}\|^{2}+\mu\cdot\mathrm{diam}(\mathcal{X})\sqrt{\sum_{i=1}^{N}\|\nabla_{\pi_{i}}G(\pi_{i}^{t},\sigma_{i}^{\prime})-\nabla_{\pi_{i}}G(\pi_{i}^{t},\sigma_{i})\|^{2}}.$

Recall that $\pi^{\mu,\sigma}$ is the stationary point of (37). Therefore, by
setting the start point as $\pi^{0}=\pi^{\mu,\sigma}$, we have for all $t\geq
0,\pi^{t}=\pi^{\mu,\sigma}$. In this case, for all $t\geq 0$,
$\frac{d}{dt}\mathrm{KL}(\pi^{\mu,\sigma^{\prime}},\pi^{t})=0$ and then:

$\displaystyle\left(\min_{i\in[N],~{}j\in[d_{i}]}\sigma_{ij}^{\prime}\right)\|\pi^{\mu,\sigma^{\prime}}-\pi^{\mu,\sigma}\|^{2}\leq\mathrm{diam}(\mathcal{X})\sqrt{\sum_{i=1}^{N}\|\nabla_{\pi_{i}}G(\pi_{i}^{t},\sigma_{i}^{\prime})-\nabla_{\pi_{i}}G(\pi_{i}^{t},\sigma_{i})\|^{2}}.$

For a given $\varepsilon>0$, let us define
$\varepsilon^{\prime}=\frac{\left(\min_{i\in[N],~{}j\in[d_{i}]}\sigma_{ij}^{\prime}\right)}{\mathrm{diam}(\mathcal{X})}\varepsilon^{2}$.
Since $\nabla_{\pi_{i}}G(\pi_{i}^{\mu,\sigma},\cdot)$ is continuous on
$\mathcal{X}_{i}$, for $\varepsilon^{\prime}$, there exists $\delta>0$ such
that
$\|\sigma^{\prime}-\sigma\|<\delta\Rightarrow\sqrt{\sum_{i=1}^{N}\|\nabla_{\pi_{i}}G(\pi_{i}^{\mu,\sigma},\sigma_{i}^{\prime})-\nabla_{\pi_{i}}G(\pi_{i}^{\mu,\sigma},\sigma_{i})\|^{2}}<\varepsilon^{\prime}=\frac{\left(\min_{i\in[N],~{}j\in[d_{i}]}\sigma_{ij}^{\prime}\right)}{\mathrm{diam}(\mathcal{X})}\varepsilon^{2}$.
Thus, for every $\varepsilon>0$, there exists $\delta>0$ such that

$\displaystyle\|\sigma^{\prime}-\sigma\|<\delta$
$\displaystyle\Rightarrow\|\pi^{\mu,\sigma^{\prime}}-\pi^{\mu,\sigma}\|\leq\sqrt{\frac{\mathrm{diam}(\mathcal{X})}{\left(\min_{i\in[N],~{}j\in[d_{i}]}\sigma_{ij}^{\prime}\right)}}\left(\sum_{i=1}^{N}\|\nabla_{\pi_{i}}G(\pi_{i}^{\mu,\sigma},\sigma_{i}^{\prime})-\nabla_{\pi_{i}}G(\pi_{i}^{\mu,\sigma},\sigma_{i})\|^{2}\right)^{\frac{1}{4}}<\varepsilon.$

This implies that $F(\cdot)$ is a continuous function on $\mathcal{X}$ when
$G$ is $\alpha$-divergence. A similar argument can be applied to Rényi-
divergence and reverse KL divergence. ∎

###  F.12 Proof of Lemma F.1

###### Proof of Lemma F.1.

Since $\frac{1}{\kappa s+2\theta}$ is a decreasing function for $s\geq 0$, for
all $s\geq 1$,

$\frac{1}{\kappa s+2\theta}\leq\int_{s-1}^{s}\frac{1}{\kappa x+2\theta}dx.$

Using this inequality, we can upper bound the sum as follows.

$\displaystyle\sum_{s=0}^{t}\frac{1}{\kappa s+2\theta}$
$\displaystyle=\frac{1}{2\theta}+\sum_{s=1}^{t}\frac{1}{\kappa s+2\theta}$
$\displaystyle\leq\frac{1}{2\theta}+\sum_{s=1}^{t}\int_{s-1}^{s}\frac{1}{\kappa
x+2\theta}dx$ $\displaystyle=\frac{1}{2\theta}+\int_{0}^{t}\frac{1}{\kappa
x+2\theta}dx$
$\displaystyle=\frac{1}{2\theta}+\frac{1}{\kappa}\int_{0}^{t}\frac{1}{x+\frac{2\theta}{\kappa}}dx$
$\displaystyle=\frac{1}{2\theta}+\frac{1}{\kappa}\int_{\frac{2\theta}{\kappa}}^{t+\frac{2\theta}{\kappa}}\frac{1}{u}du\qquad\left(u=x+\frac{2\theta}{\kappa}\right)$
$\displaystyle=\frac{1}{2\theta}+\frac{1}{\kappa}\log\left(\frac{\kappa}{2\theta}t+1\right).$

This concludes the proof. ∎

###  F.13 Proof of Lemma F.2

###### Proof of Lemma F.2.

From the first-order optimality condition for $\pi^{\mu,\sigma^{k}}$ and
$\pi^{\mu,\sigma^{k-1}}$, we have for any $k\geq 1$:

$\displaystyle\langle\nabla_{\pi_{i}}v_{i}(\pi_{i}^{\mu,\sigma^{k}},\pi_{-i}^{\mu,\sigma^{k}})-\mu\left(\pi_{i}^{\mu,\sigma^{k}}-\sigma_{i}^{k}\right),\sigma_{i}^{k}-\pi_{i}^{\mu,\sigma^{k}}\rangle\leq
0,$
$\displaystyle\langle\nabla_{\pi_{i}}v_{i}(\pi_{i}^{\mu,\sigma^{k-1}},\pi_{-i}^{\mu,\sigma^{k-1}})-\mu\left(\pi_{i}^{\mu,\sigma^{k-1}}-\sigma_{i}^{k-1}\right),\pi_{i}^{\mu,\sigma^{k}}-\pi_{i}^{\mu,\sigma^{k-1}}\rangle\leq
0.$

Summing up these inequalities yields:

$\displaystyle 0$
$\displaystyle\geq\sum_{i=1}^{N}\langle\nabla_{\pi_{i}}v_{i}(\pi_{i}^{\mu,\sigma^{k}},\pi_{-i}^{\mu,\sigma^{k}})-\mu\left(\pi_{i}^{\mu,\sigma^{k}}-\sigma_{i}^{k}\right),\sigma_{i}^{k}-\pi_{i}^{\mu,\sigma^{k}}\rangle$
$\displaystyle+\sum_{i=1}^{N}\langle\nabla_{\pi_{i}}v_{i}(\pi_{i}^{\mu,\sigma^{k-1}},\pi_{-i}^{\mu,\sigma^{k-1}})-\mu\left(\pi_{i}^{\mu,\sigma^{k-1}}-\sigma_{i}^{k-1}\right),\pi_{i}^{\mu,\sigma^{k}}-\pi_{i}^{\mu,\sigma^{k-1}}\rangle$
$\displaystyle=\sum_{i=1}^{N}\langle\nabla_{\pi_{i}}v_{i}(\pi_{i}^{\mu,\sigma^{k}},\pi_{-i}^{\mu,\sigma^{k}}),\pi_{i}^{\mu,\sigma^{k-1}}-\pi_{i}^{\mu,\sigma^{k}}\rangle+\sum_{i=1}^{N}\langle\nabla_{\pi_{i}}v_{i}(\pi_{i}^{\mu,\sigma^{k}},\pi_{-i}^{\mu,\sigma^{k}}),\sigma_{i}^{k}-\pi_{i}^{\mu,\sigma^{k-1}}\rangle+\mu\|\pi^{\mu,\sigma^{k}}-\sigma^{k}\|^{2}$
$\displaystyle+\sum_{i=1}^{N}\langle\nabla_{\pi_{i}}v_{i}(\pi_{i}^{\mu,\sigma^{k-1}},\pi_{-i}^{\mu,\sigma^{k-1}}),\pi_{i}^{\mu,\sigma^{k}}-\pi_{i}^{\mu,\sigma^{k-1}}\rangle+\mu\sum_{i=1}^{N}\langle\sigma_{i}^{k-1}-\pi_{i}^{\mu,\sigma^{k-1}},\pi_{i}^{\mu,\sigma^{k}}-\pi_{i}^{\mu,\sigma^{k-1}}\rangle$
$\displaystyle=\sum_{i=1}^{N}\langle\nabla_{\pi_{i}}v_{i}(\pi_{i}^{\mu,\sigma^{k}},\pi_{-i}^{\mu,\sigma^{k}})-\nabla_{\pi_{i}}v_{i}(\pi_{i}^{\mu,\sigma^{k-1}},\pi_{-i}^{\mu,\sigma^{k-1}}),\pi_{i}^{\mu,\sigma^{k-1}}-\pi_{i}^{\mu,\sigma^{k}}\rangle$
$\displaystyle+\sum_{i=1}^{N}\langle\nabla_{\pi_{i}}v_{i}(\pi_{i}^{\mu,\sigma^{k}},\pi_{-i}^{\mu,\sigma^{k}}),\sigma_{i}^{k}-\pi_{i}^{\mu,\sigma^{k-1}}\rangle+\mu\|\pi^{\mu,\sigma^{k}}-\sigma^{k}\|^{2}+\mu\sum_{i=1}^{N}\langle\sigma_{i}^{k-1}-\pi_{i}^{\mu,\sigma^{k-1}},\pi_{i}^{\mu,\sigma^{k}}-\pi_{i}^{\mu,\sigma^{k-1}}\rangle$
$\displaystyle\geq\sum_{i=1}^{N}\langle\nabla_{\pi_{i}}v_{i}(\pi_{i}^{\mu,\sigma^{k}},\pi_{-i}^{\mu,\sigma^{k}}),\sigma_{i}^{k}-\pi_{i}^{\mu,\sigma^{k-1}}\rangle+\mu\|\pi^{\mu,\sigma^{k}}-\sigma^{k}\|^{2}+\mu\sum_{i=1}^{N}\langle\sigma_{i}^{k-1}-\pi_{i}^{\mu,\sigma^{k-1}},\pi_{i}^{\mu,\sigma^{k}}-\pi_{i}^{\mu,\sigma^{k-1}}\rangle,$

where the last inequality follows from (1). Then, since

$\displaystyle\langle\sigma_{i}^{k-1}-\pi_{i}^{\mu,\sigma^{k-1}},\pi_{i}^{\mu,\sigma^{k-1}}-\pi_{i}^{\mu,\sigma^{k}}\rangle=\frac{\|\pi_{i}^{\mu,\sigma^{k}}-\sigma_{i}^{k-1}\|^{2}}{2}-\frac{\|\pi_{i}^{\mu,\sigma^{k}}-\pi_{i}^{\mu,\sigma^{k-1}}\|^{2}}{2}-\frac{\|\pi_{i}^{\mu,\sigma^{k-1}}-\sigma_{i}^{k-1}\|^{2}}{2},$

we have:

