In recent years, the success of deep learning has inspired many researchers
to study the optimization of general smooth non-convex functions. However,
recent works have established pessimistic worst-case complexities for this
class functions, which is in stark contrast with their superior performance in
real-world applications (e.g. training deep neural networks). On the other
hand, it is found that many popular non-convex optimization problems enjoy
certain structured properties which bear some similarities to convexity. In
this paper, we study the class of \textit{quasar-convex functions} to close the
gap between theory and practice. We study the convergence of first order
methods in a variety of different settings and under different optimality
criterions. We prove complexity upper bounds that are similar to standard
results established for convex functions and much better that state-of-the-art
convergence rates of non-convex functions. Overall, this paper suggests that
\textit{quasar-convexity} allows efficient optimization procedures, and we are
looking forward to seeing more problems that demonstrate similar properties in
practice.

本文研究了一类名为 'quasar-convex function' 的函数，该函数在实践中证明具有类似于凸性的结构性质，并且通过比较其最优解的收敛速率得到了类似于凸函数的复杂度上界，从而表明该函数类能够实现有效的优化过程。

关于夸脱 - 凸优化第一阶段算法收敛性的研究

On The Convergence of First Order Methods for Quasar-Convex Optimization

We study the question of testing structured properties (classes) of discrete
distributions. Specifically, given sample access to an arbitrary distribution
$D$ over $[n]$ and a property $\mathcal{P}$, the goal is to distinguish between
$D\in\mathcal{P}$ and $\ell_1(D,\mathcal{P})>\varepsilon$. We develop a general
algorithm for this question, which applies to a large range of
"shape-constrained" properties, including monotone, log-concave, $t$-modal,
piecewise-polynomial, and Poisson Binomial distributions. Moreover, for all
cases considered, our algorithm has near-optimal sample complexity with regard
to the domain size and is computationally efficient. For most of these classes,
we provide the first non-trivial tester in the literature. In addition, we also
describe a generic method to prove lower bounds for this problem, and use it to
show our upper bounds are nearly tight. Finally, we extend some of our
techniques to tolerant testing, deriving nearly-tight upper and lower bounds
for the corresponding questions.

本文主要研究了离散分布的结构性质（类），提出了一个通用的算法，可以测试各种形状约束的属性，包括单调、对数凹、t - 模态、分段多项式和泊松二项式分布。此外，对于所有考虑的情况，算法在领域大小方面具有近乎最优的样本复杂度，并且计算效率高。