We study the problem of controlling linear time-invariant systems with known
noisy dynamics and adversarially chosen quadratic losses. We present the first
efficient online learning algorithms in this setting that guarantee
$O(\sqrt{T})$ regret under mild assumptions, where $T$ is the time horizon. Our
algorithms rely on a novel SDP relaxation for the steady-state distribution of
the system. Crucially, and in contrast to previously proposed relaxations, the
feasible solutions of our SDP all correspond to "strongly stable" policies that
mix exponentially fast to a steady state.

我们研究了控制具有已知嘈杂动力学和对抗选择二次损失的线性时不变系统的问题，并提出了第一种在这种情况下保证 O（sqrt（T））遗憾的有效在线学习算法。我们的算法依赖于对系统稳态分布的新型 SDP 松弛。与以前提出的松弛相反，我们的 SDP 的可行解都对应于 “强稳定” 策略，这些策略混合到稳定状态的速度呈指数增长。

在线线性二次控制

Online Linear Quadratic Control

It has been shown that for a general-valued constraint language $\Gamma$ the
following statements are equivalent: (1) any instance of
$\operatorname{VCSP}(\Gamma)$ can be solved to optimality using a constant
level of the Sherali-Adams LP hierarchy; (2) any instance of
$\operatorname{VCSP}(\Gamma)$ can be solved to optimality using the third level
of the Sherali-Adams LP hierarchy; (3) the support of $\Gamma$ satisfies the
"bounded width condition", i.e., it contains weak near-unanimity operations of
all arities.
We show that if the support of $\Gamma$ violates the bounded width condition
then not only is $\operatorname{VCSP}(\Gamma)$ not solved by a constant level
of the Sherali-Adams LP hierarchy but it is also not solved by $\Omega(n)$
levels of the Lasserre SDP hierarchy (also known as the sum-of-squares SDP
hierarchy). For $\Gamma$ corresponding to linear equations in an Abelian group,
this result follows from existing work on inapproximability of Max-CSPs. By a
breakthrough result of Lee, Raghavendra, and Steurer [STOC'15], our result
implies that for any $\Gamma$ whose support violates the bounded width
condition no SDP relaxation of polynomial-size solves
$\operatorname{VCSP}(\Gamma)$.
We establish our result by proving that various reductions preserve exact
solvability by the Lasserre SDP hierarchy (up to a constant factor in the level
of the hierarchy). Our results hold for general-valued constraint languages,
i.e., sets of functions on a fixed finite domain that take on rational or
infinite values, and thus also hold in notable special cases of
$\{0,\infty\}$-valued languages (CSPs), $\{0,1\}$-valued languages
(Min-CSPs/Max-CSPs), and $\mathbb{Q}$-valued languages (finite-valued CSPs).

通过证明各种规约保持 Lasserre SDP 层次的精确可解性，我们展示了支持破坏有限宽度条件的一般值约束语言不能被静态程序快速求解。

一般值约束满足问题的 SDP 松弛上界

The limits of SDP relaxations for general-valued CSPs

Resolving a conjecture of Abbe, Bandeira and Hall, the authors have recently
shown that the semidefinite programming (SDP) relaxation of the maximum
likelihood estimator achieves the sharp threshold for exactly recovering the
community structure under the binary stochastic block model of two equal-sized
clusters. The same was shown for the case of a single cluster and outliers.
Extending the proof techniques, in this paper it is shown that SDP relaxations
also achieve the sharp recovery threshold in the following cases: (1) Binary
stochastic block model with two clusters of sizes proportional to network size
but not necessarily equal; (2) Stochastic block model with a fixed number of
equal-sized clusters; (3) Binary censored block model with the background graph
being Erd\H{o}s-R\'enyi. Furthermore, a sufficient condition is given for an
SDP procedure to achieve exact recovery for the general case of a fixed number
of clusters plus outliers. These results demonstrate the versatility of SDP
relaxation as a simple, general purpose, computationally feasible methodology
for community detection.

通过对 Semidefinite Programming（SDP）放松的最大似然估计进行研究，本文论证了 SDP 放松技术在社区检测中的可行性和通用性。

通过半正定规划实现精确群集恢复阈值：扩展

Achieving Exact Cluster Recovery Threshold via Semidefinite Programming:  Extensions

In this paper, we study the average case complexity of the Unique Games
problem. We propose a natural semi-random model, in which a unique game
instance is generated in several steps. First an adversary selects a completely
satisfiable instance of Unique Games, then she chooses an epsilon-fraction of
all edges, and finally replaces ("corrupts") the constraints corresponding to
these edges with new constraints. If all steps are adversarial, the adversary
can obtain any (1-epsilon) satisfiable instance, so then the problem is as hard
as in the worst case. In our semi-random model, one of the steps is random, and
all other steps are adversarial. We show that known algorithms for unique games
(in particular, all algorithms that use the standard SDP relaxation) fail to
solve semi-random instances of Unique Games.
We present an algorithm that with high probability finds a solution
satisfying a (1-delta) fraction of all constraints in semi-random instances (we
require that the average degree of the graph is Omega(log k). To this end, we
consider a new non-standard SDP program for Unique Games, which is not a
relaxation for the problem, and show how to analyze it. We present a new
rounding scheme that simultaneously uses SDP and LP solutions, which we believe
is of independent interest.
Our result holds only for epsilon less than some absolute constant. We prove
that if epsilon > 1/2, then the problem is hard in one of the models, the
result assumes the 2-to-2 conjecture.
Finally, we study semi-random instances of Unique Games that are at most
(1-epsilon) satisfiable. We present an algorithm that with high probability,
distinguishes between the case when the instance is a semi-random instance and
the case when the instance is an (arbitrary) (1-delta) satisfiable instance if
epsilon > c delta.

该研究使用半随机模型研究 Unique Games 问题，提出了一种新的 SDP 算法以解决该问题，并通过比较 SDP 松弛问题和新问题的解来进行分析。