$\displaystyle 0$
$\displaystyle\geq\sum_{i=1}^{N}\langle\nabla_{\pi_{i}}v_{i}(\pi_{i}^{\mu,\sigma^{k}},\pi_{-i}^{\mu,\sigma^{k}}),\sigma_{i}^{k}-\pi_{i}^{\mu,\sigma^{k-1}}\rangle+\mu\|\pi^{\mu,\sigma^{k}}-\sigma^{k}\|^{2}$
$\displaystyle+\frac{\mu}{2}\sum_{i=1}^{N}\langle\sigma_{i}^{k-1}-\pi_{i}^{\mu,\sigma^{k-1}},\pi_{i}^{\mu,\sigma^{k}}-\pi_{i}^{\mu,\sigma^{k-1}}\rangle+\frac{\mu}{2}\sum_{i=1}^{N}\langle\sigma_{i}^{k-1}-\pi_{i}^{\mu,\sigma^{k-1}},\pi_{i}^{\mu,\sigma^{k}}-\pi_{i}^{\mu,\sigma^{k-1}}\rangle$
$\displaystyle\geq\sum_{i=1}^{N}\langle\nabla_{\pi_{i}}v_{i}(\pi_{i}^{\mu,\sigma^{k}},\pi_{-i}^{\mu,\sigma^{k}}),\sigma_{i}^{k}-\pi_{i}^{\mu,\sigma^{k-1}}\rangle+\mu\|\pi^{\mu,\sigma^{k}}-\sigma^{k}\|^{2}$
$\displaystyle-\frac{\mu}{4}\left(\|\sigma^{k-1}-\pi^{\mu,\sigma^{k-1}}\|^{2}+\|\pi^{\mu,\sigma^{k}}-\pi^{\mu,\sigma^{k-1}}\|^{2}-\|\pi^{\mu,\sigma^{k}}-\pi^{\mu,\sigma^{k-1}}+\sigma^{k-1}-\pi^{\mu,\sigma^{k-1}}\|^{2}\right)$
$\displaystyle-\frac{\mu}{4}\left(\|\pi^{\mu,\sigma^{k}}-\sigma^{k-1}\|^{2}-\|\pi^{\mu,\sigma^{k}}-\pi^{\mu,\sigma^{k-1}}\|^{2}-\|\pi^{\mu,\sigma^{k-1}}-\sigma^{k-1}\|^{2}\right)$
$\displaystyle\geq\sum_{i=1}^{N}\langle\nabla_{\pi_{i}}v_{i}(\pi_{i}^{\mu,\sigma^{k}},\pi_{-i}^{\mu,\sigma^{k}}),\sigma_{i}^{k}-\pi_{i}^{\mu,\sigma^{k-1}}\rangle+\mu\|\pi^{\mu,\sigma^{k}}-\sigma^{k}\|^{2}-\frac{\mu}{4}\|\pi^{\mu,\sigma^{k}}-\sigma^{k-1}\|^{2}$
$\displaystyle\geq\sum_{i=1}^{N}\langle\nabla_{\pi_{i}}v_{i}(\pi_{i}^{\mu,\sigma^{k}},\pi_{-i}^{\mu,\sigma^{k}}),\sigma_{i}^{k}-\pi_{i}^{\mu,\sigma^{k-1}}\rangle+\mu\|\pi^{\mu,\sigma^{k}}-\sigma^{k}\|^{2}-\frac{\mu}{2}\|\pi^{\mu,\sigma^{k}}-\sigma^{k}\|^{2}-\frac{\mu}{2}\|\sigma^{k}-\sigma^{k-1}\|^{2}$
$\displaystyle=\sum_{i=1}^{N}\langle\nabla_{\pi_{i}}v_{i}(\pi_{i}^{\mu,\sigma^{k}},\pi_{-i}^{\mu,\sigma^{k}}),\sigma_{i}^{k}-\pi_{i}^{\mu,\sigma^{k-1}}\rangle+\frac{\mu}{2}\|\pi^{\mu,\sigma^{k}}-\sigma^{k}\|^{2}-\frac{\mu}{2}\|\sigma^{k}-\sigma^{k-1}\|^{2},$

where the third inequality follows from $(a+b)^{2}\leq 2(a^{2}+b^{2})$ for
$a,b\in\mathbb{R}$. Thus,

$\displaystyle\|\pi^{\mu,\sigma^{k}}-\sigma^{k}\|^{2}$
$\displaystyle\leq\|\sigma^{k}-\sigma^{k-1}\|^{2}+\frac{2}{\mu}\sum_{i=1}^{N}\langle\nabla_{\pi_{i}}v_{i}(\pi_{i}^{\mu,\sigma^{k}},\pi_{-i}^{\mu,\sigma^{k}}),\pi_{i}^{\mu,\sigma^{k-1}}-\sigma_{i}^{k}\rangle$
$\displaystyle\leq\|\sigma^{k}-\sigma^{k-1}\|^{2}+\frac{2}{\mu}\|\pi^{\mu,\sigma^{k-1}}-\sigma^{k}\|\sqrt{\sum_{i=1}^{N}\|\nabla_{\pi_{i}}v_{i}(\pi_{i}^{\mu,\sigma^{k}},\pi_{-i}^{\mu,\sigma^{k}})\|^{2}}$
$\displaystyle\leq\|\sigma^{k}-\sigma^{k-1}\|^{2}+\frac{2\zeta}{\mu}\|\pi^{\mu,\sigma^{k-1}}-\sigma^{k}\|.$

∎

###  F.14 Proof of Lemma F.3

###### Proof of Lemma F.3.

From the first-order optimality condition for $p^{k+1}$, we have:

$\displaystyle\langle\nabla_{\pi_{i}}v_{i}(\pi_{i}^{\mu,\sigma^{k}},\pi_{-i}^{\mu,\sigma^{k}})-\mu\left(\pi_{i}^{\mu,\sigma^{k}}-\sigma_{i}^{k}\right),\pi_{i}^{\ast}-\pi_{i}^{\mu,\sigma^{k}}\rangle\leq
0.$

Thus, from the three-point identity $2\langle
a-b,c-a\rangle=\|b-c\|^{2}-\|a-b\|^{2}-\|a-c\|^{2}$ and Young’s inequality:

$\displaystyle\sum_{i=1}^{N}\langle\nabla_{\pi_{i}}v_{i}(\pi_{i}^{\mu,\sigma^{k}},\pi_{-i}^{\mu,\sigma^{k}}),\pi_{i}^{\ast}-\pi_{i}^{\mu,\sigma^{k}}\rangle\leq\mu\sum_{i=1}^{N}\langle\pi_{i}^{\mu,\sigma^{k}}-\sigma_{i}^{k},\pi_{i}^{\ast}-\pi_{i}^{\mu,\sigma^{k}}\rangle$
$\displaystyle=\frac{\mu}{2}\|\pi^{\ast}-\sigma^{k}\|^{2}-\frac{\mu}{2}\|\pi^{\mu,\sigma^{k}}-\sigma^{k}\|^{2}-\frac{\mu}{2}\|\pi^{\ast}-\pi^{\mu,\sigma^{k}}\|^{2}$
$\displaystyle=\frac{\mu}{2}\|\pi^{\ast}-\sigma^{k}\|^{2}-\frac{\mu}{2}\|\pi^{\mu,\sigma^{k}}-\sigma^{k}\|^{2}-\frac{\mu}{2}\|\pi^{\ast}-\sigma^{k+1}+\sigma^{k+1}-\pi^{\mu,\sigma^{k}}\|^{2}$
$\displaystyle=\frac{\mu}{2}\|\pi^{\ast}-\sigma^{k}\|^{2}-\frac{\mu}{2}\|\pi^{\mu,\sigma^{k}}-\sigma^{k}\|^{2}-\frac{\mu}{2}\|\pi^{\ast}-\sigma^{k+1}\|^{2}-\frac{\mu}{2}\|\sigma^{k+1}-\pi^{\mu,\sigma^{k}}\|^{2}-\mu\langle\pi^{\ast}-\sigma^{k+1},\sigma^{k+1}-\pi^{\mu,\sigma^{k}}\rangle$
$\displaystyle\leq\frac{\mu}{2}\|\pi^{\ast}-\sigma^{k}\|^{2}-\frac{\mu}{2}\|\pi^{\mu,\sigma^{k}}-\sigma^{k}\|^{2}-\frac{\mu}{2}\|\pi^{\ast}-\sigma^{k+1}\|^{2}+\mu\|\pi^{\ast}-\sigma^{k+1}\|\|\sigma^{k+1}-\pi^{\mu,\sigma^{k}}\|$
$\displaystyle\leq\frac{\mu}{2}\|\pi^{\ast}-\sigma^{k}\|^{2}-\frac{\mu}{2}\|\pi^{\mu,\sigma^{k}}-\sigma^{k}\|^{2}-\frac{\mu}{2}\|\pi^{\ast}-\sigma^{k+1}\|^{2}+\frac{\mu}{64K^{3}}\|\pi^{\ast}-\sigma^{k+1}\|^{2}+\frac{32\mu
K^{3}}{2}\|\sigma^{k+1}-\pi^{\mu,\sigma^{k}}\|^{2}.$

Here, since $T_{\sigma}=\max(\frac{3}{\ln c}\ln K+\frac{\ln 64}{\ln c},1)$, we
have $c^{2T_{\sigma}}\geq 64c^{\frac{3}{\ln c}\ln K}=64K^{3}$. Therefore, we
get:

$\displaystyle\sum_{i=1}^{N}\langle\nabla_{\pi_{i}}v_{i}(\pi_{i}^{\mu,\sigma^{k}},\pi_{-i}^{\mu,\sigma^{k}}),\pi_{i}^{\ast}-\pi_{i}^{\mu,\sigma^{k}}\rangle$
$\displaystyle\leq\frac{\mu}{2}\|\pi^{\ast}-\sigma^{k}\|^{2}-\frac{\mu}{2}\|\pi^{\mu,\sigma^{k}}-\sigma^{k}\|^{2}-\frac{\mu}{2}\|\pi^{\ast}-\sigma^{k+1}\|^{2}+\frac{\mu}{64K^{3}}\|\pi^{\ast}-\sigma^{k+1}\|^{2}+\frac{32\mu
K^{3}}{2}\|\sigma^{k}-\pi^{\mu,\sigma^{k}}\|^{2}\left(\frac{1}{c}\right)^{2T_{\sigma}}$
$\displaystyle\leq\frac{\mu}{2}\|\pi^{\ast}-\sigma^{k}\|^{2}-\frac{\mu}{2}\|\pi^{\mu,\sigma^{k}}-\sigma^{k}\|^{2}-\frac{\mu}{2}\|\pi^{\ast}-\sigma^{k+1}\|^{2}+\frac{\mu}{64K^{3}}\|\pi^{\ast}-\sigma^{k+1}\|^{2}+\frac{\mu}{4}\|\sigma^{k}-\pi^{\mu,\sigma^{k}}\|^{2}$
$\displaystyle=\frac{\mu}{2}\|\pi^{\ast}-\sigma^{k}\|^{2}-\frac{\mu}{2}\|\pi^{\ast}-\sigma^{k+1}\|^{2}-\frac{\mu}{4}\|\pi^{\mu,\sigma^{k}}-\sigma^{k}\|^{2}+\frac{\mu}{64K^{3}}\|\pi^{\ast}-\sigma^{k+1}\|^{2}.$

Summing up this inequality from $k=0$ to $K-1$ yields:

$\displaystyle\frac{\mu}{2}\|\pi^{\ast}-\sigma^{0}\|^{2}+\frac{\mu}{64K^{3}}\sum_{k=0}^{K-1}\|\pi^{\ast}-\sigma^{k+1}\|^{2}-\frac{\mu}{4}\sum_{k=0}^{K-1}\|\pi^{\mu,\sigma^{k}}-\sigma^{k}\|^{2}$
$\displaystyle\geq\sum_{k=0}^{K-1}\sum_{i=1}^{N}\langle\nabla_{\pi_{i}}v_{i}(\pi_{i}^{\mu,\sigma^{k}},\pi_{-i}^{\mu,\sigma^{k}}),\pi_{i}^{\ast}-\pi_{i}^{\mu,\sigma^{k}}\rangle$
$\displaystyle\geq\sum_{k=0}^{K-1}\sum_{i=1}^{N}\langle\nabla_{\pi_{i}}v_{i}(\pi_{i}^{\ast},\pi_{-i}^{\ast}),\pi_{i}^{\ast}-\pi_{i}^{\mu,\sigma^{k}}\rangle$
$\displaystyle\geq 0.$

Then, from Cauchy–Schwarz inequality, we have:

$\displaystyle\frac{\mu}{4}\sum_{k=0}^{K-1}\|\pi^{\mu,\sigma^{k}}-\sigma^{k}\|^{2}$
$\displaystyle\leq\frac{\mu}{2}\|\pi^{\ast}-\sigma^{0}\|^{2}+\frac{\mu}{64K^{3}}\sum_{k=0}^{K-1}\|\pi^{\ast}-\sigma^{k+1}\|^{2}$
$\displaystyle\leq\frac{\mu}{2}\|\pi^{\ast}-\sigma^{0}\|^{2}+\frac{\mu}{64K^{3}}\sum_{k=0}^{K-1}\left(\sum_{\tau=0}^{k}\|\sigma^{\tau+1}-\sigma^{\tau}\|+\|\pi^{\ast}-\sigma^{0}\|\right)^{2}$
$\displaystyle\leq\frac{\mu}{2}\|\pi^{\ast}-\sigma^{0}\|^{2}+\frac{\mu}{32K^{3}}\sum_{k=0}^{K-1}\left(\left(\sum_{\tau=0}^{k}\|\sigma^{\tau+1}-\sigma^{\tau}\|\right)^{2}+\|\pi^{\ast}-\sigma^{0}\|^{2}\right)$
$\displaystyle\leq\frac{\mu}{2}\|\pi^{\ast}-\sigma^{0}\|^{2}+\frac{\mu}{32K^{3}}\sum_{k=0}^{K-1}\left(K\sum_{\tau=0}^{k}\|\sigma^{\tau+1}-\sigma^{\tau}\|^{2}+\|\pi^{\ast}-\sigma^{0}\|^{2}\right)$
$\displaystyle\leq\frac{\mu}{2}\|\pi^{\ast}-\sigma^{0}\|^{2}+\frac{\mu}{32K^{3}}\left(K^{2}\sum_{k=0}^{K-1}\|\sigma^{k+1}-\sigma^{k}\|^{2}+K\|\pi^{\ast}-\sigma^{0}\|^{2}\right)$
$\displaystyle\leq\frac{\mu}{2}\|\pi^{\ast}-\sigma^{0}\|^{2}+\frac{\mu}{32K^{3}}\left(K^{2}\sum_{k=0}^{K-1}\left(\|\sigma^{k+1}-\pi^{\mu,\sigma^{k}}\|+\|\pi^{\mu,\sigma^{k}}-\sigma^{k}\|\right)^{2}+K\|\pi^{\ast}-\sigma^{0}\|^{2}\right).$

