Stochastic optimization is a widely used approach for optimization under
uncertainty, where uncertain input parameters are modeled by random variables.
Exact or approximation algorithms have been obtained for several fundamental
problems in this area. However, a significant limitation of this approach is
that it requires full knowledge of the underlying probability distributions.
Can we still get good (approximation) algorithms if these distributions are
unknown, and the algorithm needs to learn them through repeated interactions?
In this paper, we resolve this question for a large class of "monotone"
stochastic problems, by providing a generic online learning algorithm with
$\sqrt{T \log T}$ regret relative to the best approximation algorithm (under
known distributions). Importantly, our online algorithm works in a semi-bandit
setting, where in each period, the algorithm only observes samples from the
r.v.s that were actually probed. Our framework applies to several fundamental
problems in stochastic optimization such as prophet inequality, Pandora's box,
stochastic knapsack, stochastic matchings and stochastic submodular
optimization.

通过提供一种具有与最佳近似算法（在已知分布下）相对于平方根的 T 乘以 log T 束缚的通用在线学习算法，在半探测器环境中解决了在一大类 “单调” 随机问题中对于未知分布是否能够获得良好（近似）算法进行学习的问题。我们的框架适用于随机优化的若干基本问题，如先知不等式、潘多拉盒、随机背包、随机匹配和随机次模优化。

单调随机优化的半强盗学习

Semi-Bandit Learning for Monotone Stochastic Optimization

Deep Learning is having a remarkable impact on the design of Reduced Order
Models (ROMs) for Partial Differential Equations (PDEs), where it is exploited
as a powerful tool for tackling complex problems for which classical methods
might fail. In this respect, deep autoencoders play a fundamental role, as they
provide an extremely flexible tool for reducing the dimensionality of a given
problem by leveraging on the nonlinear capabilities of neural networks. Indeed,
starting from this paradigm, several successful approaches have already been
developed, which are here referred to as Deep Learning-based ROMs (DL-ROMs).
Nevertheless, when it comes to stochastic problems parameterized by random
fields, the current understanding of DL-ROMs is mostly based on empirical
evidence: in fact, their theoretical analysis is currently limited to the case
of PDEs depending on a finite number of (deterministic) parameters. The purpose
of this work is to extend the existing literature by providing some theoretical
insights about the use of DL-ROMs in the presence of stochasticity generated by
random fields. In particular, we derive explicit error bounds that can guide
domain practitioners when choosing the latent dimension of deep autoencoders.
We evaluate the practical usefulness of our theory by means of numerical
experiments, showing how our analysis can significantly impact the performance
of DL-ROMs.

深度学习在设计偏微分方程的降阶模型（ROMs）方面产生了显著影响，特别是在处理复杂问题以及基于随机领域参数化的随机问题中，深度自动编码器作为一种灵活工具提供了降低问题维数的手段，该研究通过理论分析为深度学习基于 ROMs 在随机领域参数化问题中的应用提供了一些实用的指导方法，这对于领域专家在选择深度自动编码器的潜在维度时具有重要意义，并通过数值实验证明了理论分析对于 DL-ROMs 性能的重大影响。