We study chance constrained optimization problems $\min_x f(x)$ s.t.
$P(\left\{ \theta: g(x,\theta)\le 0 \right\})\ge 1-\epsilon$ where $\epsilon\in
(0,1)$ is the violation probability, when the distribution $P$ is not known to
the decision maker (DM). When the DM has access to a set of distributions
$\mathcal{U}$ such that $P$ is contained in $\mathcal{U}$, then the problem is
known as the ambiguous chance-constrained problem \cite{erdougan2006ambiguous}.
We study ambiguous chance-constrained problem for the case when $\mathcal{U}$
is of the form $\left\{\mu:\frac{\mu (y)}{\nu(y)}\leq C, \forall y\in\Theta,
\mu(y)\ge 0\right\}$, where $\nu$ is a ``reference distribution.'' We show that
in this case the original problem can be ``well-approximated'' by a sampled
problem in which $N$ i.i.d. samples of $\theta$ are drawn from $\nu$, and the
original constraint is replaced with $g(x,\theta_i)\le 0,~i=1,2,\ldots,N$. We
also derive the sample complexity associated with this approximation, i.e., for
$\epsilon,\delta>0$ the number of samples which must be drawn from $\nu$ so
that with a probability greater than $1-\delta$ (over the randomness of $\nu$),
the solution obtained by solving the sampled program yields an
$\epsilon$-feasible solution for the original chance constrained problem.

研究了具有未知分布约束的机会约束优化问题，并通过使用具有参考分布的样本问题的近似方法，导出了解决原始机会约束问题的采样程序的样本复杂度。

学习模糊的机会约束问题

On Learning for Ambiguous Chance Constrained Problems

We point out a limitation of the mutual information neural estimation (MINE)
where the network fails to learn at the initial training phase, leading to slow
convergence in the number of training iterations. To solve this problem, we
propose a faster method called the mutual information neural entropic
estimation (MI-NEE). Our solution first generalizes MINE to estimate the
entropy using a custom reference distribution. The entropy estimate can then be
used to estimate the mutual information. We argue that the seemingly redundant
intermediate step of entropy estimation allows one to improve the convergence
by an appropriate reference distribution. In particular, we show that MI-NEE
reduces to MINE in the special case when the reference distribution is the
product of marginal distributions, but faster convergence is possible by
choosing the uniform distribution as the reference distribution instead.
Compared to the product of marginals, the uniform distribution introduces more
samples in low-density regions and fewer samples in high-density regions, which
appear to lead to an overall larger gradient for faster convergence.

提出了一种更快的方法，称为互信息神经熵估计（MI-NEE），其将互信息估计变为估计熵的过程，通过使用自定义参考分布和选择均匀分布作为参考分布来实现更快的收敛。