By applying
$\|\pi^{\mu,\sigma^{k}}-\sigma^{k+1}\|\leq\|\pi^{\mu,\sigma^{k}}-\sigma^{k}\|\left(\frac{1}{c}\right)^{T_{\sigma}}$
to the above inequality, we get:

$\displaystyle\frac{\mu}{4}\sum_{k=0}^{K-1}\|\pi^{\mu,\sigma^{k}}-\sigma^{k}\|^{2}$
$\displaystyle\leq\frac{\mu}{2}\|\pi^{\ast}-\sigma^{0}\|^{2}+\frac{\mu}{32K^{3}}\left(K^{2}\sum_{k=0}^{K-1}\left(\frac{\|\sigma^{k}-\pi^{\mu,\sigma^{k}}\|}{c^{T_{\sigma}}}+\|\pi^{\mu,\sigma^{k}}-\sigma^{k}\|\right)^{2}+K\|\pi^{\ast}-\sigma^{0}\|^{2}\right)$
$\displaystyle\leq\frac{\mu}{2}\|\pi^{\ast}-\sigma^{0}\|^{2}+\frac{\mu}{32K^{3}}\left(K^{2}\sum_{k=0}^{K-1}\left(2\|\pi^{\mu,\sigma^{k}}-\sigma^{k}\|\right)^{2}+K\|\pi^{\ast}-\sigma^{0}\|^{2}\right)$
$\displaystyle=\frac{\mu}{2}\|\pi^{\ast}-\sigma^{0}\|^{2}+\frac{\mu}{32K^{3}}\left(4K^{2}\sum_{k=0}^{K-1}\|\pi^{\mu,\sigma^{k}}-\sigma^{k}\|^{2}+K\|\pi^{\ast}-\sigma^{0}\|^{2}\right)$
$\displaystyle=\mu\left(\frac{1}{2}+\frac{1}{K^{2}}\right)\|\pi^{\ast}-\sigma^{0}\|^{2}+\frac{\mu}{8K}\sum_{k=0}^{K-1}\|\pi^{\mu,\sigma^{k}}-\sigma^{k}\|^{2}.$

Therefore, for $K\geq 1$, we get:

$\displaystyle\sum_{k=0}^{K-1}\|\pi^{\mu,\sigma^{k}}-\sigma^{k}\|^{2}$
$\displaystyle\leq 16\|\pi^{\ast}-\sigma^{0}\|^{2}.$

∎

###  F.15 Proof of Lemma F.4

###### Proof of Lemma F.4.

From the first-order optimality condition for $p^{k+1}$, we have:

$\displaystyle\langle\nabla_{\pi_{i}}v_{i}(\pi_{i}^{\mu,\sigma^{k}},\pi_{-i}^{\mu,\sigma^{k}})-\mu\left(\pi_{i}^{\mu,\sigma^{k}}-\sigma_{i}^{k}\right),\pi_{i}^{\ast}-\pi_{i}^{\mu,\sigma^{k}}\rangle\leq
0.$

Thus, from the three-point identity $2\langle
a-b,c-a\rangle=\|b-c\|^{2}-\|a-b\|^{2}-\|a-c\|^{2}$ and Young’s inequality:

$\displaystyle\sum_{i=1}^{N}\langle\nabla_{\pi_{i}}v_{i}(\pi_{i}^{\mu,\sigma^{k}},\pi_{-i}^{\mu,\sigma^{k}}),\pi_{i}^{\ast}-\pi_{i}^{\mu,\sigma^{k}}\rangle\leq\mu\sum_{i=1}^{N}\langle\pi_{i}^{\mu,\sigma^{k}}-\sigma_{i}^{k},\pi_{i}^{\ast}-\pi_{i}^{\mu,\sigma^{k}}\rangle$
$\displaystyle=\frac{\mu}{2}\|\pi^{\ast}-\sigma^{k}\|^{2}-\frac{\mu}{2}\|\pi^{\mu,\sigma^{k}}-\sigma^{k}\|^{2}-\frac{\mu}{2}\|\pi^{\ast}-\pi^{\mu,\sigma^{k}}\|^{2}$
$\displaystyle=\frac{\mu}{2}\|\pi^{\ast}-\sigma^{k}\|^{2}-\frac{\mu}{2}\|\pi^{\mu,\sigma^{k}}-\sigma^{k}\|^{2}-\frac{\mu}{2}\|\pi^{\ast}-\sigma^{k+1}+\sigma^{k+1}-\pi^{\mu,\sigma^{k}}\|^{2}$
$\displaystyle=\frac{\mu}{2}\|\pi^{\ast}-\sigma^{k}\|^{2}-\frac{\mu}{2}\|\pi^{\mu,\sigma^{k}}-\sigma^{k}\|^{2}-\frac{\mu}{2}\|\pi^{\ast}-\sigma^{k+1}\|^{2}-\frac{\mu}{2}\|\sigma^{k+1}-\pi^{\mu,\sigma^{k}}\|^{2}-\mu\langle\pi^{\ast}-\sigma^{k+1},\sigma^{k+1}-\pi^{\mu,\sigma^{k}}\rangle$
$\displaystyle\leq\frac{\mu}{2}\|\pi^{\ast}-\sigma^{k}\|^{2}-\frac{\mu}{2}\|\pi^{\mu,\sigma^{k}}-\sigma^{k}\|^{2}-\frac{\mu}{2}\|\pi^{\ast}-\sigma^{k+1}\|^{2}+\mu\|\pi^{\ast}-\sigma^{k+1}\|\|\sigma^{k+1}-\pi^{\mu,\sigma^{k}}\|$
$\displaystyle\leq\frac{\mu}{2}\|\pi^{\ast}-\sigma^{k}\|^{2}-\frac{\mu}{2}\|\pi^{\mu,\sigma^{k}}-\sigma^{k}\|^{2}-\frac{\mu}{2}\|\pi^{\ast}-\sigma^{k+1}\|^{2}+\mu\cdot\mathrm{diam}(\mathcal{X})\|\sigma^{k+1}-\pi^{\mu,\sigma^{k}}\|$

Here, since $T_{\sigma}\geq\max(K^{4},3)$, we have $\frac{\ln
T_{\sigma}}{T_{\sigma}}\leq\frac{4\ln K}{K^{4}}$. Therefore, we get:

$\displaystyle\mathbb{E}\left[\sum_{i=1}^{N}\langle\nabla_{\pi_{i}}v_{i}(\pi_{i}^{\mu,\sigma^{k}},\pi_{-i}^{\mu,\sigma^{k}}),\pi_{i}^{\ast}-\pi_{i}^{\mu,\sigma^{k}}\rangle\
|\ \sigma_{k}\right]$
$\displaystyle\leq\mathbb{E}\left[\frac{\mu}{2}\|\pi^{\ast}-\sigma^{k}\|^{2}-\frac{\mu}{2}\|\pi^{\mu,\sigma^{k}}-\sigma^{k}\|^{2}-\frac{\mu}{2}\|\pi^{\ast}-\sigma^{k+1}\|^{2}\
|\ \sigma^{k}\right]+\mu\cdot\mathrm{diam}(\mathcal{X})\cdot c\sqrt{\frac{\ln
T_{\sigma}}{T_{\sigma}}}$
$\displaystyle\leq\mathbb{E}\left[\frac{\mu}{2}\|\pi^{\ast}-\sigma^{k}\|^{2}-\frac{\mu}{2}\|\pi^{\mu,\sigma^{k}}-\sigma^{k}\|^{2}-\frac{\mu}{2}\|\pi^{\ast}-\sigma^{k+1}\|^{2}\
|\ \sigma^{k}\right]+\mu c\cdot\mathrm{diam}(\mathcal{X})\cdot\frac{2\sqrt{\ln
K}}{K^{2}}.$

Summing up this inequality from $k=0$ to $K-1$ and taking its expectation
yields:

$\displaystyle\frac{\mu}{2}\|\pi^{\ast}-\sigma^{0}\|^{2}-\mathbb{E}\left[\frac{\mu}{2}\sum_{k=0}^{K-1}\|\pi^{\mu,\sigma^{k}}-\sigma^{k}\|^{2}\right]+\mu
c\cdot\mathrm{diam}(\mathcal{X})\cdot\frac{2\sqrt{\ln K}}{K^{2}}$
$\displaystyle\geq\mathbb{E}\left[\sum_{k=0}^{K-1}\sum_{i=1}^{N}\langle\nabla_{\pi_{i}}v_{i}(\pi_{i}^{\mu,\sigma^{k}},\pi_{-i}^{\mu,\sigma^{k}}),\pi_{i}^{\ast}-\pi_{i}^{\mu,\sigma^{k}}\rangle\right]$
$\displaystyle\geq\mathbb{E}\left[\sum_{k=0}^{K-1}\sum_{i=1}^{N}\langle\nabla_{\pi_{i}}v_{i}(\pi_{i}^{\ast},\pi_{-i}^{\ast}),\pi_{i}^{\ast}-\pi_{i}^{\mu,\sigma^{k}}\rangle\right]$
$\displaystyle\geq 0.$

Therefore, for $K\geq 1$, we get:

$\displaystyle\mathbb{E}\left[\sum_{k=0}^{K-1}\|\pi^{\mu,\sigma^{k}}-\sigma^{k}\|^{2}\right]\leq\|\pi^{\ast}-\sigma^{0}\|^{2}+\mathrm{diam}(\mathcal{X})\cdot\frac{4c\sqrt{\ln
K}}{K^{2}}.$

∎

###  F.16 Proof of Lemma F.5

###### Proof of Lemma F.5.

By using the first-order optimality condition for $\sigma_{i}^{k+1}$, we have
for all $\pi\in\mathcal{X}$:

$\displaystyle\sum_{i=1}^{N}\langle\nabla_{\pi_{i}}v_{i}(\sigma_{i}^{k+1},\sigma_{-i}^{k+1})-\mu\nabla_{\pi_{i}}G(\sigma_{i}^{k+1},\sigma_{i}^{k}),\pi_{i}-\sigma_{i}^{k+1}\rangle\leq
0,$

and then

$\displaystyle\sum_{i=1}^{N}\langle\nabla_{\pi_{i}}v_{i}(\sigma_{i}^{k+1},\sigma_{-i}^{k+1}),\pi_{i}-\sigma_{i}^{k+1}\rangle\leq\mu\sum_{i=1}^{N}\langle\nabla_{\pi_{i}}G(\sigma_{i}^{k+1},\sigma_{i}^{k}),\pi_{i}-\sigma_{i}^{k+1}\rangle.$

When $G$ is $\alpha$-divergence, we have for all $\pi\in\mathcal{X}$:

$\displaystyle\sum_{i=1}^{N}\langle\nabla_{\pi_{i}}G(\sigma_{i}^{k+1},\sigma_{i}^{k}),\pi_{i}-\sigma_{i}^{k+1}\rangle$
$\displaystyle=\frac{1}{1-\alpha}\sum_{i=1}^{N}\sum_{j=1}^{d_{i}}(\sigma_{ij}^{k+1}-\pi_{ij})\left(\frac{\sigma_{ij}^{k}}{\sigma_{ij}^{k+1}}\right)^{1-\alpha}$
$\displaystyle=\frac{1}{1-\alpha}\sum_{i=1}^{N}\sum_{j=1}^{d_{i}}(\sigma_{ij}^{k+1}-\pi_{ij})=0,$

where we use the assumption that $\sigma^{k+1}=\sigma^{k}$ and
$\mathcal{X}_{i}=\Delta^{d_{i}}$. Similarly, when $G$ is Rényi-divergence, we
have for all $\pi\in\mathcal{X}$:

