We study the problem of learning to stabilize unknown noisy Linear
Time-Invariant (LTI) systems on a single trajectory. It is well known in the
literature that the learn-to-stabilize problem suffers from exponential blow-up
in which the state norm blows up in the order of $\Theta(2^n)$ where $n$ is the
state space dimension. This blow-up is due to the open-loop instability when
exploring the $n$-dimensional state space. To address this issue, we develop a
novel algorithm that decouples the unstable subspace of the LTI system from the
stable subspace, based on which the algorithm only explores and stabilizes the
unstable subspace, the dimension of which can be much smaller than $n$. With a
new singular-value-decomposition(SVD)-based analytical framework, we prove that
the system is stabilized before the state norm reaches $2^{O(k \log n)}$, where
$k$ is the dimension of the unstable subspace. Critically, this bound avoids
exponential blow-up in state dimension in the order of $\Theta(2^n)$ as in the
previous works, and to the best of our knowledge, this is the first paper to
avoid exponential blow-up in dimension for stabilizing LTI systems with noise.

我们研究了在单一轨迹上学习稳定未知噪声线性时不变系统的问题，并提出了一种新算法，通过解耦系统的不稳定子空间和稳定子空间来避免状态空间维度的指数级增长。使用基于奇异值分解的分析框架，我们证明了在状态范数达到 2^(O (k log n)) 之前，系统能够被稳定，避免了传统方法中状态维度指数级增长的问题。本文是第一篇避免状态空间维度指数级增长的稳定带噪声线性时不变系统的文章。

学习在单轨迹下稳定未知线性时不变系统的方法在随机噪声下

Learning to Stabilize Unknown LTI Systems on a Single Trajectory under  Stochastic Noise

Modern recommender systems may output considerably different recommendations
due to small perturbations in the training data. Changes in the data from a
single user will alter the recommendations as well as the recommendations of
other users. In applications like healthcare, housing, and finance, this
sensitivity can have adverse effects on user experience. We propose a method to
stabilize a given recommender system against such perturbations. This is a
challenging task due to (1) the lack of a ``reference'' rank list that can be
used to anchor the outputs; and (2) the computational challenges in ensuring
the stability of rank lists with respect to all possible perturbations of
training data. Our method, FINEST, overcomes these challenges by obtaining
reference rank lists from a given recommendation model and then fine-tuning the
model under simulated perturbation scenarios with rank-preserving
regularization on sampled items. Our experiments on real-world datasets
demonstrate that FINEST can ensure that recommender models output stable
recommendations under a wide range of different perturbations without
compromising next-item prediction accuracy.

我们提出了一种名为 FINEST 的方法，通过在模拟扰动场景中对抽样项目进行秩序保持正则化，从而获取给定推荐模型的参考秩序列表，进而使推荐模型在各种扰动下输出稳定建议，同时不损害下一个项目预测的准确性。

FINEST：通过保持排序微调来稳定推荐

FINEST: Stabilizing Recommendations by Rank-Preserving Fine-Tuning

Inspired by the work of Tsiamis et al. \cite{tsiamis2022learning}, in this
paper we study the statistical hardness of learning to stabilize linear
time-invariant systems. Hardness is measured by the number of samples required
to achieve a learning task with a given probability. The work in
\cite{tsiamis2022learning} shows that there exist system classes that are hard
to learn to stabilize with the core reason being the hardness of
identification. Here we present a class of systems that can be easy to
identify, thanks to a non-degenerate noise process that excites all modes, but
the sample complexity of stabilization still increases exponentially with the
system dimension. We tie this result to the hardness of co-stabilizability for
this class of systems using ideas from robust control.

通过对 Tsiamis 等人的工作进行启发，本文研究了学习线性时不变系统的统计困难性，难度通过实现给定概率下所需样本数来衡量；作者通过展示一类系统可以容易地进行识别，但系统稳定化的样本复杂度仍然随系统维度呈指数增加，并利用强控制中的理念将此结果与该系统类别的共稳定性困难性相关联。