$\displaystyle\sum_{i=1}^{N}\langle\nabla_{\pi_{i}}G(\sigma_{i}^{k+1},\sigma_{i}^{k}),\pi_{i}-\sigma_{i}^{k+1}\rangle$
$\displaystyle=\frac{\alpha}{1-\alpha}\sum_{i=1}^{N}\frac{1}{\sum_{j=1}^{d_{i}}(\sigma_{ij}^{k+1})^{\alpha}(\sigma_{ij}^{k})^{1-\alpha}}\sum_{j=1}^{d_{i}}(\sigma_{ij}^{k+1}-\pi_{ij})\left(\frac{\sigma_{ij}^{k}}{\sigma_{ij}^{k+1}}\right)^{1-\alpha}$
$\displaystyle=\frac{\alpha}{1-\alpha}\sum_{i=1}^{N}\frac{1}{\sum_{j=1}^{d_{i}}(\sigma_{ij}^{k+1})^{\alpha}(\sigma_{ij}^{k})^{1-\alpha}}\sum_{j=1}^{d_{i}}(\sigma_{ij}^{k+1}-\pi_{ij})=0.$

Furthermore, if $G$ is reverse KL divergence, we have for all
$\pi\in\mathcal{X}$:

$\displaystyle\sum_{i=1}^{N}\langle\nabla_{\pi_{i}}G(\sigma_{i}^{k+1},\sigma_{i}^{k}),\pi_{i}-\sigma_{i}^{k+1}\rangle$
$\displaystyle=\sum_{i=1}^{N}\sum_{j=1}^{d_{i}}(\sigma_{ij}^{k+1}-\pi_{ij})\frac{\sigma_{ij}^{k}}{\sigma_{ij}^{k+1}}$
$\displaystyle=\sum_{i=1}^{N}\sum_{j=1}^{d_{i}}(\sigma_{ij}^{k+1}-\pi_{ij})=0,$

Thus, we have for all $\pi\in\mathcal{X}$:

$\displaystyle\sum_{i=1}^{N}\langle\nabla_{\pi_{i}}v_{i}(\sigma_{i}^{k+1},\sigma_{-i}^{k+1}),\pi_{i}-\sigma_{i}^{k+1}\rangle\leq
0.$

This is equivalent to the first-order optimality condition for
$\pi^{\ast}\in\Pi^{\ast}$. Therefore, $\sigma^{k+1}=\sigma^{k}$ is a Nash
equilibrium of the original game. ∎

###  F.17 Proof of Lemma F.6

###### Proof of Lemma F.6.

First, we prove the statement for $\alpha$-divergence:
$G(\sigma_{i}^{k+1},\sigma_{i}^{k})=\frac{1}{\alpha(1-\alpha)}\left(1-\sum_{j=1}^{d_{i}}(\sigma_{ij}^{k+1})^{\alpha}(\sigma_{ij}^{k})^{1-\alpha}\right)$.
From the definition of $\alpha$-divergence, we have for all
$\pi^{\ast}\in\Pi^{\ast}$:

$\displaystyle\sum_{i=1}^{N}\langle\nabla_{\pi_{i}}G(\sigma_{i}^{k+1},\sigma_{i}^{k}),\sigma_{i}^{k+1}-\pi_{i}^{\ast}\rangle$
$\displaystyle=\frac{1}{1-\alpha}\sum_{i=1}^{N}\sum_{j=1}^{d_{i}}(\pi_{ij}^{\ast}-\sigma_{ij}^{k+1})\left(\frac{\sigma_{ij}^{k}}{\sigma_{ij}^{k+1}}\right)^{1-\alpha}$
$\displaystyle=\frac{1}{1-\alpha}\sum_{i=1}^{N}\sum_{j=1}^{d_{i}}\pi_{ij}^{\ast}\left(\frac{\sigma_{ij}^{k}}{\sigma_{ij}^{k+1}}\right)^{1-\alpha}-\frac{1}{1-\alpha}\sum_{i=1}^{N}\sum_{j=1}^{d_{i}}(\sigma_{ij}^{k+1})^{\alpha}(\sigma_{ij}^{k})^{1-\alpha}.$

Here, when $\alpha\in(0,1)$, we get
$\sum_{j=1}^{d_{i}}(\sigma_{ij}^{k+1})^{\alpha}(\sigma_{ij}^{k})^{1-\alpha}\leq
1$. Thus,

$\displaystyle\sum_{i=1}^{N}\langle\nabla_{\pi_{i}}G(\sigma_{i}^{k+1},\sigma_{i}^{k}),\sigma_{i}^{k+1}-\pi_{i}^{\ast}\rangle$
$\displaystyle\geq\frac{1}{1-\alpha}\sum_{i=1}^{N}\sum_{j=1}^{d_{i}}\pi_{ij}^{\ast}\left(\frac{\sigma_{ij}^{k}}{\sigma_{ij}^{k+1}}\right)^{1-\alpha}-\frac{N}{1-\alpha}$
$\displaystyle=\frac{N}{1-\alpha}\exp\left(\ln\left(\frac{1}{N}\sum_{i=1}^{N}\sum_{j=1}^{d_{i}}\pi_{ij}^{\ast}\left(\frac{\sigma_{ij}^{k}}{\sigma_{ij}^{k+1}}\right)^{1-\alpha}\right)\right)-\frac{N}{1-\alpha}$
$\displaystyle\geq\frac{N}{1-\alpha}\exp\left(\frac{1-\alpha}{N}\sum_{i=1}^{N}\sum_{j=1}^{d_{i}}\pi_{ij}^{\ast}\ln\frac{\sigma_{ij}^{k}}{\sigma_{ij}^{k+1}}\right)-\frac{N}{1-\alpha}$
$\displaystyle=\frac{N}{1-\alpha}\exp\left(\frac{1-\alpha}{N}\left(\mathrm{KL}(\pi^{\ast},\sigma^{k+1})-\mathrm{KL}(\pi^{\ast},\sigma^{k})\right)\right)-\frac{N}{1-\alpha},$

where the second inequality follows from the concavity of the $\ln(\cdot)$
function and Jensen’s inequality for concave functions. Since $\ln(\cdot)$ is
strictly concave, the equality holds if and only if $\sigma^{k+1}=\sigma^{k}$.
Therefore, under the assumption that $\sigma^{k+1}\neq\sigma^{k}$, we get:

$\displaystyle\mathrm{KL}(\pi^{\ast},\sigma^{k+1})-\mathrm{KL}(\pi^{\ast},\sigma^{k})$
$\displaystyle<\frac{N}{1-\alpha}\ln\left(1+\frac{1-\alpha}{N}\sum_{i=1}^{N}\langle\nabla_{\pi_{i}}G(\sigma_{i}^{k+1},\sigma_{i}^{k}),\sigma_{i}^{k+1}-\pi_{i}^{\ast}\rangle\right)$
$\displaystyle\leq\sum_{i=1}^{N}\langle\nabla_{\pi_{i}}G(\sigma_{i}^{k+1},\sigma_{i}^{k}),\sigma_{i}^{k+1}-\pi_{i}^{\ast}\rangle,$
(41)

where the second inequality follows from $\ln(1+x)\leq x$ for $x>-1$. From the
first-order optimality condition for $\sigma_{i}^{k+1}$, we have for all
$\pi^{\ast}\in\Pi^{\ast}$:

$\displaystyle\sum_{i=1}^{N}\langle\nabla_{\pi_{i}}v_{i}(\sigma_{i}^{k+1},\sigma_{-i}^{k+1})-\mu\nabla_{\pi_{i}}G(\sigma_{i}^{k+1},\sigma_{i}^{k}),\pi_{i}^{\ast}-\sigma_{i}^{k+1}\rangle\leq
0.$

Then,

$\displaystyle\sum_{i=1}^{N}\langle\nabla_{\pi_{i}}G(\sigma_{i}^{k+1},\sigma_{i}^{k}),\sigma_{i}^{k+1}-\pi_{i}^{\ast}\rangle$
$\displaystyle\leq\frac{1}{\mu}\sum_{i=1}^{N}\langle\nabla_{\pi_{i}}v_{i}(\sigma_{i}^{k+1},\sigma_{-i}^{k+1}),\sigma_{i}^{k+1}-\pi_{i}^{\ast}\rangle$
$\displaystyle\leq\frac{1}{\mu}\sum_{i=1}^{N}\langle\nabla_{\pi_{i}}v_{i}(\pi_{i}^{\ast},\pi_{-i}^{\ast}),\sigma_{i}^{k+1}-\pi_{i}^{\ast}\rangle,$
(42)

where the second inequality follows from (1). Moreover, since $\pi^{\ast}$ is
the Nash equilibrium, from the first-order optimality condition, we get:

$\displaystyle\sum_{i=1}^{N}\langle\sigma_{i}^{k+1}-\pi_{i}^{\ast},\nabla_{\pi_{i}}v_{i}(\pi_{i}^{\ast},\pi_{-i}^{\ast})\rangle\leq
0.$ (43)

By combining (41), (F.17), and (43), if $\sigma^{k+1}\neq\sigma^{k}$, we have
any $\pi^{\ast}\in\Pi^{\ast}$:

$\displaystyle\mathrm{KL}(\pi^{\ast},\sigma^{k+1})-\mathrm{KL}(\pi^{\ast},\sigma^{k})<0.$

Next, we prove the statement for Rényi-divergence:
$G(\sigma_{i}^{k+1},\sigma_{i}^{k})=\frac{1}{\alpha-1}\ln\left(\sum_{j=1}^{d_{i}}(\sigma_{ij}^{k+1})^{\alpha}(\sigma_{ij}^{k})^{1-\alpha}\right)$.
We have for all $\pi^{\ast}\in\Pi^{\ast}$:

$\displaystyle\sum_{i=1}^{N}\langle\nabla_{\pi_{i}}G(\sigma_{i}^{k+1},\sigma_{i}^{k}),\sigma_{i}^{k+1}-\pi_{i}^{\ast}\rangle$
$\displaystyle=\frac{\alpha}{1-\alpha}\sum_{i=1}^{N}\frac{1}{\sum_{j=1}^{d_{i}}(\sigma_{ij}^{k+1})^{\alpha}(\sigma_{ij}^{k})^{1-\alpha}}\sum_{j=1}^{d_{i}}(\pi_{ij}^{\ast}-\sigma_{ij}^{k+1})\left(\frac{\sigma_{ij}^{k}}{\sigma_{ij}^{k+1}}\right)^{1-\alpha}$
$\displaystyle=\frac{\alpha}{1-\alpha}\sum_{i=1}^{N}\frac{1}{\sum_{j=1}^{d_{i}}(\sigma_{ij}^{k+1})^{\alpha}(\sigma_{ij}^{k})^{1-\alpha}}\sum_{j=1}^{d_{i}}\pi_{ij}^{\ast}\left(\frac{\sigma_{ij}^{k}}{\sigma_{ij}^{k+1}}\right)^{1-\alpha}-\frac{N\alpha}{1-\alpha}.$

Again, by using
$\sum_{j=1}^{d_{i}}(\sigma_{ij}^{k+1})^{\alpha}(\sigma_{ij}^{k})^{1-\alpha}\leq
1$ when $\alpha\in(0,1)$, we get:

$\displaystyle\sum_{i=1}^{N}\langle\nabla_{\pi_{i}}G(\sigma_{i}^{k+1},\sigma_{i}^{k}),\sigma_{i}^{k+1}-\pi_{i}^{\ast}\rangle$
$\displaystyle\geq\frac{\alpha}{1-\alpha}\sum_{i=1}^{N}\sum_{j=1}^{d_{i}}\pi_{ij}^{\ast}\left(\frac{\sigma_{ij}^{k}}{\sigma_{ij}^{k+1}}\right)^{1-\alpha}-\frac{N\alpha}{1-\alpha}$
$\displaystyle=\frac{N\alpha}{1-\alpha}\exp\left(\ln\left(\frac{1}{N}\sum_{i=1}^{N}\sum_{j=1}^{d_{i}}\pi_{ij}^{\ast}\left(\frac{\sigma_{ij}^{k}}{\sigma_{ij}^{k+1}}\right)^{1-\alpha}\right)\right)-\frac{N\alpha}{1-\alpha}$
$\displaystyle\geq\frac{N\alpha}{1-\alpha}\exp\left(\frac{1-\alpha}{N}\sum_{i=1}^{N}\sum_{j=1}^{d_{i}}\pi_{ij}^{\ast}\ln\frac{\sigma_{ij}^{k}}{\sigma_{ij}^{k+1}}\right)-\frac{N\alpha}{1-\alpha}$
$\displaystyle=\frac{N\alpha}{1-\alpha}\exp\left(\frac{1-\alpha}{N}\left(\mathrm{KL}(\pi^{\ast},\sigma^{k+1})-\mathrm{KL}(\pi^{\ast},\sigma^{k})\right)\right)-\frac{N\alpha}{1-\alpha},$

where the second inequality follows from Jensen’s inequality for $\ln(\cdot)$
function. Since $\ln(\cdot)$ is strictly concave, the equality holds if and
only if $\sigma^{k+1}=\sigma^{k}$. Therefore, under the assumption that
$\sigma^{k+1}\neq\sigma^{k}$, we get:

$\displaystyle\mathrm{KL}(\pi^{\ast},\sigma^{k+1})-\mathrm{KL}(\pi^{\ast},\sigma^{k})$
$\displaystyle<\frac{N\alpha}{1-\alpha}\ln\left(1+\frac{1-\alpha}{N\alpha}\sum_{i=1}^{N}\langle\nabla_{\pi_{i}}G(\sigma_{i}^{k+1},\sigma_{i}^{k}),\sigma_{i}^{k+1}-\pi_{i}^{\ast}\rangle\right)$
$\displaystyle\leq\sum_{i=1}^{N}\langle\nabla_{\pi_{i}}G(\sigma_{i}^{k+1},\sigma_{i}^{k}),\sigma_{i}^{k+1}-\pi_{i}^{\ast}\rangle,$
(44)

where the second inequality follows from $\ln(1+x)\leq x$ for $x>-1$. Thus, by
combining (F.17), (43), and (44), if $\sigma^{k+1}\neq\sigma^{k}$, we have any
$\pi^{\ast}\in\Pi^{\ast}$:

$\displaystyle\mathrm{KL}(\pi^{\ast},\sigma^{k+1})-\mathrm{KL}(\pi^{\ast},\sigma^{k})<0.$

Finally, we prove the statement for reverse KL divergence:
$G(\sigma_{i}^{k+1},\sigma_{i}^{k})=\sum_{j=1}^{d_{i}}\sigma_{ij}^{k}\ln\frac{\sigma_{ij}^{k}}{\sigma_{ij}^{k+1}}$.
We have for all $\pi^{\ast}\in\Pi^{\ast}$:

$\displaystyle\sum_{i=1}^{N}\langle\nabla_{\pi_{i}}G(\sigma_{i}^{k+1},\sigma_{i}^{k}),\sigma_{i}^{k+1}-\pi_{i}^{\ast}\rangle$
$\displaystyle=\sum_{i=1}^{N}\sum_{j=1}^{d_{i}}(\pi_{ij}^{\ast}-\sigma_{ij}^{k+1})\frac{\sigma_{ij}^{k}}{\sigma_{ij}^{k+1}}$
$\displaystyle=\sum_{i=1}^{N}\sum_{j=1}^{d_{i}}\pi_{ij}^{\ast}\frac{\sigma_{ij}^{k}}{\sigma_{ij}^{k+1}}-N$
$\displaystyle=N\exp\left(\ln\left(\frac{1}{N}\sum_{i=1}^{N}\sum_{j=1}^{d_{i}}\pi_{ij}^{\ast}\frac{\sigma_{ij}^{k}}{\sigma_{ij}^{k+1}}\right)\right)-N$
$\displaystyle\geq
N\exp\left(\frac{1}{N}\sum_{i=1}^{N}\sum_{j=1}^{d_{i}}\pi_{ij}^{\ast}\ln\frac{\sigma_{ij}^{k}}{\sigma_{ij}^{k+1}}\right)-N$
$\displaystyle=N\exp\left(\frac{1}{N}\left(\mathrm{KL}(\pi^{\ast},\sigma^{k+1})-\mathrm{KL}(\pi^{\ast},\sigma^{k})\right)\right)-N,$

where the inequality follows from Jensen’s inequality for $\ln(\cdot)$
function. Thus, under the assumption that $\sigma^{k+1}\neq\sigma^{k}$, we
get:

$\displaystyle\mathrm{KL}(\pi^{\ast},\sigma^{k+1})-\mathrm{KL}(\pi^{\ast},\sigma^{k})$
$\displaystyle<N\ln\left(1+\frac{1}{N}\sum_{i=1}^{N}\langle\nabla_{\pi_{i}}G(\sigma_{i}^{k+1},\sigma_{i}^{k}),\sigma_{i}^{k+1}-\pi_{i}^{\ast}\rangle\right)$
$\displaystyle\leq\sum_{i=1}^{N}\langle\nabla_{\pi_{i}}G(\sigma_{i}^{k+1},\sigma_{i}^{k}),\sigma_{i}^{k+1}-\pi_{i}^{\ast}\rangle,$
(45)

where the second inequality follows from $\ln(1+x)\leq x$ for $x>-1$. Thus, by
combining (F.17), (43), and (45), if $\sigma^{k+1}\neq\sigma^{k}$, we have any
$\pi^{\ast}\in\Pi^{\ast}$:

$\displaystyle\mathrm{KL}(\pi^{\ast},\sigma^{k+1})-\mathrm{KL}(\pi^{\ast},\sigma^{k})<0.$

∎

##  Appendix G Convergence Results with Other Divergence Functions

In this section, we establish the convergence results for our algorithm where
$G$ is not squared $\ell^{2}$-distance, and $T_{\sigma}$ is sufficiently
large. From Theorems 4.2 and 4.9, when $T_{\sigma}$ is large enough, updating
$\sigma^{k}$ becomes equivalent to setting $\sigma^{k+1}$ to
$\pi^{\mu,\sigma^{k}}$. First, we provide convergence results for the case
where $G$ is a Bregman divergence:

######  Theorem G.1.

Assume that $G$ is a Bregman divergence $D_{\psi^{\prime}}$ for some strongly
convex function $\psi^{\prime}$, and $\sigma^{k+1}=\pi^{\mu,\sigma^{k}}$ for
$k\geq 0$. Then, there exists $\pi^{\ast}\in\Pi^{\ast}$ such that
$\sigma^{K}\to\pi^{\ast}$ as $K\to\infty$.

Next, we consider divergence functions $G$ other than Bregman divergence for
games with probability simplex strategy spaces, i.e.,
$\mathcal{X}_{i}=\Delta^{d_{i}}$. Specifically, we provide the convergence
results when $G$ is one of the following divergences; 1) $\alpha$-divergence
$G(\pi_{i},\sigma_{i})=\frac{1}{\alpha(1-\alpha)}\left(1-\sum_{j=1}^{d_{i}}(\pi_{ij})^{\alpha}(\sigma_{ij})^{1-\alpha}\right)$;
2) Rényi-divergence
$G(\pi_{i},\sigma_{i})=\frac{1}{\alpha-1}\ln\left(\sum_{j=1}^{d_{i}}(\pi_{ij})^{\alpha}(\sigma_{ij})^{1-\alpha}\right)$;
3) reverse KL divergence.

######  Theorem G.2.

Let us define $\mathcal{X}_{i}=\Delta^{d_{i}}$. Assume that
$\sigma^{k+1}=\pi^{\mu,\sigma^{k}}$ for $k\geq 0$, and $G$ is one of the
following divergence: 1) $\alpha$-divergence with $\alpha\in(0,1)$; 2) Rényi-
divergence with $\alpha\in(0,1)$; 3) reverse KL divergence. If the initial
slingshot strategy $\sigma^{0}$ is in the interior of $\mathcal{X}$, the
sequence $\\{\sigma^{k}\\}_{k\geq 1}$ converges to the set of Nash equilibria
$\Pi^{\ast}$ of the original game.

##  Appendix H Additional experimental results and details

###  H.1 Payoff Matrix in Three-Player Biased RPS Game

Table 2: Three-Player Biased RPS game matrix. | R | P | S
---|---|---|---
R | $0$ | $-1/3$ | $1$
P | $1/3$ | $0$ | $-1/3$
S | $-1$ | $1/3$ | $0$

###  H.2 Experimental setting for Section 6

The experiments in Section 6 are conducted in Ubuntu 20.04.2 LTS with Intel(R)
Core(TM) i9-10850K CPU @ 3.60GHz and 64GB RAM.

In the full feedback setting, we use a constant learning rate $\eta=0.1$ for
MWU and OMWU, and FTRL-SP in all three games. For FTRL-SP, we set $\mu=0.1$
and $T_{\sigma}=100$ for KL and reverse KL divergence perturbation, and set
$\mu=0.1$ and $T_{\sigma}=20$ for squared $\ell^{2}$-distance perturbation. As
an exception, $\eta=0.01$, $\mu=1.0$, and $T_{\sigma}=200$ are used for FTRL-
SP with squared $\ell^{2}$-distance perturbation in the random payoff games
with $50$ actions.

For the noisy feedback setting, we use the lower learning rate $\eta=0.01$ for
all algorithms, except FTRL-SP with squared $\ell^{2}$-distance perturbation
for the random payoff games with $50$ actions. We update the slingshot
strategy $\sigma^{k}$ every $T_{\sigma}=1000$ iterations in FTRL-SP with KL
and reverse KL divergence perturbation, and update it every $T_{\sigma}=200$
iterations in FTRL-SP with squared $\ell^{2}$-distance perturbation. For FTRL-
SP with $\ell^{2}$-distance perturbation in the random payoff games with $50$
actions, we set $\eta=0.001$ and $T_{\sigma}=2000$.

###  H.3 Additional experiments

In this section, we compare the performance of FTRL-SP and MD-SP to MWU, OMWU,
and optimistic gradient descent (OGD) [Daskalakis et al., 2018, Wei et al.,
2021] in the full/noisy feedback setting. The parameter settings for MWU,
OMWU, and FTRL-SP are the same as Section 6. For MD-SP, we use the squared
$\ell^{2}$-distance and the parameter is the same as FTRL-SP with squared
$\ell^{2}$-distance perturbation. For OGD, we use the same learning rate as
FTRL-SP with squared $\ell^{2}$-distance perturbation.

Figure 3 shows the logarithmic exploitability of $\pi^{t}$ averaged over $100$
instances with full feedback. We observe that FTRL-SP and MD-SP with squared
$\ell^{2}$-distance perturbation exhibit competitive performance to OGD. The
experimental results in the noisy feedback setting are presented in Figure 4.
Surprisingly, in the noisy feedback setting, all FTRL-SP-based algorithms and
the MD-SP-based algorithm exhibit overwhelmingly superior performance to OGD
in all three games.

Figure 3:  Exploitability of $\pi^{t}$ for FTRL-SP, MD-SP, MWU, OMWU, and OGD
with full feedback. The shaded area represents the standard errors. Note that
the KL divergence, reverse KL divergence, and squared $\ell^{2}$-distance are
abbreviated to KL, RKL, and L2, respectively.  Figure 4:  Exploitability of
$\pi^{t}$ for FTRL-SP, MD-SP, MWU, OMWU, and OGD with noisy feedback. The
shaded area represents the standard errors.

###  H.4 Sensitivity analysis of update interval for the slingshot strategy

In this section, we investigate the performance when changing the update
interval of the slingshot strategy. We vary the $T_{\sigma}$ of FTRL-SP with
KL perturbation in 3BRPS with full feedback to be
$T_{\sigma}\in\\{10,100,1000,10000\\}$, and with noisy feedback to be
$T_{\sigma}\in\\{10,100,1000,10000\\}$. All other parameters are the same as
in Section 6. Figure 5 shows the logarithmic exploitability of $\pi^{t}$
averaged over $100$ instances in 3BRPS with full/noisy feedback. We observe
that the smaller the $T_{\sigma}$, the faster the exploitability converges.
However, if $T_{\sigma}$ is too small, exploitability does not converge (See
$T_{\sigma}=10$ with full feedback, and $T_{\sigma}=100$ with noisy feedback
in Figure 5).

(a) Full feedback

(b) Noisy feedback

Figure 5: Exploitability of $\pi^{t}$ for FTRL-SP with varying $T_{\sigma}$ in
3BRPS with full/noisy feedback. The shaded area represents the standard
errors.


# X-ray observations of the isolated black hole candidate OGLE-2011-BLG-0462,
and other collapsed objects discovered through gravitational microlensing

S. Mereghetti INAF, Istituto di Astrofisica Spaziale e Fisica Cosmica, via A.
Corti 12, I-20133 Milano, Italy  L. Sidoli INAF, Istituto di Astrofisica
Spaziale e Fisica Cosmica, via A. Corti 12, I-20133 Milano, Italy  G. Ponti
INAF, Osservatorio Astronomico di Brera, via E. Bianchi 46, I-23807 Merate
(LC), Italy  Max-Planck-Institut für extraterrestrische Physik,
Giessenbachstrasse, D-85748, Garching, Germany  A. Treves Università
dell’Insubria, via Valleggio 11, I-22100 Como, Italy  INAF, Osservatorio
Astronomico di Brera, via E. Bianchi 46, I-23807 Merate (LC), Italy

(Received April 21, 2022 ; Revised June 13, 2022; Accepted June 14, 2022 )

###### Abstract

Isolated black holes and neutron stars can be revealed through the observation
of long duration gravitational microlensing events. A few candidates have been
found in surveys of stars in the direction of the Galactic bulge. Recently,
thanks to the addition of astrometric information at milliarcsecond level, it
has been possible to reduce the uncertainties in the masses and distances for
some of these “dark” gravitational lenses and select the most promising
candidates. These isolated compact objects might emit X-rays powered by
accretion from the interstellar medium. Using data of the _Chandra_ , _XMM-
Newton_ , and INTEGRAL satellites, we searched for X-ray emission in the
isolated black hole candidate OGLE-2011-BLG-0462, and in several other
putative collapsed objects found with gravitational microlensing.
OGLE-2011-BLG-0462 has been recently interpreted as a 7.1 ${\rm M}_{\odot}$
black hole at a distance of 1.6 kpc, although a different group obtained a
mass range (1.6-4.4 ${\rm M}_{\odot}$) that cannot exclude a massive neutron
star. We have derived upper limits on the flux from OGLE-2011-BLG-0462 of
9$\times 10^{-15}$ erg cm-2 s-1 in the 0.5-7 keV range and $\sim 2\times
10^{-12}$ erg cm-2 s-1 in the 17-60 keV range. The implied X-ray luminosity is
consistent with the small radiative efficiency expected for a black hole and
disfavour a neutron star interpretation. Limits down to a factor about five
lower are obtained for the soft X-ray flux of other candidates, but their
interpretation is affected by larger uncertainties in the masses, distances
and spatial velocities.

X-ray sources, Stellar mass black holes, Gravitational lensing, Bondi
accretion

##  1 Introduction

Our galaxy is thought to contain of the order of 107-109 black holes (BH), but
only a few tens have been discovered so far in X-ray binaries (e.g., Corral-
Santana et al. (2016)). In fact, X-ray emission powered by accretion of matter
from the binary companion star allows us to select candidates, which can be
eventually confirmed as BH by subsequent dynamical mass measurements. The
presence of BH has been claimed also in a few non-interacting binaries, based
on radial velocity measurements of their companions (Liu et al., 2019;
Thompson et al., 2019; Gomez & Grindlay, 2021), but their true BH nature is
debated (see, e.g., van den Heuvel & Tauris (2020); El-Badry & Quataert
(2021), and references therein). A significant number of the BH in the Galaxy
should be isolated, either because they originated from single stars or
because the binary in which they formed was disrupted by the natal supernova
kick. In principle, such isolated BH can be detected if they accrete matter
from the interstellar medium (e.g., Shvartsman (1971); Ipser & Price (1982);
Campana & Pardi (1993); Agol & Kamionkowski (2002)), as it was also proposed
for isolated neutron stars (Ostriker et al., 1970). However, several searches
for isolated neutron stars (NS) and BH fed by accretion from the interstellar
medium (ISM) gave negative results (Stocke et al., 1995; Schwope et al., 1999;
Chisholm et al., 2003; Muno et al., 2006), despite the initially optimistic
estimates on the number of detectable objects (Treves & Colpi, 1991; Blaes &
Madau, 1993).

Isolated BH can also be detected through gravitational microlensing, when they
pass in front of background stars (e.g., Paczynski, 1986). Several long-
duration microlensing events found in surveys targeted to stars in the
Galactic bulge should be due to BH, because the relative rarity of these
objects is compensated by their large cross section. For most microlensing
events only the crossing time of the Einstein radius can be measured. This
timescale depends not only on the lens mass, but also on the relative
velocities and distances of the lens and the lensed star. Therefore, on the
basis of the sole crossing time, it is not possible to determine the
individual values of these parameters. More information on the mass of the
lens can be obtained in sufficiently long events, in which the effects caused
by the non-linear motion of the Earth can be detected in the light curve of
the magnified star (Gould, 1992). The first BH candidates selected with this
method were reported by Bennett et al. (2002) and Mao et al. (2002). Note
that, despite the “parallax microlensing” name of this phenomenon, it involves
only photometric observations to measure the time-dependent deviations of the
source magnification from the light curve expected with a simple model based
on linear motions of observer, lens and lensed star. Due to the intrinsic
degeneracy between the parameters that characterize parallax microlensing (in
particular the one between lens mass and relative velocities) the masses
derived in this way are subject to large uncertainties. This problem has
usually been tackled by statistical analysis based on our best knowledge of
the distributions of masses, positions and velocities of the lens and of the
lensed star (Agol et al., 2002). The nature of the lens (i.e. BH, NS, white
dwarf, normal star) can thus be assessed in probabilistic terms.

A step forward to reduce such a degeneracy can be done if also the deviations
in the apparent position of the magnified star caused by the gravitational
lens are detected. This requires very accurate astrometric measurements at the
milliarcsecond level. Recently, using this “astrometric microlensing” method,
Sahu et al. (2022) derived a mass of 7.1$\pm$1.3 ${\rm M}_{\odot}$ and
distance of 1.58$\pm$0.18 kpc for the lens object in OGLE-2011-BLG-0462. Based
on the lack of detectable light from the lens, these authors concluded that
this is the first unambiguous detection of an isolated stellar-mass BH.
Somewhat different results have been reported for the same event by Lam et al.
(2022), who inferred a lens mass in the range 1.6 - 4.4 ${\rm M}_{\odot}$
which leaves open the possibility that OGLE-2011-BLG-0462 be an isolated NS.

X-ray observations of the compact objects found with gravitational
microlensing can provide useful constraints on the physics of accretion. This
is possibly the only way to derive information on accretion of ISM onto
isolated compact objects of stellar mass. A systematic comparative study
between microlensing events and ROSAT sources was presented by Sartore &
Treves (2012), finding no meaningful matching. Up to now, deep searches for
X-ray emission have been reported only for MACHO-96-BLG-5 (Maeda et al., 2005;
Nucita et al., 2006). The negative results imply accretion on the putative BH
at a rate lower than $\sim 10^{-9}$ of the Eddington value.

Here we report on new and archival _Chandra_ observations of the isolated BH
candidate OGLE-2011-BLG-0462 111 Our results supersede the limits quoted in
Lam et al. (2022), which were obtained from a simple inspection of X-ray
catalogues.. We also carried out a search for X-ray emission using all the
available _Chandra_ and _XMM-Newton_ archival data for several other
gravitational microlensing sources proposed, with different levels of
confidence, as BH and NS candidates. To our knowledge, only the results for
MACHO-98-BLG-5 have been previously reported in the literature All the
considered targets are briefly described in Section 2 while the data analysis
and results are given in Section 3. The implications for the physics of
accretion and future prospects are discussed in Sections 4 and 5.

##  2 Candidate BH and NS found with gravitational microlensing

###  2.1  OGLE-2011-BLG-0462

OGLE-2011-BLG-0462 (also known as MOA-2011-BLG-191) is probably the most
extensively studied BH candidate found through astrometric microlensing.
Combining numerous data sets, including accurate astrometry carried out with
the Hubble Space Telescope, Sahu et al. (2022) derived a lens mass
$M_{L}=7.1\pm 1.3$ ${\rm M}_{\odot}$. They also obtained stringent limits on
the lens luminosity that rule out a non-degenerate star. These authors could
exclude the possibility that the lens is a close222 A wide binary could not be
excluded. binary system, thus concluding that it is an isolated BH. The
derived value of the lens distance is $D_{L}=1.58\pm 0.18$ kpc, only slightly
dependent on the distance of the lensed star, and the lens transverse velocity
is $V_{T}\sim$45 km s-1. They also noted that the region surrounding the lens
is affected by severe differential extinction, which may indicate that locally
the density of the ISM is large.

This microlensing event was studied also by Lam et al. (2022) who confirmed
the dark nature of the lens. However, depending on the relative weights given
to the astrometric and photometric data used in the fits, these authors found
two solutions leading to different values for the lens mass, distance and
velocity (see Table 1). While one solution gives $M_{L}=3.79^{+0.62}_{-0.57}$
${\rm M}_{\odot}$, hence a high probability for a BH, the other one gives
$M_{L}=2.15^{+0.67}_{-0.54}$ ${\rm M}_{\odot}$, corresponding to comparable NS
and BH probabilities. In both solutions, the transverse velocity of the lens
($V_{T}<$25 km s-1) is smaller than that obtained by Sahu et al. (2022).

###  2.2  MACHO-99-BLG-22

MACHO-99-BLG-22 (also known as OGLE-1999-BUL-32) was first pointed out as a
possible BH by Mao et al. (2002), on the basis of a small parallax
microlensing effect. The derived lens mass depends on the lens and source
distances. For example, $M_{L}\sim$13 ${\rm M}_{\odot}$ for a source at the
Galactic center distance of 8 kpc and a midway lens. The high likelihood of a
BH nature was supported by a Bayesan statistical analysis that considered the
mass functions of the disk and bulge stars (Agol et al., 2002). Poindexter et
al. (2005) made a systematic analysis of 22 parallax microlensing events,
taking into account various subtle effects that can lead to degenerate
solutions and using a likelihood analysis to derive the masses.
MACHO-99-BLG-22 was found to be the strongest BH candidate (78% probability)
in their sample. In the following, we adopt their most likely values of
$M_{L}\sim 7.5$ ${\rm M}_{\odot}$ and $D_{L}\sim 4.8$ kpc.

###  2.3  MACHO-96-BLG-5

MACHO-96-BLG-5 is one of the two massive ($\sim$6 ${\rm M}_{\odot}$) and dark
(L$<1~{}L_{\odot}$) lenses found in an analysis of the longest microlensing
events in the MACHO survey (Bennett et al., 2002). The lens mass-distance
relation, assuming a source at the bulge distance of 8 kpc, gives most likely
values in the ranges $M_{L}\sim 3-16~{}{\rm M}_{\odot}$ and $D_{L}\sim$0.5–2.2
kpc. The subsequent analysis of Poindexter et al. (2005) found a somewhat
reduced probability of only 37% for a BH nature. A recent analysis of deep
optical and near infrared obervations has strongly reduced the possible
parameter space for a non BH lens in MACHO-96-BLG-5 (Abdurrahman et al.,
2021).

A 10 ks long observation of MACHO-96-BLG-5 was carried out with the _Chandra_
ACIS-S instrument, but no X-rays were detected (Maeda et al., 2005), with a
99% confidence level upper limits on the 0.3-8 keV absorbed flux of $\sim
5\times 10^{-15}$ erg cm-2 s-1. Also a much longer (100 ks) observation with
_XMM-Newton_ could not detect any X-ray emission (Nucita et al., 2006) and
provided slightly worse upper limits.

###  2.4  MACHO-98-BLG-6

MACHO-98-BLG-6 is the second BH candidate proposed by Bennett et al. (2002).
With the assumption that the magnified star is at 8 kpc, Soto et al. (2007)
derived $M_{L}=6^{+9}_{-4}$ ${\rm M}_{\odot}$ and a distance $D_{L}$=1.9 kpc.
However, according to the analysis of Poindexter et al. (2005) MACHO-98-BLG-6
has a very small probability of being a BH (2%).

###  2.5  OGLE-2011-BLG-0310

OGLE-2011-BLG-0310 (also known as MOA-2011-BLG-332) is one of the five
microlensing events for which an astrometric study has been carried out by Lam
et al. (2022). However, these authors found an astrometric signal consistent
with zero. According to their analysis, OGLE-2011-BLG-0310 is most likely a
white dwarf (65% probability) or a NS (22%) with mass of
0.78${}^{+2.98}_{-0.68}$ ${\rm M}_{\odot}$.

###  2.6 OGLE3-ULENS-PAR-01, PAR-02, and PAR-05

Wyrzykowski et al. (2016) made an extensive search for BH, NS and white dwarf
candidates in the OGLE-III database. They found 13 out of 59 parallax
microlensing events consistent with a compact object lens. Their masses and
distances were estimated assuming standard proper motions for stars in the
Galaxy. Although it cannot be excluded that some of them are main sequence
stars in the disk that move in parallel with bulge sources giving very small
relative proper motions, a few interesting candidates were selected.
OGLE3-ULENS-PAR-02 and OGLE3-ULENS-PAR-05 are those with the highest estimated
masses, consistent with being BH. Both of them have two possible solutions
yielding slightly different parameters (see Table 1).

###  2.7 MOA-2009-BLG-260

MOA-2009-BLG-260 is an astrometric parallax event with mass
$M_{L}=1.37^{+0.74}_{-0.60}$ (Lam et al., 2022), most likely consisting of a
NS (44% probability). No _Chandra_ or _XMM-Newton_ observations of its
position are available.

###  2.8  OGLE-00-BUL-43

Soto et al. (2007) performed spectroscopy of 16 microlensing events to add
radial velocity information and determine spectral types and estimated lens
masses based on a model of the Galaxy. OGLE-00-BUL-43 was found as a possible
BH candidate, but no _Chandra_ or _XMM-Newton_ observations are available.

Table 1: Isolated BH and NS candidates found with microlensing

Source | Mass | Distance | Comments | References
---|---|---|---|---
| (${\rm M}_{\odot}$) | (kpc) |  |
OGLE-2011-BLG-0462 | 7.1$\pm$1.3 | 1.58$\pm$0.18 |  $V_{T}$=45 km s-1 | [S22]
| 3.79${}^{+0.62}_{-0.57}$ | [1.47-1.92] |  $V_{T}$=21-27 km s-1, 100% BH | [L22]
| 2.12${}^{+0.67}_{-0.54}$ | [0.70-1.30] |  $V_{T}$=2-12 km s-1, 44%, BH, 50% NS, 6% WD | [L22]
MACHO-99-BLG-22 | 7.5 | 4.8 | 78% BH, 7% NS, 4% WD | [M02,P05]
MACHO-96-BLG-5 | 6${}^{+10}_{-3}$ | 1.3${}^{+0.9}_{-0.8}$ | 37% BH, 14% NS, 19% WD | [B02,P05]
MACHO-98-BLG-6 | 6${}^{+9}_{-4}$ | 1.9 | 2% BH, 13% NS, 26% WD | [B02,P05,S07]
OGLE-2011-BLG-0310 | 0.78${}^{+0.71}_{-0.39}$ | 4.3${}^{+1.9}_{-1.6}$ | 5% BH, 22% NS, 65% WD | [L22]
OGLE3-ULENS-PAR-01 | 1.0${}^{+1.8}_{-0.7}$ | 1.3${}^{+1.1}_{-0.7}$ |  | [W16]
OGLE3-ULENS-PAR-02 | 8.7${}^{+8.1}_{-4.7}$ | 1.8${}^{+1.1}_{-0.8}$ |  | [W16]
| 9.3${}^{+8.7}_{-4.3}$ | 2.4${}^{+1.1}_{-1.0}$ |  |
OGLE3-ULENS-PAR-05 | 3.3${}^{+2.7}_{-1.5}$ | 2.9${}^{+1.1}_{-0.9}$ |  | [W16]
| 4.8${}^{+4.0}_{-2.5}$ | 1.8${}^{+1.2}_{-0.7}$ |  |
MOA-2009-BLG-260 | 1.37${}^{+2.72}_{-1.16}$ | 5.0${}^{+1.7}_{-1.3}$ | 14% BH, 44% NS, 38% WD | [L22]
OGLE-00-BUL-43 | 2.9${}^{+6.1}_{-1.9}$ | 0.5 | for disk lensed source | [S07]
| 2${}^{+6}_{-1.5}$ | 0.8 | for bulge lensed source |

References:  [S22] Sahu et al. (2022), [L22] Lam et al. (2022), [W16]
Wyrzykowski et al. (2016), [P05] Poindexter et al. (2005), [B02] Bennett et
al. (2002), [S07] Soto et al. (2007) , [M02] Mao et al. (2002).

##  3 X-ray data analysis and results

_Chandra_ observations are available for six of the microlensing events listed
in Table 1: OGLE-2011-BLG-0462, MACHO-99-BLG-22, MACHO-98-BLG-5,
OGLE-2011-BLG-0310, OGLE3-ULENS-PAR-05, and OGLE3-ULENS-PAR-01. Four of them
have also _XMM-Newton_ observations, but these data have a worse sensitivity
compared to the _Chandra_ ones and will not be considered in the following. We
report the results obtained with _XMM-Newton_ only for the two sources not
observed with _Chandra_ : MACHO-98-BLG-6 and OGLE3-ULENS-PAR-02. A log of the
observations used in our analysis is given in Table 2.

###  3.1 _Chandra_

_Chandra_ data reprocessing and analysis were performed with standard
procedures using the most recent version of the _Chandra_ Interactive Analysis
of Observation (CIAO 4.14) and CALDB (4.9.6). Images and exposure maps were
produced using fluximage in the energy range 0.5-7.0 keV. The script srcflux
was used to estimate the radii of the circular regions enclosing 90% of the
point spread function at 1.0 keV. Such radii range from 0.9 arcsec on-axis to
9 arcsec for the most off-axis sources. The background for each observation
was estimated using annular regions centered on the source positions, with
inner and outer radii of one and five times source extraction radius. These
source-free background regions contain zero or, at most, a few counts. The
effective exposures used for the flux computation take into account the off-
axis positions of the sources and have been computed taking the average
exposure time inside the source extraction circle.

No X-ray sources were detected at the sky positions of the microlensing
events. Given the small number of detected counts, the upper limits on the
count rates (95% c.l.) were computed following Kraft et al. (1991). These
limits were then converted to fluxes using the appropriate calibration files
for each source and observation. Such conversions depend on the assumed
spectral shape and absorption. We used a power law spectrum with photon index
of 2 and interstellar medium abundances from Wilms et al. (2000). The total
hydrogen column densities in the directions of our targets are in the range
$[2-6]\times 10^{21}$ cm-2 (HI4PI Collaboration et al., 2016). Given that the
lenses are at intermediate distances between us and the Galactic bulge, these
values can be regarded as upper limits. We inspected the run of extinction
versus distance in the direction of each source derived by Green et al. (2019)
and, based on the lens distance, we rescaled the total hydrogen column
densities to obtain the $N_{H}$ values listed in Table 3. These values were
used to compute the unabsorbed fluxes. When HRC-I data were analysed, the
upper limit on the net count rate was estimated in the energy range 0.1-10
keV, but the fluxes have been converted to the 0.5-7 keV band. All the results
are reported in Table 3.

###  3.2 _XMM-Newton_

An _XMM-Newton_ observation was targeted at the event OGLE3-ULENS-PAR-02 in
September 2016. We used the data obtained with the EPIC pn camera that was
operated in Prime Full Window mode with the thin optical filter. _XMM-Newton_
data were analysed with the Science Analysis Software (SAS) adopting standard
procedures. A faint source (4XMM J175723.4-284627 in the _XMM-Newton_
Serendipitous Source Catalog, Webb et al. 2020) is detected at an angular
distance of 6.6 arcsec from the position of OGLE3-ULENS-PAR-02. Its net count
rate, measured using an extraction radius of 20 arcsec, is
$(2.55\pm{0.75})\times 10^{-3}$ counts s-1 in the 0.2-12 keV range. The error
on its coordinates (0.98 arcsec statistical, 0.45 arcsec systematic, $1\sigma$
c.l.) is sufficiently small to exclude that this source is related to
OGLE3-ULENS-PAR-02.

The instrument sensitivity at the position of the gravitational lens is
reduced due to the presence of this nearby contaminating source. Therefore, we
conservatively assume as an upper limit the count rate of 4XMM
J175723.4-284627. For a power law spectrum with photon index of 2 and
$N_{H}=10^{21}$ cm-2 this count rate corresponds to an unabsorbed 0.2-12 keV
flux of $1.1\times 10^{-14}$ erg cm-2 s-1 (or $7.1\times 10^{-15}$ erg cm-2
s-1 in the 0.5-7 keV range used above for _Chandra_).

The same _XMM-Newton_ observation covers also the sky position of the event
MACHO-98-BLG-6, lying about 4.3 arcmin offset. For this event we analysed the
EPIC MOS data (effective exposure Texp=30.7 ks, MOS1), because its sky
position lies very close to the gap between CCDs in the pn camera, hampering a
proper determination of the upper limit. We have measured a 95% upper limit to
the count rate of $9.0\times 10^{-4}$ counts s-1 (0.2-12 keV), using the SAS
script eupper. With the same assumptions on the spectrum used above, this
translates into an upper limit to the unabsorbed flux of $1.78\times 10^{-14}$
erg cm-2 s-1 ($1.14\times 10^{-14}$ erg cm-2 s-1, 0.5-7 keV).

Table 2: X-ray observations of microlensing events. VF and F in column 5 indicate very faint and faint mode, respectively.  Source | Satellite | ObsID | Date | Instrument | Duration | Effective | Off-axis
---|---|---|---|---|---|---|---
|  |  |  |  |  | exposure | angle
|  |  |  |  | (ks) | (ks) | (′)
OGLE-2011-BLG-0462 | Chandra | 8764 | 2008-05-13 | ACIS-I (VF) | 2.15 | 1.92 | 8.8
| Chandra | 13540 | 2011-11-01 | ACIS-I (VF) | 1.95 | 1.83 | 5.1
| Chandra | 21628 | 2019-07-29 | ACIS-I (VF) | 1.89 | 1.74 | 6.6
| Chandra | 23951 | 2022-06-02 | ACIS-I (VF) | 5.29 | 4.84 | 5.1
MACHO-99-BLG-22 | Chandra | 3818 | 2003-08-01 | ACIS-S (VF) | 9.7 | 9.36 | 0.0
MACHO-96-BLG-5 | Chandra | 3789 | 2003-02-18 | ACIS-S (F) | 9.83 | 9.83 | 0.0
| Chandra | 14677 | 2013-10-03 | ACIS-I (F) | 1.0 | 0.95 | 7.6
MACHO-98-BLG-6 | XMM | 0782770201 | 2016-09-03 | EPIC MOS1 | 40 | 30.7 | 4.3
OGLE-2011-BLG-0310 | Chandra | 13542 | 2011-11-01 | ACIS-I (VF) | 1.95 | 1.864 | 8.7
| Chandra | 13564 | 2011-11-01 | ACIS-I (VF) | 1.95 | 1.785 | 10
| Chandra | 13565 | 2011-11-01 | ACIS-I (VF) | 1.95 | 1.744 | 5.9
| Chandra | 23957 | 2021-03-18 | ACIS-I (VF) | 5.1 | 4.52 | 7.5
OGLE3-ULENS-PAR-01 | Chandra | 7541 | 2007-10-23 | HRC-I | 1.19 | 1.17 | 13.8
OGLE3-ULENS-PAR-02 | XMM | 0782770201 | 2016-09-03 | EPIC pn | 35 | 32 | 0.0
OGLE3-ULENS-PAR-05 | Chandra | 19723 | 2018-05-10 | ACIS-I (VF) | 19.8 | 19.8 | 0.0
Table 3: X-ray upper limits (95% c.l.). All count rates are in the energy range 0.5-7 keV, except _Chandra_ HRC-I rates, which are in the energy range 0.1-10 keV, and _XMM-Newton_ count rates (0.2-12 keV).  Source | ObsID | Count rate | Observed flux | NH | Unabsorbed flux | Unabsorbed flux
---|---|---|---|---|---|---
|  |  | 0.5-7 keV |  | 0.5-7.0 keV | 0.1-100 keV
|  | (count s-1) | (erg cm-2 s-1) | (cm-2) | (erg cm-2 s-1) | (erg cm-2 s-1)
OGLE-2011-BLG-0462 | 8764 |  $<$0.0016 |  $<$2.20$\times$10-14 | 1021 |  $<$2.57$\times$10-14 |  $<$6.76$\times$10-14
| 13540 |  $<$0.0016 |  $<$1.93$\times$10-14 |  |  $<$2.25$\times$10-14 |  $<$5.92$\times$10-14
| 21628 |  $<$0.0017 |  $<$3.16$\times$10-14 |  |  $<$3.69$\times$10-14 |  $<$9.71$\times$10-14
| 23951 |  $<$0.00126 |  $<$2.72$\times$10-14 |  |  $<$3.17$\times$10-14 |  $<$8.31$\times$10-14
| combined |  $<$0.00051 |  $<$8.6$\times$10-15 |  |  $<$1.0$\times$10-14 |  $<$2.6$\times$10-14
MACHO-99-BLG-22 | 3818 |  $<$3.2$\times$10-4 |  $<$1.99$\times$10-15 | $3\times 10^{21}$ |  $<$2.83$\times$10-15 |  $<$7.43$\times$10-15
MACHO-96-BLG-5 | 3789 |  $<$3.1$\times$10-4 |  $<$1.86$\times$10-15 | 1021 |  $<$2.17$\times$10-15 |  $<$5.71$\times$10-15
| 14677 |  $<$0.0032 |  $<$5.63$\times$10-14 |  |  $<$6.57$\times$10-14 |  $<$1.73$\times$10-13
MACHO-98-BLG-6 | 0782770201 |  $<$9$\times$10-4 |  $<$9.77$\times$10-15 | 1021 |  $<$1.14$\times$10-14 |  $<$3.0$\times$10-14
OGLE-2011-BLG-0310 | 13542 |  $<$0.0031 |  $<$4.37$\times$10-14 | $5\times 10^{21}$ |  $<$7.1$\times$10-14 |  $<$1.9$\times$10-13
| 13564 |  $<$0.0032 |  $<$4.97$\times$10-14 |  |  $<$8.0$\times$10-14 |  $<$2.1$\times$10-13
| 13565 |  $<$0.0026 |  $<$3.60$\times$10-14 |  |  $<$5.8$\times$10-14 |  $<$1.5$\times$10-13
| 23957 |  $<$0.00066 |  $<$1.42$\times$10-14 |  |  $<$2.3$\times$10-14 |  $<$6.0$\times$10-14
| combined |  $<$0.0009 |  $<$1.54$\times$10-14 |  |  $<$2.5$\times$10-14 |  $<$6.6$\times$10-14
OGLE3-ULENS-PAR-01 | 7541 |  $<$0.0178 |  $<$4.37$\times$10-13 | 1021 |  $<$5.1$\times$10-13 |  $<$1.3$\times$10-12
OGLE3-ULENS-PAR-02 | 0782770201 |  $<$2.55$\times$10-3 |  $<$6.08$\times$10-15 | 1021 |  $<$7.1$\times$10-15 |  $<$1.9$\times$10-14
OGLE3-ULENS-PAR-05 | 19723 |  $<$0.00015 |  $<$2.11$\times$10-15 | $2\times 10^{21}$ |  $<$2.76$\times$10-15 |  $<$7.24$\times$10-15
Figure 1: Constraints on the product of BH efficiency $\eta(\equiv
L/\dot{M}c^{2}$) and ISM density derived from the X-ray flux upper limits of
OGLE-2011-BLG-0462 (red) and other BH candidates found with gravitational
microlensing (see text for the definition of $\lambda$ and $\epsilon$). The
red diamonds and triangles refer to the BH properties derived by Sahu et al.
(2022) and Lam et al. (2022), respectively. The curves for the other BH
candidates are derived using for each mass value the corresponding distance
and transverse velocity as given by the microlensing relations. Figure 2:
Constraints on the NS velocity and ISM density for candidate isolated NS found
with gravitational microlensing. A sound speed $c_{s}$=10 km s-1 has been
assumed. The regions above the solid lines are excluded by the upper limits on
the soft X-ray flux obtained with _Chandra_ , while those above the dashed
lines are excluded by the INTEGRAL limits in the hard X-ray range (17-60 keV).
Figure 3: Image in the 17-60 keV range obtained with the INTEGRAL IBIS
instrument (adapted from Krivonos et al. (2022)). The image is in Galactic
coordinates and covers a region of about 10$\times$10 deg2. The positions of
the BH/NS candidates found with gravitational microlensing are indicated by
circles with radius of 10 arcmin. The source on the border of the
MACHO-96-BLG-5 circle, IGR J18044–2739, is identified with a magnetic
cataclysmic variable (Masetti et al., 2012). Its coordinates are incompatible
with those of the microlens source.

##  4 Discussion

The accretion-powered luminosity of an isolated compact star of mass $M$
moving with velocity $V$ in the interstellar medium of density $n$ depends on
the mass accretion rate $\dot{M}$, that, in the Bondi-Hoyle description, is
given by

$\dot{M}=4\pi nm_{p}\frac{(GM)^{2}}{(V^{2}+c_{s}^{2})^{3/2}}\lambda~{}~{}{\rm
g~{}s^{-1}}.$ (1)

Here $G$ is the gravitational constant, $m_{p}$ the proton mass, and $c_{s}$
the sound speed, which we will neglect in the following because, for the most
common phases of the ISM, it is much smaller than the typical space velocity
of isolated NS and BH. The parameter $\lambda$ accounts for the fact that not
all the matter crossing the Bondi-Hoyle radius is accreted. As discussed in
Tsuna et al. (2018), its value is subject to a large uncertainty. Therefore,
we keep it as an explicit scaling factor in all the results given below and in
Fig. 1.

According to the analysis of Lam et al. (2022), there is a non negligible
probability that OGLE-2011-BLG-0462 is a NS rather than a BH; also other
objects listed in Table 1 are most likely NS. Therefore, in the following we
discuss both the BH and NS case.

###  4.1 Black hole

Sahu et al. (2022), derived for OGLE-2011-BLG-0462 a transverse velocity of 45
km s-1. We can thus neglect $c_{s}$ in eq. 1 and, with the 7.1 ${\rm
M}_{\odot}$ mass given by these authors, estimate an accretion rate of
$2\times 10^{11}~{}n\lambda$ g s-1. This is actually an upper limit because it
does not take into account the unknown component of the velocity along the
line of sight. Then, for the BH distance $d$=1.58 kpc, the upper limit on the
X-ray flux derived above, $F_{X}<8.9\times 10^{-15}$ erg cm-2 s-1, implies a
luminosity

$L<\frac{F_{X}}{\epsilon}4\pi d^{2}=2.7\times 10^{30}/\epsilon~{}~{}~{}~{}{\rm
erg~{}s^{-1}},$ (2)

and an efficiency of conversion of gravitational energy to electromagnetic
radiation

$\eta\equiv
L/(\dot{M}c^{2})<\frac{0.015}{\epsilon\lambda}\left(\frac{n}{1~{}{\rm
cm^{-3}}}\right)^{-1},$ (3)

where we introduced the factor $\epsilon$ to account for the fraction of
accretion luminosity falling outside the observed X-ray band. Assuming that
the luminosity is emitted in the 0.1-100 keV range with a power-law spectrum
of photon index $\alpha$=2, the 0.5–7 keV range used above corresponds to
$\epsilon$ = $F_{0.5-7}/F_{0.1-100}$ = 0.38 ($\epsilon$= 0.2 / 0.34 for
$\alpha$ = 1.5 / 2.5).

Lam et al. (2022) found for the BH case a most likely mass of 3.8 ${\rm
M}_{\odot}$ and a transverse velocity in the range 21-27 km s-1. The mass
accretion rate has a strong dependence on the BH velocity. Thus, despite the
smaller mass, the limit on $\eta$ obtained using these values is more
constraining than that derived above. This is shown in Fig. 1, where the upper
limit on the product $\eta\lambda\epsilon n$ is plotted as a function of the
BH mass for both the Sahu et al. (2022) and Lam et al. (2022) parameters.

We plot in Fig. 1 also the constraints obtained in a similar way from our
upper limits on the X-ray fluxes of the other candidates of Table 1 that are
likely to be BH. To derive the limits for these sources, which, contrary to
OGLE-2011-BLG-0462, do not have an astrometric parallax measurement, we took
into account the distance dependence of the mass and transverse velocity. Our
constraints for MACHO-96-BLG-5 agree with those derived by Maeda et al. (2005)
using the same data. These constraints are below those of OGLE-2011-BLG-0462,
but note that, besides having a large distance uncertainty, after the analysis
of Poindexter et al. (2005) MACHO-96-BLG-5 is no more considered a strong BH
candidate.

###  4.2 Neutron star

The expected accretion luminosity for a NS of mass $M_{NS}$ and radius
$R_{NS}$ is

$L_{NS}=\frac{GM_{NS}}{R_{NS}}\dot{M}=\frac{(GM_{NS})^{3}}{R_{NS}V^{3}}4\pi
m_{p}n\lambda,$ (4)

which, for $M_{NS}=1.4~{}{\rm M}_{\odot}$ and $R_{NS}$=10 km, gives a
luminosity

$L_{NS}=1.3\times 10^{32}\left(\frac{10~{}{\rm
km~{}s^{-1}}}{V}\right)^{3}n\lambda~{}~{}{\rm erg~{}s^{-1}}.$ (5)

For the small distance (0.7–1.3 kpc at 99% c.l.) and transverse velocity (2–12
km s-1) derived for OGLE-2011-BLG-0462 in the NS case (Lam et al., 2022), the
resulting flux is in the broad range $F_{X}=[0.6-140]\times
10^{-12}n\lambda/d^{2}_{kpc}$ erg cm-2 s-1. This expected flux is close to, or
higher than, the upper limit of $\sim 10^{-14}$ erg cm-2 s-1 derived above,
thus possibly disfavoring a NS nature.

Taking booth $\lambda$ and $d_{kpc}\sim$1, implies that the NS is in a low
density ( $n\lesssim 10^{-2}$ cm-3) environment, but unfortunately, no strong
conclusions can be drawn because the radial component of the NS is unknown.
Isolated NS have relatively high space velocities. Hobbs et al. (2005) found
that radio pulsars have a velocity distribution consistent with a Maxwellian
with rms $\sigma$=265 km s-1, while a more recent analysis found that the sum
of two Maxwellian distributions with $\sigma_{1}$=128 km s-1 and
$\sigma_{2}$=298 km s-1 gives a better description of the pulsar velocity
distribution (Igoshev, 2020).

We show in Fig. 2 how the upper limits on the X-ray flux we derived for the
microlensing NS candidates translate into constraints on the ISM density as a
function of their space velocities. For all the candidates we assumed
$M_{NS}=1.4~{}{\rm M}_{\odot}$, $R_{NS}$=10 km, the best fit distances given
in Table 1, $\lambda=1$, and an ISM sound speed of $c_{s}$= 10 km s-1. For
example, if OGLE-2011-BLG-0462 has a space velocity below 100 km s-1, it
cannot be in a region of the ISM with density higher than $\sim$10 cm-3. This
excludes that OGLE-2011-BLG-0462 is either inside or within the envelope of a
molecular cloud.

###  4.3 Limits in the hard X-ray band

If these compact objects are in dense molecular clouds, their flux can be
severely absorbed in the softest part of the energy range explored with
_Chandra_ and _XMM-Newton_. It is therefore interesting to search also for
X-ray emission in the hard X-ray range, that is unaffected by interstellar
absorption. All the microlensing events considered here are in the direction
of the Galactic bulge, a region that has been extensively observed with the
INTEGRAL satellite. The positions of the microlensing candidates are marked in
Fig. 3 on a sky map obtained with the INTEGRAL IBIS instrument (Ubertini et
al., 2003) in the 17-60 keV energy range. This image has been derived by
summing observations carried out from 2002 to 2017 and corresponds to a total
net exposure time of more than 13 millions of seconds (Krivonos et al., 2022).
The resulting limiting sensitivity is $\sim 2\times 10^{-12}$ erg cm-2 s-1.

Since no hard X-ray sources are detected at the positions of the microlensing
events, we used this flux value to derive the limits shown by the dashed lines
in Fig. 2. Although these limits are above those obtained in soft X-rays, they
do not depend on the absorption and can be applied also for sources inside
dense molecular clouds.

##  5 Conclusions

We derived upper limits on the X-ray flux from OGLE-2011-BLG-0462. This is
currently the strongest BH candidate selected through astrometric
gravitational microlensing (Sahu et al., 2022), although there is some tension
in its parameters as derived by different groups and a NS nature cannot be
completely excluded (Lam et al., 2022). Our limits are consistent with an
accreting isolated BH with low radiative efficiency, as observed in quiescent
X-ray binaries and in AGNs, and do not support a NS nature for the lens in
OGLE-2011-BLG-0462. However, we cannot exclude the possibility of a NS in a
relatively low density environment and/or with a space velocity much larger
than the projected value obtained from the gravitational microlensing
analysis.

Using _Chandra_ _XMM-Newton_ and INTEGRAL data, we derived upper limits on the
soft and hard X-ray fluxes also for several other candidate compact objects
found with gravitational microlensing. The limits reported here are the best
ones currently available for these objects, but the resulting constraints on
the physical parameters governing the accretion process depend on several
poorly known factors. Among these, a major role is played by the space
velocity of the compact object, since even a factor of two yields a difference
of about one order of magnitude in the expected accretion rate. In fact, the
relative positions of the limits plotted in Fig. 1 are determined mainly by
the different velocities of these objects.

Further observations will provide better estimates of the masses and distances
of these candidate collapsed objects and possibly discriminate between BH and
NS, but the radial velocity component of dark lenses cannot be directly
measured. This problem will always affect the interpretation of X-ray
observations of individual sources. However, many more isolated BH and NS will
be discovered with microlensing surveys in the future. Thanks to the
availability of a large sample of reliable candidates observed with high-
sensitivity X-ray telescopes, it will be possible to perform statistical
analysis exploiting our best knowledge of the distribution of spatial
velocities, thus deriving meaningful constraints on the physics of accretion
from the ISM onto isolated compact objects.

We thank the anonymous referee for useful suggestions. We acknowledge
financial support from the Italian Ministry for University and Research
through grant 2017LJ39LM “UnIAM” and the INAF “Main-streams” funding grant (DP
n.43/18). GP acknowledges funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(grant agreement No 865637). This research has made use of data obtained from
the _Chandra_ Data Archive and software provided by the _Chandra_ X-ray Center
(CXC) in the application package CIAO. This research is based on observations
obtained with _XMM-Newton_ , an ESA science mission with instruments and
contributions directly funded by ESA Member States and NASA. The _XMM-Newton_
data were downloaded by means of the _XMM-Newton_ Science Archive (XSA v14.1)
and of the High Energy Astrophysics Science Archive Research Center (HEASARC),
a service of the Astrophysics Science Division at NASA/GSFC.

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